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/-
 - Created in 2024 by GaΓ«tan SerrΓ©
 -/

/-
- https://github.com/gaetanserre/Hirsch-Analysis
-/


import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.NumberTheory.Padics.PadicVal.Basic

open Set Filter Topology Classical

set_option trace.Meta.Tactic.simp.rewrite true

set_option maxHeartbeats 1000000

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y)

----------------------------------COUNTABLE-----------------------------------

/--
A set E is countable if it exists a surjection from β„• to E.
-/
def countable {u : Type} (E : Set u) := βˆƒ (f : β„• β†’ u), SurjOn f (univ) E

/--
β„• is countable.
-/
lemma N_countable : countable (univ : Set β„•) := 
Goals accomplished! πŸ™
countable univ
h
SurjOn (fun n => n) univ univ
countable univ
n: β„•
n_in: n ∈ univ
h
n ∈ (fun n => n) '' univ
countable univ
Goals accomplished! πŸ™


/--
β„€ is countable
-/
lemma Z_countable : countable (univ : Set β„€) := 
Goals accomplished! πŸ™
countable univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
countable univ
countable univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
countable univ
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: Β¬0 < z
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: Β¬0 < z
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
pos
z ∈ f '' univ
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
pos
z ∈ f '' univ
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h
n ∈ univ ∧ f n = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h
f n = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h
f n = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h
f n = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h
(if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
[Meta.Tactic.simp.rewrite] dif_pos:1000, if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1 ==> -(↑n / 2)
[Meta.Tactic.simp.rewrite] dif_neg:1000, if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1 ==> (↑n - 1) / 2 + 1
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: Even n
h.isTrue
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
(↑n - 1) / 2 + 1 = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: Even n
h.isTrue
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
(↑n - 1) / 2 + 1 = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h_if: Even n
False
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h_if: Even n
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h_if: Even n
False
Warning: `Nat.odd_iff_not_even` has been deprecated, use `Nat.not_even_iff_odd` instead
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h_if: Even n
False
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
h.isFalse
(↑n - 1) / 2 + 1 = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
↑n = 2 * (z - 1) + 1
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
↑n = 2 * (z - 1) + 1
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
↑n = 2 * (z - 1) + 1
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe_tmp: ↑(z - 1).toNat = z - 1
↑n = 2 * (z - 1) + 1
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
↑n = 2 * (z - 1) + 1
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe_tmp: ↑(z - 1).toNat = z - 1
↑n = 2 * (z - 1) + 1
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe_tmp: ↑(z - 1).toNat = z - 1
↑n = 2 * ↑(z - 1).toNat + 1
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe_tmp: ↑(z - 1).toNat = z - 1
↑n = 2 * ↑(z - 1).toNat + 1
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
↑n = 2 * (z - 1) + 1
Goals accomplished! πŸ™
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe: ↑n = 2 * (z - 1) + 1
h.isFalse
(↑n - 1) / 2 + 1 = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe: ↑n = 2 * (z - 1) + 1
h.isFalse
(2 * (z - 1) + 1 - 1) / 2 + 1 = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe: ↑n = 2 * (z - 1) + 1
h.isFalse
(2 * (z - 1) + 1 - 1) / 2 + 1 = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
pos
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe: ↑n = 2 * (z - 1) + 1
h.isFalse
(2 * (z - 1) + 1 - 1) / 2 + 1 = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe: ↑n = 2 * (z - 1) + 1
(2 * (z - 1) + 1 - 1) / 2 + 1 = 2 * (z - 1) / 2 + 1
[Meta.Tactic.simp.rewrite] add_sub_cancel_right:1000, 2 * (z - 1) + 1 - 1 ==> 2 * (z - 1)
[Meta.Tactic.simp.rewrite] ne_eq:1000, 2 β‰  0 ==> Β¬2 = 0
[Meta.Tactic.simp.rewrite] OfNat.ofNat_ne_zero:1000, 2 = 0 ==> False
[Meta.Tactic.simp.rewrite] not_false_eq_true:1000, Β¬False ==> True
[Meta.Tactic.simp.rewrite] mul_div_cancel_leftβ‚€:1000, 2 * (z - 1) / 2 ==> z - 1
[Meta.Tactic.simp.rewrite] sub_add_cancel:1000, z - 1 + 1 ==> z
[Meta.Tactic.simp.rewrite] ne_eq:1000, 2 β‰  0 ==> Β¬2 = 0
[Meta.Tactic.simp.rewrite] OfNat.ofNat_ne_zero:1000, 2 = 0 ==> False
[Meta.Tactic.simp.rewrite] not_false_eq_true:1000, Β¬False ==> True
[Meta.Tactic.simp.rewrite] mul_div_cancel_leftβ‚€:1000, 2 * (z - 1) / 2 ==> z - 1
[Meta.Tactic.simp.rewrite] sub_add_cancel:1000, z - 1 + 1 ==> z
[Meta.Tactic.simp.rewrite] eq_self:1000, z = z ==> True
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe: ↑n = 2 * (z - 1) + 1
h.isFalse
(2 * (z - 1) + 1 - 1) / 2 + 1 = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe: ↑n = 2 * (z - 1) + 1
2 * (z - 1) / 2 + 1 = z - 1 + 1
[Meta.Tactic.simp.rewrite] ne_eq:1000, 2 β‰  0 ==> Β¬2 = 0
[Meta.Tactic.simp.rewrite] OfNat.ofNat_ne_zero:1000, 2 = 0 ==> False
[Meta.Tactic.simp.rewrite] not_false_eq_true:1000, Β¬False ==> True
[Meta.Tactic.simp.rewrite] mul_div_cancel_leftβ‚€:1000, 2 * (z - 1) / 2 ==> z - 1
[Meta.Tactic.simp.rewrite] sub_add_cancel:1000, z - 1 + 1 ==> z
[Meta.Tactic.simp.rewrite] sub_add_cancel:1000, z - 1 + 1 ==> z
[Meta.Tactic.simp.rewrite] eq_self:1000, z = z ==> True
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe: ↑n = 2 * (z - 1) + 1
h.isFalse
(2 * (z - 1) + 1 - 1) / 2 + 1 = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: 0 < z
n:= 2 * (z - 1).toNat + 1: β„•
odd_n: Odd n
h✝: ¬Even n
coe: ↑n = 2 * (z - 1) + 1
z - 1 + 1 = z
[Meta.Tactic.simp.rewrite] sub_add_cancel:1000, z - 1 + 1 ==> z
[Meta.Tactic.simp.rewrite] eq_self:1000, z = z ==> True
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
neg
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
neg
z ∈ f '' univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
neg
z ∈ f '' univ
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
Even n
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
Even n
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
Even n
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
βˆƒ r, n = r + r
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
Even n
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
h
n = (-z).toNat + (-z).toNat
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
Even n
Goals accomplished! πŸ™
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n: Even n
h
n ∈ univ ∧ f n = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n: Even n
h
f n = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n: Even n
h
f n = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n: Even n
h
f n = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n: Even n
h
(if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
[Meta.Tactic.simp.rewrite] dif_pos:1000, if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1 ==> -(↑n / 2)
[Meta.Tactic.simp.rewrite] dif_pos:1000, if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1 ==> -(↑n / 2)
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
h.isTrue
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n: Even n
h✝: ¬Even n
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
h.isTrue
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
h.isTrue
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n: Even n
h✝: ¬Even n
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
h.isTrue
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
h.isTrue
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
h.isTrue
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
h.isTrue
-(↑n / 2) = z
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
h.isTrue
-(↑n / 2) = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
-(↑n / 2) = -(2 * ↑(-z).toNat / 2)
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
-(↑n / 2) = -(2 * ↑(-z).toNat / 2)
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
-(↑n / 2) = -(2 * ↑(-z).toNat / 2)
Goals accomplished! πŸ™
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
h.isTrue
-(↑n / 2) = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
-(2 * ↑(-z).toNat / 2) = -(2 * m_z / 2)
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
-(2 * ↑(-z).toNat / 2) = -(2 * m_z / 2)
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
-(2 * -z / 2) = -(2 * m_z / 2)
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
h.isTrue
-(↑n / 2) = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
-(2 * m_z / 2) = -m_z
[Meta.Tactic.simp.rewrite] ne_eq:1000, 2 β‰  0 ==> Β¬2 = 0
[Meta.Tactic.simp.rewrite] OfNat.ofNat_ne_zero:1000, 2 = 0 ==> False
[Meta.Tactic.simp.rewrite] not_false_eq_true:1000, Β¬False ==> True
[Meta.Tactic.simp.rewrite] mul_div_cancel_leftβ‚€:1000, 2 * m_z / 2 ==> m_z
[Meta.Tactic.simp.rewrite] eq_self:1000, -m_z = -m_z ==> True
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
h.isTrue
-(↑n / 2) = z
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
z: β„€
a✝: z ∈ univ
h: z ≀ 0
n:= 2 * (-z).toNat: β„•
even_n, h✝: Even n
coe: ↑(-z).toNat = -z
m_z:= -z: β„€
- -z = z
[Meta.Tactic.simp.rewrite] neg_neg:1000, - -z ==> z
[Meta.Tactic.simp.rewrite] eq_self:1000, z = z ==> True
Goals accomplished! πŸ™
f:= fun n => if h : Even n then -(↑n / 2) else (↑n - 1) / 2 + 1: β„• β†’ β„€
SurjOn f univ univ
Goals accomplished! πŸ™
Goals accomplished! πŸ™
countable univ
Goals accomplished! πŸ™


