Theorem: Finset.map_truncatedInf
File: Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeInf α
inst✝⁴ : BoundedOrder α
inst✝³ : DecidableRel fun x x_1 => x ≤ x_1
inst✝² : SemilatticeInf β
inst✝¹ : BoundedOrder β
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s✝ t : Finset α
a✝ : α
e : α ≃o β
s : Finset α
a : α
⊢ e (s.truncatedInf a) = (map e.toEmbedding s).truncatedInf (e a)
state_after: α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeInf α
inst✝⁴ : BoundedOrder α
inst✝³ : DecidableRel fun x x_1 => x ≤ x_1
inst✝² : SemilatticeInf β
inst✝¹ : BoundedOrder β
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s✝ t : Finset α
a✝ : α
e : α ≃o β
s : Finset α
a : α
this : e a ∈ upperClosure ↑(map e.toEmbedding s) ↔ a ∈ upperClosure ↑s
⊢ e (s.truncatedInf a) = (map e.toEmbedding s).truncatedInf (e a)
tactic: have : e a ∈ upperClosure (s.map e.toEquiv.toEmbedding) ↔ a ∈ upperClosure s := by simp
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeInf α
inst✝⁴ : BoundedOrder α
inst✝³ : DecidableRel fun x x_1 => x ≤ x_1
inst✝² : SemilatticeInf β
inst✝¹ : BoundedOrder β
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s✝ t : Finset α
a✝ : α
e : α ≃o β
s : Finset α
a : α
this : e a ∈ upperClosure ↑(map e.toEmbedding s) ↔ a ∈ upperClosure ↑s
⊢ e (s.truncatedInf a) = (map e.toEmbedding s).truncatedInf (e a)
state_after: α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeInf α
inst✝⁴ : BoundedOrder α
inst✝³ : DecidableRel fun x x_1 => x ≤ x_1
inst✝² : SemilatticeInf β
inst✝¹ : BoundedOrder β
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s✝ t : Finset α
a✝ : α
e : α ≃o β
s : Finset α
a : α
this : e a ∈ upperClosure ↑(map e.toEmbedding s) ↔ a ∈ upperClosure ↑s
⊢ (if h : a ∈ upperClosure ↑s then (filter (fun x => x ≤ a) s).inf' ⋯ (⇑e ∘ id) else ⊥) =
    if h : a ∈ upperClosure ↑s then (filter (fun x => x ≤ e a) (map e.toEmbedding s)).inf' ⋯ id else ⊥
tactic: simp_rw [truncatedInf, apply_dite e, map_finset_inf', map_bot, this]
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeInf α
inst✝⁴ : BoundedOrder α
inst✝³ : DecidableRel fun x x_1 => x ≤ x_1
inst✝² : SemilatticeInf β
inst✝¹ : BoundedOrder β
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s✝ t : Finset α
a✝ : α
e : α ≃o β
s : Finset α
a : α
this : e a ∈ upperClosure ↑(map e.toEmbedding s) ↔ a ∈ upperClosure ↑s
⊢ (if h : a ∈ upperClosure ↑s then (filter (fun x => x ≤ a) s).inf' ⋯ (⇑e ∘ id) else ⊥) =
    if h : a ∈ upperClosure ↑s then (filter (fun x => x ≤ e a) (map e.toEmbedding s)).inf' ⋯ id else ⊥
state_after: case e_t.h
α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeInf α
inst✝⁴ : BoundedOrder α
inst✝³ : DecidableRel fun x x_1 => x ≤ x_1
inst✝² : SemilatticeInf β
inst✝¹ : BoundedOrder β
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s✝ t : Finset α
a✝ : α
e : α ≃o β
s : Finset α
a : α
this : e a ∈ upperClosure ↑(map e.toEmbedding s) ↔ a ∈ upperClosure ↑s
h : a ∈ upperClosure ↑s
⊢ (filter (fun x => x ≤ a) s).inf' ⋯ (⇑e ∘ id) = (filter (fun x => x ≤ e a) (map e.toEmbedding s)).inf' ⋯ id
tactic: congr with h
state_before: case e_t.h
α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeInf α
inst✝⁴ : BoundedOrder α
inst✝³ : DecidableRel fun x x_1 => x ≤ x_1
inst✝² : SemilatticeInf β
inst✝¹ : BoundedOrder β
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s✝ t : Finset α
a✝ : α
e : α ≃o β
s : Finset α
a : α
this : e a ∈ upperClosure ↑(map e.toEmbedding s) ↔ a ∈ upperClosure ↑s
h : a ∈ upperClosure ↑s
⊢ (filter (fun x => x ≤ a) s).inf' ⋯ (⇑e ∘ id) = (filter (fun x => x ≤ e a) (map e.toEmbedding s)).inf' ⋯ id
state_after: no goals
tactic: simp only [filter_map, Function.comp, Equiv.coe_toEmbedding, RelIso.coe_fn_toEquiv,
    OrderIso.le_iff_le, id, inf'_map]