Theorem: Profinite.NobelingProof.Products.eval_πs_image
File: Mathlib/Topology/Category/Profinite/Nobeling.lean

Proof steps:
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
⊢ eval C '' {m | m < l} = ⇑(πs C o) '' (eval (π C fun x => ord I x < o) '' {m | m < l})
state_after: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
⊢ f ∈ eval C '' {m | m < l} ↔ f ∈ ⇑(πs C o) '' (eval (π C fun x => ord I x < o) '' {m | m < l})
tactic: ext f
state_before: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
⊢ f ∈ eval C '' {m | m < l} ↔ f ∈ ⇑(πs C o) '' (eval (π C fun x => ord I x < o) '' {m | m < l})
state_after: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
⊢ (∃ x < l, eval C x = f) ↔ ∃ a < l, (πs C o) (eval (π C fun x => ord I x < o) a) = f
tactic: simp only [Set.mem_image, Set.mem_setOf_eq, exists_exists_and_eq_and]
state_before: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
⊢ (∃ x < l, eval C x = f) ↔ ∃ a < l, (πs C o) (eval (π C fun x => ord I x < o) a) = f
state_after: case h.h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
⊢ ∀ (a : Products I), a < l ∧ eval C a = f ↔ a < l ∧ (πs C o) (eval (π C fun x => ord I x < o) a) = f
tactic: apply exists_congr
state_before: case h.h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
⊢ ∀ (a : Products I), a < l ∧ eval C a = f ↔ a < l ∧ (πs C o) (eval (π C fun x => ord I x < o) a) = f
state_after: case h.h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
m : Products I
⊢ m < l ∧ eval C m = f ↔ m < l ∧ (πs C o) (eval (π C fun x => ord I x < o) m) = f
tactic: intro m
state_before: case h.h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
m : Products I
⊢ m < l ∧ eval C m = f ↔ m < l ∧ (πs C o) (eval (π C fun x => ord I x < o) m) = f
state_after: case h.h.h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
m : Products I
⊢ m < l → (eval C m = f ↔ (πs C o) (eval (π C fun x => ord I x < o) m) = f)
tactic: apply and_congr_right
state_before: case h.h.h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
m : Products I
⊢ m < l → (eval C m = f ↔ (πs C o) (eval (π C fun x => ord I x < o) m) = f)
state_after: case h.h.h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
m : Products I
hm : m < l
⊢ eval C m = f ↔ (πs C o) (eval (π C fun x => ord I x < o) m) = f
tactic: intro hm
state_before: case h.h.h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
m : Products I
hm : m < l
⊢ eval C m = f ↔ (πs C o) (eval (π C fun x => ord I x < o) m) = f
state_after: no goals
tactic: rw [eval_πs C (lt_ord_of_lt hm hl)]