/--
Let E a set. E is countable iff it exists a surjection from a countable set A to E.
-/
lemma countable_trans {u : Type} {E : Set u} : countable E ↔ (βˆƒ (v : Type), βˆƒ (A : Set v), (countable A ∧ βˆƒ (f : v β†’ u), SurjOn f A E)) := 
Goals accomplished! πŸ™
u: Type
E: Set u
countable E ↔ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
mp
countable E β†’ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
(βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E) β†’ countable E
u: Type
E: Set u
countable E ↔ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
mp
countable E β†’ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
mp
countable E β†’ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
(βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E) β†’ countable E
u: Type
E: Set u
h_denomb: countable E
mp
βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
mp
countable E β†’ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
h_denomb: countable E
h
countable univ ∧ βˆƒ f, SurjOn f univ E
u: Type
E: Set u
mp
countable E β†’ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
h_denomb: countable E
h.left
countable univ
u: Type
E: Set u
h_denomb: countable E
βˆƒ f, SurjOn f univ E
u: Type
E: Set u
mp
countable E β†’ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
h_denomb: countable E
h.left
countable univ
u: Type
E: Set u
h_denomb: countable E
h.left
countable univ
u: Type
E: Set u
h_denomb: countable E
βˆƒ f, SurjOn f univ E
Goals accomplished! πŸ™
u: Type
E: Set u
mp
countable E β†’ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
f: β„• β†’ u
f_surj: SurjOn f univ E
h.right.intro
βˆƒ f, SurjOn f univ E
u: Type
E: Set u
mp
countable E β†’ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
Goals accomplished! πŸ™
u: Type
E: Set u
countable E ↔ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
h: βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
mpr
countable E
u: Type
E: Set u
countable E ↔ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
mpr.intro.intro.intro.intro.intro
countable E
u: Type
E: Set u
countable E ↔ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
mpr.intro.intro.intro.intro.intro
countable E
Goals accomplished! πŸ™
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
e: u
e_in_E: e ∈ E
e ∈ f ∘ g '' univ
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
e: u
e_in_E: e ∈ E
f_surj: e ∈ f '' A
e ∈ f ∘ g '' univ
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
intro.intro
e ∈ f ∘ g '' univ
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
g_surj: a ∈ g '' univ
intro.intro
e ∈ f ∘ g '' univ
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
intro.intro.intro.intro
e ∈ f ∘ g '' univ
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
h
n ∈ univ ∧ (f ∘ g) n = e
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
h.left
n ∈ univ
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
(f ∘ g) n = e
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
h.left
n ∈ univ
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
h.left
n ∈ univ
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
(f ∘ g) n = e
Goals accomplished! πŸ™
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
h.right
f (g n) = e
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
g_surj: SurjOn g univ A
f: v β†’ u
f_surj: SurjOn f A E
SurjOn (f ∘ g) univ E
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
h.right
f (g n) = e
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
h.right
f a = e
u: Type
E: Set u
v: Type
A: Set v
g: β„• β†’ v
f: v β†’ u
e: u
e_in_E: e ∈ E
a: v
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
h.right
f a = e
Goals accomplished! πŸ™
u: Type
E: Set u
countable E ↔ βˆƒ v A, countable A ∧ βˆƒ f, SurjOn f A E
Goals accomplished! πŸ™


/--
Let E be a countable set. Then any A βŠ† E is countable.
-/
lemma subset_of_countable_set {u : Type} {E : Set u} (A : Set u) (h : A βŠ† E) (he : countable E) : countable A := 
Goals accomplished! πŸ™
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
countable A
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
f:= fun e => e: u β†’ u
countable A
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
countable A
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
f:= fun e => e: u β†’ u
countable A
Goals accomplished! πŸ™
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
f:= fun e => e: u β†’ u
SurjOn f E A
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
f:= fun e => e: u β†’ u
SurjOn f E A
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
f:= fun e => e: u β†’ u
SurjOn f E A
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
f:= fun e => e: u β†’ u
a: u
a_in_A: a ∈ A
a ∈ f '' E
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
f:= fun e => e: u β†’ u
SurjOn f E A
Goals accomplished! πŸ™
Goals accomplished! πŸ™
u: Type
E, A: Set u
h: A βŠ† E
he: countable E
countable A
Goals accomplished! πŸ™


theorem prod_pow_primes_left {p q : β„•} [hp : Fact p.Prime] [hq : Fact q.Prime]
  (n m : β„•) (neq : p β‰  q) : padicValNat p (p^n * q^m) = n := 
Goals accomplished! πŸ™
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (p ^ n * q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (p ^ n * q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (p ^ n) + padicValNat p (q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (p ^ n) + padicValNat p (q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (p ^ n * q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (p ^ n) + padicValNat p (q ^ m) = n
Goals accomplished! πŸ™
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (p ^ n) + padicValNat p (q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (q ^ m) = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
m * padicValNat p q = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (q ^ m) = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
m * 0 = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (q ^ m) = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
0 = 0
Goals accomplished! πŸ™
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padicValNat p (p ^ n * q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padic_p_qm_eq_0: padicValNat p (q ^ m) = 0
padicValNat p (p ^ n) + padicValNat p (q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padic_p_qm_eq_0: padicValNat p (q ^ m) = 0
n + padicValNat p (q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padic_p_qm_eq_0: padicValNat p (q ^ m) = 0
padicValNat p (p ^ n) + padicValNat p (q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padic_p_qm_eq_0: padicValNat p (q ^ m) = 0
n + 0 = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padic_p_qm_eq_0: padicValNat p (q ^ m) = 0
padicValNat p (p ^ n) + padicValNat p (q ^ m) = n
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: p β‰  q
padic_p_qm_eq_0: padicValNat p (q ^ m) = 0
n = n
Goals accomplished! πŸ™


