Theorem: isClosedMap_smul_left
File: Mathlib/Topology/Algebra/Module/FiniteDimension.lean

tactic: by_cases hc : c = 0
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : T2Space E
inst✝⁸ : TopologicalAddGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : ContinuousSMul 𝕜 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : T2Space F
inst✝² : TopologicalAddGroup F
inst✝¹ : Module 𝕜 F
inst✝ : ContinuousSMul 𝕜 F
c : E
⊢ IsClosedMap fun x => x • c
state_after: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : T2Space E
inst✝⁸ : TopologicalAddGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : ContinuousSMul 𝕜 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : T2Space F
inst✝² : TopologicalAddGroup F
inst✝¹ : Module 𝕜 F
inst✝ : ContinuousSMul 𝕜 F
c : E
hc : c = 0
⊢ IsClosedMap fun x => x • c

case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : T2Space E
inst✝⁸ : TopologicalAddGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : ContinuousSMul 𝕜 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : T2Space F
inst✝² : TopologicalAddGroup F
inst✝¹ : Module 𝕜 F
inst✝ : ContinuousSMul 𝕜 F
c : E
hc : ¬c = 0
⊢ IsClosedMap fun x => x • c

tactic: simp_rw [hc, smul_zero]
state_before: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : T2Space E
inst✝⁸ : TopologicalAddGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : ContinuousSMul 𝕜 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : T2Space F
inst✝² : TopologicalAddGroup F
inst✝¹ : Module 𝕜 F
inst✝ : ContinuousSMul 𝕜 F
c : E
hc : c = 0
⊢ IsClosedMap fun x => x • c
state_after: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : T2Space E
inst✝⁸ : TopologicalAddGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : ContinuousSMul 𝕜 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : T2Space F
inst✝² : TopologicalAddGroup F
inst✝¹ : Module 𝕜 F
inst✝ : ContinuousSMul 𝕜 F
c : E
hc : c = 0
⊢ IsClosedMap fun x => 0

tactic: exact isClosedMap_const
state_before: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : T2Space E
inst✝⁸ : TopologicalAddGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : ContinuousSMul 𝕜 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : T2Space F
inst✝² : TopologicalAddGroup F
inst✝¹ : Module 𝕜 F
inst✝ : ContinuousSMul 𝕜 F
c : E
hc : c = 0
⊢ IsClosedMap fun x => 0
state_after: no goals

tactic: exact (closedEmbedding_smul_left hc).isClosedMap
state_before: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : T2Space E
inst✝⁸ : TopologicalAddGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : ContinuousSMul 𝕜 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : T2Space F
inst✝² : TopologicalAddGroup F
inst✝¹ : Module 𝕜 F
inst✝ : ContinuousSMul 𝕜 F
c : E
hc : ¬c = 0
⊢ IsClosedMap fun x => x • c
state_after: no goals