Theorem: intrinsicClosure_idem
File: Mathlib/Analysis/Convex/Intrinsic.lean

Proof steps:
state_before: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t : Set P
x : P
s : Set P
⊢ intrinsicClosure 𝕜 (intrinsicClosure 𝕜 s) = intrinsicClosure 𝕜 s
state_after: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t : Set P
x : P
s : Set P
⊢ IsClosed (Subtype.val ⁻¹' intrinsicClosure 𝕜 s)
tactic: refine IsClosed.intrinsicClosure ?_
state_before: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t : Set P
x : P
s : Set P
⊢ IsClosed (Subtype.val ⁻¹' intrinsicClosure 𝕜 s)
state_after: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t✝ : Set P
x : P
s : Set P
t : AffineSubspace 𝕜 P := affineSpan 𝕜 (intrinsicClosure 𝕜 s)
ht : t = affineSpan 𝕜 (intrinsicClosure 𝕜 s)
⊢ IsClosed (Subtype.val ⁻¹' intrinsicClosure 𝕜 s)
tactic: set t := affineSpan 𝕜 (intrinsicClosure 𝕜 s) with ht
state_before: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t✝ : Set P
x : P
s : Set P
t : AffineSubspace 𝕜 P := affineSpan 𝕜 (intrinsicClosure 𝕜 s)
ht : t = affineSpan 𝕜 (intrinsicClosure 𝕜 s)
⊢ IsClosed (Subtype.val ⁻¹' intrinsicClosure 𝕜 s)
state_after: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t✝ : Set P
x : P
s : Set P
t : AffineSubspace 𝕜 P
ht : t = affineSpan 𝕜 (intrinsicClosure 𝕜 s)
⊢ IsClosed (Subtype.val ⁻¹' intrinsicClosure 𝕜 s)
tactic: clear_value t
state_before: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t✝ : Set P
x : P
s : Set P
t : AffineSubspace 𝕜 P
ht : t = affineSpan 𝕜 (intrinsicClosure 𝕜 s)
⊢ IsClosed (Subtype.val ⁻¹' intrinsicClosure 𝕜 s)
state_after: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t : Set P
x : P
s : Set P
ht : affineSpan 𝕜 s = affineSpan 𝕜 (intrinsicClosure 𝕜 s)
⊢ IsClosed (Subtype.val ⁻¹' intrinsicClosure 𝕜 s)
tactic: obtain rfl := ht.trans (affineSpan_intrinsicClosure _)
state_before: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t : Set P
x : P
s : Set P
ht : affineSpan 𝕜 s = affineSpan 𝕜 (intrinsicClosure 𝕜 s)
⊢ IsClosed (Subtype.val ⁻¹' intrinsicClosure 𝕜 s)
state_after: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t : Set P
x : P
s : Set P
ht : affineSpan 𝕜 s = affineSpan 𝕜 (intrinsicClosure 𝕜 s)
⊢ IsClosed (closure (Subtype.val ⁻¹' s))
tactic: rw [intrinsicClosure, preimage_image_eq _ Subtype.coe_injective]
state_before: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s✝ t : Set P
x : P
s : Set P
ht : affineSpan 𝕜 s = affineSpan 𝕜 (intrinsicClosure 𝕜 s)
⊢ IsClosed (closure (Subtype.val ⁻¹' s))
state_after: no goals
tactic: exact isClosed_closure