theorem prod_pow_primes_right {p q : β„•} [hp : Fact p.Prime] [hq : Fact q.Prime]
  (n m : β„•) (neq : q β‰  p) : padicValNat q (p^n * q^m) = m := 
Goals accomplished! πŸ™
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n * q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n * q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n) + padicValNat q (q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n) + padicValNat q (q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n * q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n) + padicValNat q (q ^ m) = m
Goals accomplished! πŸ™
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n) + padicValNat q (q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n) = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
n * padicValNat q p = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n) = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
n * 0 = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n) = 0
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
0 = 0
Goals accomplished! πŸ™
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padicValNat q (p ^ n * q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padic_q_pn_eq_0: padicValNat q (p ^ n) = 0
padicValNat q (p ^ n) + padicValNat q (q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padic_q_pn_eq_0: padicValNat q (p ^ n) = 0
padicValNat q (p ^ n) + m = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padic_q_pn_eq_0: padicValNat q (p ^ n) = 0
padicValNat q (p ^ n) + padicValNat q (q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padic_q_pn_eq_0: padicValNat q (p ^ n) = 0
0 + m = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padic_q_pn_eq_0: padicValNat q (p ^ n) = 0
padicValNat q (p ^ n) + padicValNat q (q ^ m) = m
p, q: β„•
hp: Fact (Nat.Prime p)
hq: Fact (Nat.Prime q)
n, m: β„•
neq: q β‰  p
padic_q_pn_eq_0: padicValNat q (p ^ n) = 0
m = m
Goals accomplished! πŸ™


/--
β„•Β² is countable.
-/
lemma N2_countable : countable (univ : Set (β„• Γ— β„•)) := 
Goals accomplished! πŸ™
countable univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
countable univ
countable univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
countable univ
countable univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
countable univ
Goals accomplished! πŸ™
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
x ∈ f '' A
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
x ∈ f '' A
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
h
n ∈ A ∧ f n = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A: n ∈ A
h
n ∈ A ∧ f n = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A: n ∈ A
h
f n = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A: n ∈ A
h
f n = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A: n ∈ A
h
f n = x
Goals accomplished! πŸ™
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A: n ∈ A
h
(if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0)) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
[Meta.Tactic.simp.rewrite] if_pos:1000, if n ∈ A then (n.factorization 2, n.factorization 3)
    else (0, 0) ==> (n.factorization 2, n.factorization 3)
[Meta.Tactic.simp.rewrite] if_pos:1000, if n ∈ A then (n.factorization 2, n.factorization 3)
    else (0, 0) ==> (n.factorization 2, n.factorization 3)
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(n.factorization 2, n.factorization 3) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A: n ∈ A
h✝: n βˆ‰ A
(n.factorization 2, n.factorization 3) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(n.factorization 2, n.factorization 3) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(n.factorization 2, n.factorization 3) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A: n ∈ A
h✝: n βˆ‰ A
(n.factorization 2, n.factorization 3) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(n.factorization 2, n.factorization 3) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(n.factorization 2, n.factorization 3) = x
Goals accomplished! πŸ™
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(padicValNat 2 n, n.factorization 3) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(n.factorization 2, n.factorization 3) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(padicValNat 2 n, n.factorization 3) = x
Goals accomplished! πŸ™
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(padicValNat 2 n, padicValNat 3 n) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(n.factorization 2, n.factorization 3) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(padicValNat 2 n, padicValNat 3 n) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(padicValNat 2 n, padicValNat 3 n) = x
[Meta.Tactic.simp.rewrite] ne_eq:1000, 2 β‰  3 ==> Β¬2 = 3
[Meta.Tactic.simp.rewrite] not_false_eq_true:1000, Β¬False ==> True
Goals accomplished! πŸ™
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(x.1, padicValNat 3 n) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(padicValNat 2 n, padicValNat 3 n) = x
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(x.1, padicValNat 3 n) = x
[Meta.Tactic.simp.rewrite] ne_eq:1000, 3 β‰  2 ==> Β¬3 = 2
[Meta.Tactic.simp.rewrite] Nat.succ_ne_self:1000, 3 = 2 ==> False
[Meta.Tactic.simp.rewrite] not_false_eq_true:1000, Β¬False ==> True
Goals accomplished! πŸ™
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
x: β„• Γ— β„•
a✝: x ∈ univ
n:= 2 ^ x.1 * 3 ^ x.2: β„•
n_in_A, h✝: n ∈ A
h.isTrue
(x.1, x.2) = x
Goals accomplished! πŸ™
A:= {n | βˆƒ a b, n = 2 ^ a * 3 ^ b}: Set β„•
f:= fun n => if n ∈ A then (n.factorization 2, n.factorization 3) else (0, 0): β„• β†’ β„• Γ— β„•
SurjOn f A univ
Goals accomplished! πŸ™
Goals accomplished! πŸ™
countable univ
Goals accomplished! πŸ™


