Theorem: LinearPMap.closureHasCore
File: Mathlib/Topology/Algebra/Module/LinearPMap.lean

tactic: refine' ⟨f.le_closure.1, _⟩
state_before: R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
⊢ HasCore (closure f) f.domain

tactic: congr
state_before: R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
⊢ closure (domRestrict (closure f) f.domain) = closure f

tactic: ext x
state_before: case e_f
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
⊢ domRestrict (closure f) f.domain = f

tactic: intro y hxy
state_before: case e_f.h'
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : { x // x ∈ (domRestrict (closure f) f.domain).domain }
⊢ ∀ ⦃y : { x // x ∈ f.domain }⦄, ↑x = ↑y → ↑(domRestrict (closure f) f.domain) x = ↑f y

tactic: let z : f.closure.domain := ⟨y.1, f.le_closure.1 y.2⟩
state_before: case e_f.h'
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : { x // x ∈ (domRestrict (closure f) f.domain).domain }
y : { x // x ∈ f.domain }
hxy : ↑x = ↑y
⊢ ↑(domRestrict (closure f) f.domain) x = ↑f y

tactic: have hyz : (y : E) = z := by simp
state_before: case e_f.h'
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : { x // x ∈ (domRestrict (closure f) f.domain).domain }
y : { x // x ∈ f.domain }
hxy : ↑x = ↑y
z : { x // x ∈ (closure f).domain } := { val := ↑y, property := (_ : ↑y ∈ (closure f).domain) }
⊢ ↑(domRestrict (closure f) f.domain) x = ↑f y

tactic: rw [f.le_closure.2 hyz]
state_before: case e_f.h'
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : { x // x ∈ (domRestrict (closure f) f.domain).domain }
y : { x // x ∈ f.domain }
hxy : ↑x = ↑y
z : { x // x ∈ (closure f).domain } := { val := ↑y, property := (_ : ↑y ∈ (closure f).domain) }
hyz : ↑y = ↑z
⊢ ↑(domRestrict (closure f) f.domain) x = ↑f y

tactic: exact domRestrict_apply (hxy.trans hyz)
state_before: case e_f.h'
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : { x // x ∈ (domRestrict (closure f) f.domain).domain }
y : { x // x ∈ f.domain }
hxy : ↑x = ↑y
z : { x // x ∈ (closure f).domain } := { val := ↑y, property := (_ : ↑y ∈ (closure f).domain) }
hyz : ↑y = ↑z
⊢ ↑(domRestrict (closure f) f.domain) x = ↑(closure f) z

tactic: simp only [domRestrict_domain, Submodule.mem_inf, and_iff_left_iff_imp]
state_before: case e_f.h.h
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : E
⊢ x ∈ (domRestrict (closure f) f.domain).domain ↔ x ∈ f.domain

tactic: intro hx
state_before: case e_f.h.h
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : E
⊢ x ∈ f.domain → x ∈ (closure f).domain

tactic: exact f.le_closure.1 hx
state_before: case e_f.h.h
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : E
hx : x ∈ f.domain
⊢ x ∈ (closure f).domain

tactic: simp
state_before: R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : { x // x ∈ (domRestrict (closure f) f.domain).domain }
y : { x // x ∈ f.domain }
hxy : ↑x = ↑y
z : { x // x ∈ (closure f).domain } := { val := ↑y, property := (_ : ↑y ∈ (closure f).domain) }
⊢ ↑y = ↑z