/--
We use a definition of a set being countable different to Mathlib's to match the book as much as possible. For completeness, we show that our definition is equivalent to Mathlib's. It corresponds to Corollary 1.5.
-/
lemma countable_iff {u : Type} {E : Set u} (hE : Set.Nonempty E) : countable E ↔ Set.Countable E := 
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
mp
countable E β†’ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
E.Countable β†’ countable E
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
mp
countable E β†’ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
mp
countable E β†’ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
E.Countable β†’ countable E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
h: countable E
mp
E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
mp
countable E β†’ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
mp.intro
E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
mp
countable E β†’ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
mp.intro
E.Countable
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
βˆ€ e ∈ E, βˆƒ n, f n = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
βˆ€ e ∈ E, βˆƒ n, f n = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
βˆ€ e ∈ E, βˆƒ n, f n = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
e: u
e_in_E: e ∈ E
βˆƒ n, f n = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
βˆ€ e ∈ E, βˆƒ n, f n = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
e: u
e_in_E: e ∈ E
f_surj: e ∈ f '' univ
βˆƒ n, f n = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
βˆ€ e ∈ E, βˆƒ n, f n = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
e: u
e_in_E: e ∈ E
n: β„•
left✝: n ∈ univ
f_n: f n = e
intro.intro
βˆƒ n, f n = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
βˆ€ e ∈ E, βˆƒ n, f n = e
Goals accomplished! πŸ™
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
mp
countable E β†’ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
mp.intro
E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
mp
countable E β†’ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
mp.intro
E.Countable
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
Function.Injective g
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
Function.Injective g
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
Function.Injective g
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
Function.Injective g
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
e: u
he: e ∈ E
f (g ⟨e, he⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
e: u
he: e ∈ E
f (g ⟨e, he⟩) = e
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
e: u
he: e ∈ E
f (g ⟨e, he⟩) = e
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
e: u
he: e ∈ E
h_g_e: g ⟨e, he⟩ = Nat.find β‹―
tmp: f (Nat.find β‹―) = e
f (g ⟨e, he⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
e: u
he: e ∈ E
h_g_e: g ⟨e, he⟩ = Nat.find β‹―
tmp: f (Nat.find β‹―) = e
f (g ⟨e, he⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
e: u
he: e ∈ E
h_g_e: g ⟨e, he⟩ = Nat.find β‹―
tmp: f (g ⟨e, he⟩) = e
f (g ⟨e, he⟩) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
e: u
he: e ∈ E
h_g_e: g ⟨e, he⟩ = Nat.find β‹―
tmp: f (g ⟨e, he⟩) = e
f (g ⟨e, he⟩) = e
Goals accomplished! πŸ™
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
Function.Injective g
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
f_g_id: βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
e1, e2: s_t
hg: g e1 = g e2
e1 = e2
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
Function.Injective g
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
f_g_id: βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
e1, e2: s_t
hg: g e1 = g e2
a
↑e1 = ↑e2
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
Function.Injective g
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
f_g_id: βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
e1, e2: s_t
hg: g e1 = g e2
a
↑e1 = ↑e2
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
f_g_id: βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
e1, e2: s_t
hg: g e1 = g e2
a
f (g βŸ¨β†‘e1, β‹―βŸ©) = ↑e2
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
f_g_id: βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
e1, e2: s_t
hg: g e1 = g e2
a
↑e1 = ↑e2
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
f_g_id: βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
e1, e2: s_t
hg: g e1 = g e2
a
f (g βŸ¨β†‘e1, β‹―βŸ©) = f (g βŸ¨β†‘e2, β‹―βŸ©)
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
f_g_id: βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
e1, e2: s_t
hg: g e1 = g e2
a
↑e1 = ↑e2
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: β„• β†’ u
f_surj: SurjOn f univ E
surj': βˆ€ e ∈ E, βˆƒ n, f n = e
g:= fun e => Nat.find β‹―: s_t β†’ β„•
f_g_id: βˆ€ (e : u) (h : e ∈ E), f (g ⟨e, h⟩) = e
e1, e2: s_t
hg: g e1 = g e2
a
f (g e2) = f (g βŸ¨β†‘e2, β‹―βŸ©)
Goals accomplished! πŸ™
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
mp
countable E β†’ E.Countable
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
h: E.Countable
mpr
countable E
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
mpr.mk.intro
countable E
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
mpr.mk.intro
countable E
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
mpr.mk.intro
countable E
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
mpr.mk.intro
countable E
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
βˆ€ n ∈ fE, {e | f e = n}.Nonempty
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
βˆ€ n ∈ fE, {e | f e = n}.Nonempty
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
βˆ€ n ∈ fE, {e | f e = n}.Nonempty
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
n: β„•
hn: n ∈ fE
{e | f e = n}.Nonempty
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
βˆ€ n ∈ fE, {e | f e = n}.Nonempty
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
n: β„•
e: s_t
fe_n: f e = n
intro
{e | f e = n}.Nonempty
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Goals accomplished! πŸ™
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
mpr.mk.intro
countable E
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
mpr.mk.intro
countable E
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
mpr.mk.intro
countable E
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e ∈ f' '' fE
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
e ∈ f' '' fE
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
h
f e' ∈ fE ∧ f' (f e') = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
h
f e' ∈ fE ∧ f' (f e') = e
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
h
f e' ∈ fE ∧ f' (f e') = e
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE: f e' ∈ fE
h
f' (f e') = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE: f e' ∈ fE
h
f' (f e') = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE: f e' ∈ fE
h
f' (f e') = e
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE: f e' ∈ fE
h
(if h : f e' ∈ fE then ↑(Ο• (f e') h) else hE.some) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
[Meta.Tactic.simp.rewrite] dif_pos:1000, if h : f e' ∈ fE then ↑(Ο• (f e') h) else hE.some ==> ↑(Ο• (f e') h✝)
[Meta.Tactic.simp.rewrite] dif_pos:1000, if h : f e' ∈ fE then ↑(Ο• (f e') h) else hE.some ==> ↑(Ο• (f e') fe'_in_fE)
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE, h✝: f e' ∈ fE
h.isTrue
↑(Ο• (f e') h✝) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE: f e' ∈ fE
h✝: f e' βˆ‰ fE
↑(Ο• (f e') fe'_in_fE) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE, h✝: f e' ∈ fE
h.isTrue
↑(Ο• (f e') h✝) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE, h✝: f e' ∈ fE
h.isTrue
↑(Ο• (f e') h✝) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE: f e' ∈ fE
h✝: f e' βˆ‰ fE
↑(Ο• (f e') fe'_in_fE) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE, h✝: f e' ∈ fE
ff'e'_eq_fe': f (Ο• (f e') fe'_in_fE) = f e'
h.isTrue
↑(Ο• (f e') h✝) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE, h✝: f e' ∈ fE
h.isTrue
↑(Ο• (f e') h✝) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE, h✝: f e' ∈ fE
ff'e'_eq_fe': f (Ο• (f e') fe'_in_fE) = f e'
f_inj: Ο• (f e') fe'_in_fE = e'
h.isTrue
↑(Ο• (f e') h✝) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE, h✝: f e' ∈ fE
h.isTrue
↑(Ο• (f e') h✝) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE, h✝: f e' ∈ fE
ff'e'_eq_fe': f (Ο• (f e') fe'_in_fE) = f e'
f_inj: Ο• (f e') fe'_in_fE = e'
eq_subtype: βˆ€ {e1 e2 : s_t}, e1 = e2 β†’ ↑e1 = ↑e2
h.isTrue
↑(Ο• (f e') h✝) = e
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
e: u
e_in_E: e ∈ E
e':= ⟨e, e_in_E⟩: s_t
fe'_in_fE, h✝: f e' ∈ fE
h.isTrue
↑(Ο• (f e') h✝) = e
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
s_t:= { e // e ∈ E }: Type
f: ↑E β†’ β„•
f_inj: Function.Injective f
fE:= {n | βˆƒ e, f e = n}: Set β„•
fE_countable: countable fE
fE_dual_non_empty: βˆ€ n ∈ fE, {e | f e = n}.Nonempty
Ο•:= fun n h => β‹―.some: (n : β„•) β†’ n ∈ fE β†’ ↑E
f':= fun n => if h : n ∈ fE then ↑(Ο• n h) else hE.some: β„• β†’ u
SurjOn f' fE E
Goals accomplished! πŸ™
Goals accomplished! πŸ™
u: Type
E: Set u
hE: E.Nonempty
countable E ↔ E.Countable
Goals accomplished! πŸ™


/--
The product of two countable sets is countable. (Proposition 1.6 p.2)
-/
lemma product_countable_set {u v : Type} {E : Set u} {A : Set v} (he : countable E) (ha : countable A) : countable {x : u Γ— v | x.1 ∈ E ∧ x.2 ∈ A} := 
Goals accomplished! πŸ™
u, v: Type
E: Set u
A: Set v
he: countable E
ha: countable A
countable {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
ha: countable A
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
intro
countable {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
he: countable E
ha: countable A
countable {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
intro.intro
countable {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
he: countable E
ha: countable A
countable {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
intro.intro
countable {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
he: countable E
ha: countable A
countable {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
intro.intro
countable {x | x.1 ∈ E ∧ x.2 ∈ A}
Goals accomplished! πŸ™
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
SurjOn Ο• univ {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
SurjOn Ο• univ {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
SurjOn Ο• univ {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
x ∈ Ο• '' univ
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
SurjOn Ο• univ {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝: e ∈ univ
hfe: fe e = x.1
intro.intro
x ∈ Ο• '' univ
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
SurjOn Ο• univ {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
intro.intro.intro.intro
x ∈ Ο• '' univ
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
SurjOn Ο• univ {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
h
(e, a) ∈ univ ∧ Ο• (e, a) = x
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
SurjOn Ο• univ {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
h
Ο• (e, a) = x
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
SurjOn Ο• univ {x | x.1 ∈ E ∧ x.2 ∈ A}
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
h
Ο• (e, a) = x
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
h
Ο• (e, a) = x
Goals accomplished! πŸ™
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
h
(fe e, fa a) = x
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
h
Ο• (e, a) = x
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
h
(x.1, fa a) = x
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
h
Ο• (e, a) = x
u, v: Type
E: Set u
A: Set v
fe: β„• β†’ u
fe_surj: SurjOn fe univ E
fa: β„• β†’ v
fa_surj: SurjOn fa univ A
Ο•:= fun x => (fe x.1, fa x.2): β„• Γ— β„• β†’ u Γ— v
x: u Γ— v
hx1: x.1 ∈ E
hx2: x.2 ∈ A
e: β„•
left✝¹: e ∈ univ
hfe: fe e = x.1
a: β„•
left✝: a ∈ univ
hfa: fa a = x.2
h
(x.1, x.2) = x
Goals accomplished! πŸ™
Goals accomplished! πŸ™
u, v: Type
E: Set u
A: Set v
he: countable E
ha: countable A
countable {x | x.1 ∈ E ∧ x.2 ∈ A}
Goals accomplished! πŸ™


/--
β„š is countable. (Example 1 p.3)
-/
lemma Q_countable : countable (univ : Set β„š) := 
Goals accomplished! πŸ™
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
countable univ
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
countable univ
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
countable univ
Goals accomplished! πŸ™
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
q ∈ f '' A
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
q ∈ f '' A
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h
x ∈ A ∧ f x = q
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h
f x = q
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h
f x = q
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h
f x = q
Goals accomplished! πŸ™
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h
(if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0) = q
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
[Meta.Tactic.simp.rewrite] dif_pos:1000, if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― }
    else 0 ==> { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― }
[Meta.Tactic.simp.rewrite] dif_neg:1000, if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― }
    else 0 ==> 0
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h✝: x ∈ A
h.isTrue
{ num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } = q
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h✝: x βˆ‰ A
0 = q
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h✝: x ∈ A
h.isTrue
{ num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } = q
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h✝: x ∈ A
h.isTrue
{ num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } = q
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h✝: x βˆ‰ A
0 = q
Goals accomplished! πŸ™
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
SurjOn f A univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h✝: x βˆ‰ A
h.isFalse
0 = q
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h_if: x βˆ‰ A
False
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
q: β„š
a✝: q ∈ univ
x:= (q.num, q.den): β„€ Γ— β„•
h_if: x βˆ‰ A
0 = q
Goals accomplished! πŸ™
Goals accomplished! πŸ™
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
countable univ
Goals accomplished! πŸ™
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
countable univ
Goals accomplished! πŸ™
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
{x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
{x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
{x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
x: β„€ Γ— β„•
h
x ∈ {x | x.1 ∈ univ ∧ x.2 ∈ univ} ↔ x ∈ univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
{x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
x: β„€ Γ— β„•
h.mp
x ∈ {x | x.1 ∈ univ ∧ x.2 ∈ univ} β†’ x ∈ univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
x: β„€ Γ— β„•
x ∈ univ β†’ x ∈ {x | x.1 ∈ univ ∧ x.2 ∈ univ}
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
{x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
x: β„€ Γ— β„•
h.mp
x ∈ {x | x.1 ∈ univ ∧ x.2 ∈ univ} β†’ x ∈ univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
x: β„€ Γ— β„•
h.mp
x ∈ {x | x.1 ∈ univ ∧ x.2 ∈ univ} β†’ x ∈ univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
x: β„€ Γ— β„•
x ∈ univ β†’ x ∈ {x | x.1 ∈ univ ∧ x.2 ∈ univ}
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
x: β„€ Γ— β„•
a✝: x ∈ {x | x.1 ∈ univ ∧ x.2 ∈ univ}
h.mp
x ∈ univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
x: β„€ Γ— β„•
h.mp
x ∈ {x | x.1 ∈ univ ∧ x.2 ∈ univ} β†’ x ∈ univ
Goals accomplished! πŸ™
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
{x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
x: β„€ Γ— β„•
a✝: x ∈ univ
h.mpr
x ∈ {x | x.1 ∈ univ ∧ x.2 ∈ univ}
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
{x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
Goals accomplished! πŸ™
Goals accomplished! πŸ™
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
eq_set: {x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
univ_countable: countable {x | x.1 ∈ univ ∧ x.2 ∈ univ}
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
eq_set: {x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
univ_countable: countable {x | x.1 ∈ univ ∧ x.2 ∈ univ}
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
eq_set: {x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
univ_countable: countable univ
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
eq_set: {x | x.1 ∈ univ ∧ x.2 ∈ univ} = univ
univ_countable: countable univ
countable univ
Goals accomplished! πŸ™
Goals accomplished! πŸ™
countable univ
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
countable_prod: countable univ
countable univ
Goals accomplished! πŸ™
A:= {x | 0 < x.2 ∧ x.1.natAbs.Coprime x.2}: Set (β„€ Γ— β„•)
f:= fun x => if h : x ∈ A then { num := x.1, den := x.2, den_nz := β‹―, reduced := β‹― } else 0: β„€ Γ— β„• β†’ β„š
f_surj: SurjOn f A univ
countable_prod: countable univ
countable_A: countable A
countable univ
countable univ
Goals accomplished! πŸ™


/--
Let A and E two sets such that E is uncountable. If it exists a surjection from A to E, then A is uncountable.
-/
lemma uncountable_trans {u v : Type} (A : Set u) {E : Set v} (h : Β¬countable E) : (βˆƒ (f : u β†’ v), SurjOn f A E) β†’ Β¬countable A := 
Goals accomplished! πŸ™
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
(βˆƒ f, SurjOn f A E) β†’ Β¬countable A
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
exists_surj: βˆƒ f, SurjOn f A E
Β¬countable A
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
(βˆƒ f, SurjOn f A E) β†’ Β¬countable A
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
intro
Β¬countable A
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
(βˆƒ f, SurjOn f A E) β†’ Β¬countable A
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
h_countable: countable A
intro
False
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
(βˆƒ f, SurjOn f A E) β†’ Β¬countable A
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
intro.intro
False
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
(βˆƒ f, SurjOn f A E) β†’ Β¬countable A
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
intro.intro
False
Goals accomplished! πŸ™
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
SurjOn (f ∘ g) univ E
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
SurjOn (f ∘ g) univ E
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
SurjOn (f ∘ g) univ E
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
e ∈ f ∘ g '' univ
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
SurjOn (f ∘ g) univ E
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
a: u
a_in_A: a ∈ A
fa: f a = e
intro.intro
e ∈ f ∘ g '' univ
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
SurjOn (f ∘ g) univ E
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
a: u
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
intro.intro.intro.intro
e ∈ f ∘ g '' univ
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
SurjOn (f ∘ g) univ E
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
a: u
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
right
(f ∘ g) n = e
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
SurjOn (f ∘ g) univ E
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
a: u
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
right
(f ∘ g) n = e
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
a: u
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
right
(f ∘ g) n = e
Goals accomplished! πŸ™
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
a: u
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
right
f (g n) = e
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
a: u
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
right
(f ∘ g) n = e
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
a: u
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
right
f a = e
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
e: v
e_in_E: e ∈ E
a: u
a_in_A: a ∈ A
fa: f a = e
n: β„•
n_in_N: n ∈ univ
gn: g n = a
right
f a = e
Goals accomplished! πŸ™
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
(βˆƒ f, SurjOn f A E) β†’ Β¬countable A
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
fg_surj_N_E: SurjOn (f ∘ g) univ E
intro.intro
False
Goals accomplished! πŸ™
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
f: u β†’ v
f_surj: SurjOn f A E
g: β„• β†’ u
g_surj: SurjOn g univ A
fg_surj_N_E: SurjOn (f ∘ g) univ E
intro.intro
False
Goals accomplished! πŸ™
u, v: Type
A: Set u
E: Set v
h: Β¬countable E
(βˆƒ f, SurjOn f A E) β†’ Β¬countable A
Goals accomplished! πŸ™


/--
Corrolary: if A βŠ† E is uncountable, then E is uncountable.
-/
lemma subset_of_uncountable_set {u : Type} {E A : Set u} (h : A βŠ† E) (uncountable : Β¬ countable A) : Β¬ countable E := 
Goals accomplished! πŸ™
u: Type
E, A: Set u
h: A βŠ† E
uncountable: Β¬countable A
Β¬countable E
u: Type
E, A: Set u
h: A βŠ† E
uncountable: Β¬countable A
h_contr: countable E
False
u: Type
E, A: Set u
h: A βŠ† E
uncountable: Β¬countable A
Β¬countable E
u: Type
E, A: Set u
h: A βŠ† E
uncountable: Β¬countable A
h_contr: countable E
f:= fun e => e: u β†’ u
False
u: Type
E, A: Set u
h: A βŠ† E
uncountable: Β¬countable A
Β¬countable E
u: Type
E, A: Set u
h: A βŠ† E
uncountable: Β¬countable A
h_contr: countable E
f:= fun e => e: u β†’ u
f_surj: SurjOn f E A
False
u: Type
E, A: Set u
h: A βŠ† E
uncountable: Β¬countable A
Β¬countable E
Goals accomplished! πŸ™



------------------------------FORMALIZATION-----------------------------------
/--
The power set of β„• is uncountable. (Example 4 p.3)
-/
lemma powerset_N_uncountable : Β¬countable (𝒫 (univ : Set β„•)) :=

Goals accomplished! πŸ™
Β¬countable (𝒫 univ)
h: countable (𝒫 univ)
False
Β¬countable (𝒫 univ)
Ο•: β„• β†’ Set β„•
h: SurjOn Ο• univ (𝒫 univ)
intro
False
Β¬countable (𝒫 univ)
Ο•: β„• β†’ Set β„•
h: SurjOn Ο• univ (𝒫 univ)
A:= {n | n βˆ‰ Ο• n}: Set β„•
intro
False
Β¬countable (𝒫 univ)
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
h: A ∈ Ο• '' univ
intro
False
Β¬countable (𝒫 univ)
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
intro.intro.intro
False
Β¬countable (𝒫 univ)
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: Β¬a βˆ‰ A
False
Β¬countable (𝒫 univ)
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: Β¬a βˆ‰ A
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
a_in_phi: Β¬a βˆ‰ Ο• a
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
a_in_phi: a ∈ Ο• a
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
a_in_phi: a ∈ Ο• a
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
a_in_phi: a ∈ A
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
a_in_phi: a ∈ A
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
a_in_phi: a ∈ A
pos
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a βˆ‰ A
pos
False
Goals accomplished! πŸ™
Β¬countable (𝒫 univ)
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a ∈ A
neg
False
Β¬countable (𝒫 univ)
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a ∈ A
a_notin_phi_a: a βˆ‰ Ο• a
neg
False
Β¬countable (𝒫 univ)
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a ∈ A
a_notin_phi_a: a βˆ‰ Ο• a
neg
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a ∈ A
a_notin_phi_a: a βˆ‰ A
neg
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a ∈ A
a_notin_phi_a: a βˆ‰ A
neg
False
Ο•: β„• β†’ Set β„•
A:= {n | n βˆ‰ Ο• n}: Set β„•
a: β„•
left✝: a ∈ univ
ha: Ο• a = A
ain: a ∈ A
a_notin_phi_a: a βˆ‰ A
neg
False
Β¬countable (𝒫 univ)
Goals accomplished! πŸ™


/-
- The set of all sequences in {0, 1}
-/
def zero_one := {u: β„• β†’ β„• | βˆ€n, u n = 0 ∨ u n = 1}

/--
The set of all sequences in {0, 1} is uncountable. (Example 5 p.3)
-/
lemma zero_one_uncountable : Β¬countable zero_one :=

Goals accomplished! πŸ™
Β¬countable zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
Β¬countable zero_one
Β¬countable zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
Β¬countable zero_one
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
A ∈ f '' zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
A ∈ f '' zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
h
Ο• ∈ zero_one ∧ f Ο• = A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
h
Ο• ∈ zero_one ∧ f Ο• = A
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο• ∈ zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο• ∈ zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο• ∈ zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
Ο• n = 0 ∨ Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο• ∈ zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
pos
Ο• n = 0 ∨ Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
Ο• n = 0 ∨ Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο• ∈ zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
pos
Ο• n = 0 ∨ Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
pos
Ο• n = 0 ∨ Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
Ο• n = 0 ∨ Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
pos
Ο• n = 0 ∨ Ο• n = 1
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
n βˆ‰ A β†’ 1 = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
n βˆ‰ A β†’ 1 = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
Ο• n = 1
Goals accomplished! πŸ™
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
pos
Ο• n = 0 ∨ Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
ind_eq_1: Ο• n = 1
pos.h
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
ind_eq_1: Ο• n = 1
pos.h
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n ∈ A
pos
Ο• n = 0 ∨ Ο• n = 1
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο• ∈ zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
neg
Ο• n = 0 ∨ Ο• n = 1
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
Ο• n = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
Ο• n = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
Ο• n = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
Ο• n = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
n ∈ A β†’ 1 = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
n ∈ A β†’ 1 = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
Ο• n = 0
Goals accomplished! πŸ™
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο• ∈ zero_one
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
ind_eq_0: Ο• n = 0
neg.h
Ο• n = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
n: β„•
h: n βˆ‰ A
ind_eq_0: Ο• n = 0
neg.h
Ο• n = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο• ∈ zero_one
Goals accomplished! πŸ™
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
h
f Ο• = A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
h.h
n ∈ f Ο• ↔ n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
h.h.mp
n ∈ f Ο• β†’ n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n ∈ A β†’ n ∈ f Ο•
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
h.h.mp
n ∈ f Ο• β†’ n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
h.h.mp
n ∈ f Ο• β†’ n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n ∈ A β†’ n ∈ f Ο•
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_f_Ο•: n ∈ f Ο•
h.h.mp
n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
h.h.mp
n ∈ f Ο• β†’ n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_f_Ο•: n ∈ f Ο•
Ο•n_eq_1: Ο• n = 1
h.h.mp
n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
h.h.mp
n ∈ f Ο• β†’ n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_f_Ο•: n ∈ f Ο•
Ο•n_eq_1: Ο• n = 1
h.h.mp
n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_f_Ο•: n ∈ f Ο•
Ο•n_eq_1: n βˆ‰ A β†’ 1 = 0
h.h.mp
n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_f_Ο•: n ∈ f Ο•
Ο•n_eq_1: n βˆ‰ A β†’ 1 = 0
h.h.mp
n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_f_Ο•: n ∈ f Ο•
Ο•n_eq_1: n βˆ‰ A β†’ 1 = 0
h.h.mp
n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
h.h.mp
n ∈ f Ο• β†’ n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_f_Ο•: n ∈ f Ο•
Ο•n_eq_1: n βˆ‰ A β†’ 1 = 0
n_notin_A: ¬n ∈ A
h.h.mp
False
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
h.h.mp
n ∈ f Ο• β†’ n ∈ A
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_f_Ο•: n ∈ f Ο•
n_notin_A: ¬n ∈ A
Ο•n_eq_1: 1 = 0
h.h.mp
False
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
h.h.mp
n ∈ f Ο• β†’ n ∈ A
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
h.h.mpr
n ∈ f Ο•
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
h.h.mpr
n ∈ f Ο•
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
n βˆ‰ A β†’ 1 = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
n βˆ‰ A β†’ 1 = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
Ο• n = 1
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
n_notin_A: n βˆ‰ A
1 = 0
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
A: Set β„•
a✝: A ∈ 𝒫 univ
Ο•:= A.indicator fun x => 1: β„• β†’ β„•
Ο•_in_zero_one: Ο• ∈ zero_one
n: β„•
n_in_A: n ∈ A
Ο• n = 1
Goals accomplished! πŸ™
Goals accomplished! πŸ™
f:= fun Ο• => {n | Ο• n = 1}: (β„• β†’ β„•) β†’ Set β„•
SurjOn f zero_one (𝒫 univ)
Goals accomplished! πŸ™
Goals accomplished! πŸ™
Β¬countable zero_one
Goals accomplished! πŸ™


/--
The set of all sequences to β„• is uncountable.
-/
example : Β¬ countable (univ : Set (β„• β†’ β„•)) :=
  subset_of_uncountable_set (Ξ» _ _ ↦ trivial) zero_one_uncountable

/--
For any real number x and any positive number Ξ΅, it exists a rational q such that |x - q| < Ξ΅.
-/
lemma Q_dense (x : ℝ) : βˆ€ Ξ΅ > 0, βˆƒ (q : β„š), |x - q| < Ξ΅ :=

Goals accomplished! πŸ™
x: ℝ
βˆ€ Ξ΅ > 0, βˆƒ q, |x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ q, |x - ↑q| < Ξ΅
x: ℝ
βˆ€ Ξ΅ > 0, βˆƒ q, |x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ q, |x - ↑q| < Ξ΅
Goals accomplished! πŸ™
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ q, |x - ↑q| < Ξ΅
Goals accomplished! πŸ™
x: ℝ
βˆ€ Ξ΅ > 0, βˆƒ q, |x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
intro.intro
βˆƒ q, |x - ↑q| < Ξ΅
x: ℝ
βˆ€ Ξ΅ > 0, βˆƒ q, |x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
h
|x - ↑q| < Ξ΅
x: ℝ
βˆ€ Ξ΅ > 0, βˆƒ q, |x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
h
|x - ↑q| < Ξ΅
Warning: unused variable `x_minus_q_neg`
note: this linter can be disabled with `set_option linter.unusedVariables false`
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
h
|x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
h
|x - ↑q| < Ξ΅
Goals accomplished! πŸ™
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
h
|x - ↑q| < Ξ΅
Goals accomplished! πŸ™
x: ℝ
βˆ€ Ξ΅ > 0, βˆƒ q, |x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
h
|x - ↑q| < Ξ΅
Goals accomplished! πŸ™
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
|x - ↑q| = -(x - ↑q)
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
|x - ↑q| = -(x - ↑q)
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
|x - ↑q| = -(x - ↑q)
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
|x - ↑q| = -(x - ↑q)
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
x - ↑q ≀ 0
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
x - ↑q ≀ 0
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
|x - ↑q| = -(x - ↑q)
Goals accomplished! πŸ™
Goals accomplished! πŸ™
x: ℝ
βˆ€ Ξ΅ > 0, βˆƒ q, |x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
abs_eq_neg: |x - ↑q| = -(x - ↑q)
h
|x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
abs_eq_neg: |x - ↑q| = -(x - ↑q)
h
-(x - ↑q) < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
abs_eq_neg: |x - ↑q| = -(x - ↑q)
h
|x - ↑q| < Ξ΅
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
abs_eq_neg: |x - ↑q| = -(x - ↑q)
h
-(x - ↑q) < Ξ΅
Goals accomplished! πŸ™
x, Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h: x < x + Ξ΅
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
x_minus_q_neg: x - ↑q < 0
abs_eq_neg: |x - ↑q| = -(x - ↑q)
h
↑q - x < Ξ΅
x: ℝ
βˆ€ Ξ΅ > 0, βˆƒ q, |x - ↑q| < Ξ΅
Goals accomplished! πŸ™


/--
For any real number x, it exists a sequence of rational that tends to x.
-/
example (x : ℝ) : βˆƒ (u : β„• β†’ β„š), Tendsto (Ξ» n ↦ ((u n) : ℝ)) atTop (𝓝 x) :=

Goals accomplished! πŸ™
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
Goals accomplished! πŸ™
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
βˆ€ (n : β„•), (A n).Nonempty
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
βˆ€ (n : β„•), (A n).Nonempty
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
βˆ€ (n : β„•), (A n).Nonempty
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
n: β„•
(A n).Nonempty
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
βˆ€ (n : β„•), (A n).Nonempty
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
n: β„•
q: β„š
hq: |x - ↑q| < 1 / (↑n + 1)
intro
(A n).Nonempty
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
βˆ€ (n : β„•), (A n).Nonempty
Goals accomplished! πŸ™
Goals accomplished! πŸ™
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
h
Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
h
Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
h
βˆ€ Ξ΅ > 0, βˆƒ N, βˆ€ n > N, dist (↑(u n)) x < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
h
βˆ€ Ξ΅ > 0, βˆƒ N, βˆ€ n > N, dist (↑(u n)) x < Ξ΅
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
h
βˆ€ Ξ΅ > 0, βˆƒ N, βˆ€ n > N, dist (↑(u n)) x < Ξ΅
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
h
βˆ€ Ξ΅ > 0, βˆƒ N, βˆ€ n > N, dist (↑(u n)) x < Ξ΅
[Meta.Tactic.simp.rewrite] dst:1000, dist (↑(u n)) x ==> |x - ↑(u n)|
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
h
βˆ€ Ξ΅ > 0, βˆƒ N, βˆ€ n > N, |x - ↑(u n)| < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
h
βˆ€ Ξ΅ > 0, βˆƒ N, βˆ€ n > N, |x - ↑(u n)| < Ξ΅
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h
βˆƒ N, βˆ€ n > N, |x - ↑(u n)| < Ξ΅
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h
βˆƒ N, βˆ€ n > N, |x - ↑(u n)| < Ξ΅
Goals accomplished! πŸ™
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ N, 1 / (↑N + 1) < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ N, 1 / (↑N + 1) < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ N, 1 / (↑N + 1) < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
h
1 / (β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1) < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ N, 1 / (↑N + 1) < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
1 / (β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1) < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ N, 1 / (↑N + 1) < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
1 / (β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1) < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
1 / (β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1) < Ξ΅
Goals accomplished! πŸ™
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
(β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1)⁻¹ < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ N, 1 / (↑N + 1) < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
(β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1)⁻¹ < Ξ΅
Warning: `inv_lt` has been deprecated, use `inv_lt_commβ‚€` instead
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
(β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1)⁻¹ < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
Ρ⁻¹ < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
(β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1)⁻¹ < Ξ΅
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
Ρ⁻¹ < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
Goals accomplished! πŸ™
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
R_pos: 0 < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
h
1 / Ξ΅ < β†‘βŒˆ1 / Ξ΅βŒ‰β‚Š + 1
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
βˆƒ N, 1 / (↑N + 1) < Ξ΅
Goals accomplished! πŸ™
Goals accomplished! πŸ™
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
N: β„•
hn: 1 / (↑N + 1) < Ξ΅
h
βˆ€ n > N, |x - ↑(u n)| < Ξ΅
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
x: ℝ
A:= fun n => {q | |x - ↑q| < 1 / (↑n + 1)}: β„• β†’ Set β„š
non_empty: βˆ€ (n : β„•), (A n).Nonempty
u:= fun n => β‹―.some: β„• β†’ β„š
dst: βˆ€ (n : β„•), dist (↑(u n)) x = |x - ↑(u n)|
Ξ΅: ℝ
Ξ΅_pos: Ξ΅ > 0
N: β„•
hn: 1 / (↑N + 1) < Ξ΅
n: β„•
N_ltn: n > N
h
|x - ↑(u n)| < Ξ΅
x: ℝ
βˆƒ u, Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
Goals accomplished! πŸ™


/--
Let A be a family of disjoint, non-empty, open sets of ℝ. Then A is countable. (Example 9 p.4)
-/
example (A : NNReal β†’ Set ℝ) (h_nonempty : βˆ€ t, Set.Nonempty (A t)) (h_open : βˆ€ t, IsOpen (A t)) (h_disjoint : βˆ€ t1 t2, t1 β‰  t2 β†’ Disjoint (A t1) (A t2)) : countable {A t | t : NNReal} :=

Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
countable {x | βˆƒ t, A t = x}
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: IsOpen (A t)
intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: IsOpen (A t)
intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
intro.intro.intro
(q_A_t t).Nonempty
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: ↑q - x < Ξ΅
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: ↑q - x < Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: ↑q - x < Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: |↑q - x| < Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: ↑q - x < Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: |↑q - x| < Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: dist (↑q) x < Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: ↑q - x < Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: ↑q ∈ Metric.ball x Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: ↑q ∈ Metric.ball x Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
t: NNReal
x: ℝ
hx: x ∈ A t
h_open: βˆ€ x ∈ A t, βˆƒ Ξ΅ > 0, Metric.ball x Ξ΅ βŠ† A t
Ξ΅: ℝ
hΞ΅: Ξ΅ > 0
ball_in_At: Metric.ball x Ξ΅ βŠ† A t
q: β„š
hql: x < ↑q
hqr: ↑q < x + Ξ΅
q_dist: ↑q ∈ Metric.ball x Ξ΅
pos_abs: ↑q - x = |↑q - x|
intro.intro.intro.intro.intro
(q_A_t t).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
βˆ€ (t : NNReal), (q_A_t t).Nonempty
Goals accomplished! πŸ™
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
countable {x | βˆƒ t, A t = x}
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
q: β„š
h: βˆƒ t, q ∈ q_A_t t
(q_A q).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
q: β„š
t: NNReal
hq: q ∈ q_A_t t
intro
(q_A q).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
q: β„š
t: NNReal
hq: q ∈ q_A_t t
h
A t = A t ∧ ↑q ∈ A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
q: β„š
t: NNReal
hq: q ∈ q_A_t t
h
↑q ∈ A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Goals accomplished! πŸ™
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
countable {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
countable {x | βˆƒ t, A t = x}
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
At ∈ f '' univ
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
At ∈ f '' univ
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
A t ∈ f '' univ
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
A t ∈ f '' univ
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
intro
A t ∈ f '' univ
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h
q ∈ univ ∧ f q = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h
f q = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h
f q = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h
f q = A t
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h
(if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0) = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
[Meta.Tactic.simp.rewrite] dif_pos:1000, if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0 ==> Ο• q h✝
[Meta.Tactic.simp.rewrite] dif_neg:1000, if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0 ==> A 0
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h✝: βˆƒ t, q ∈ q_A_t t
h.isTrue
Ο• q h✝ = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h✝: Β¬βˆƒ t, q ∈ q_A_t t
A 0 = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h✝: βˆƒ t, q ∈ q_A_t t
h.isTrue
Ο• q h✝ = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h✝: Β¬βˆƒ t, q ∈ q_A_t t
A 0 = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
Ο• q h_if = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
Ο• q h_if = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
rfl_Ο•: Ο• q h_if ∈ q_A q
Ο• q h_if = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
Ο• q h_if = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
intro.intro
Ο• q h_if = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
Ο• q h_if = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
intro.intro
Ο• q h_if = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
intro.intro
A t2 = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
intro.intro
A t2 = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
Ο• q h_if = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
intro.intro
A t2 = A t
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
A t = A t2
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
A t = A t2
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
A t = A t2
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
h_neg: Β¬A t = A t2
False
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
h_neg: Β¬A t = A t2
False
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
A t = A t2
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
h_neg: A t β‰  A t2
False
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
A t = A t2
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
h_neg: A t β‰  A t2
t_neq: t β‰  t2
False
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
A t = A t2
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
h_neg: A t β‰  A t2
t_neq: t β‰  t2
disjoint: Disjoint (A t) (A t2)
False
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
A t = A t2
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
h_neg: A t β‰  A t2
t_neq: t β‰  t2
disjoint: Disjoint (A t) (A t2)
q_not_in: ↑q βˆ‰ A t2
False
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
A t = A t2
Goals accomplished! πŸ™
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
Ο• q h_if = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
At_eq_At2: A t = A t2
intro.intro
A t2 = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆƒ t, q ∈ q_A_t t
t2: NNReal
h_eq_t2: A t2 = Ο• q h_if
hq2: ↑q ∈ A t2
At_eq_At2: A t = A t2
intro.intro
A t2 = A t2
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
SurjOn f univ {x | βˆƒ t, A t = x}
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h✝: Β¬βˆƒ t, q ∈ q_A_t t
h.isFalse
A 0 = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: βˆ€ (t : NNReal), q βˆ‰ q_A_t t
A 0 = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: Β¬βˆƒ t, q ∈ q_A_t t
A 0 = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: q βˆ‰ q_A_t t
A 0 = A t
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
q_A_t:= fun t => {q | ↑q ∈ A t}: NNReal β†’ Set β„š
nonempty_q_A_t: βˆ€ (t : NNReal), (q_A_t t).Nonempty
q_A:= fun q => {x | βˆƒ t, A t = x ∧ ↑q ∈ A t}: β„š β†’ Set (Set ℝ)
nonempty_q_A: βˆ€ (q : β„š), (βˆƒ t, q ∈ q_A_t t) β†’ (q_A q).Nonempty
Ο•:= fun q hq => β‹―.some: (q : β„š) β†’ (βˆƒ t, q ∈ q_A_t t) β†’ Set ℝ
f:= fun q => if h : βˆƒ t, q ∈ q_A_t t then Ο• q h else A 0: β„š β†’ Set ℝ
At: Set ℝ
t: NNReal
h: A t = At
q: β„š
hq: q ∈ q_A_t t
h_if: Β¬βˆƒ t, q ∈ q_A_t t
A 0 = A t
Goals accomplished! πŸ™
Goals accomplished! πŸ™
A: NNReal β†’ Set ℝ
h_nonempty: βˆ€ (t : NNReal), (A t).Nonempty
h_open: βˆ€ (t : NNReal), IsOpen (A t)
h_disjoint: βˆ€ (t1 t2 : NNReal), t1 β‰  t2 β†’ Disjoint (A t1) (A t2)
countable {x | βˆƒ t, A t = x}
Goals accomplished! πŸ™