Theorem: isClosedMap_smul₀
File: Mathlib/Topology/Algebra/ConstMulAction.lean

tactic: rcases eq_or_ne c 0 with (rfl | hne)
state_before: M : Type u_1
α : Type u_2
β : Type u_3
G₀ : Type u_4
inst✝⁹ : TopologicalSpace α
inst✝⁸ : GroupWithZero G₀
inst✝⁷ : MulAction G₀ α
inst✝⁶ : ContinuousConstSMul G₀ α
inst✝⁵ : TopologicalSpace β
f : β → α
b : β
c✝ : G₀
s : Set β
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
c : G₀
⊢ IsClosedMap fun x => c • x
state_after: case inl
M : Type u_1
α : Type u_2
β : Type u_3
G₀ : Type u_4
inst✝⁹ : TopologicalSpace α
inst✝⁸ : GroupWithZero G₀
inst✝⁷ : MulAction G₀ α
inst✝⁶ : ContinuousConstSMul G₀ α
inst✝⁵ : TopologicalSpace β
f : β → α
b : β
c : G₀
s : Set β
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
⊢ IsClosedMap fun x => 0 • x

case inr
M : Type u_1
α : Type u_2
β : Type u_3
G₀ : Type u_4
inst✝⁹ : TopologicalSpace α
inst✝⁸ : GroupWithZero G₀
inst✝⁷ : MulAction G₀ α
inst✝⁶ : ContinuousConstSMul G₀ α
inst✝⁵ : TopologicalSpace β
f : β → α
b : β
c✝ : G₀
s : Set β
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
c : G₀
hne : c ≠ 0
⊢ IsClosedMap fun x => c • x

tactic: simp only [zero_smul]
state_before: case inl
M : Type u_1
α : Type u_2
β : Type u_3
G₀ : Type u_4
inst✝⁹ : TopologicalSpace α
inst✝⁸ : GroupWithZero G₀
inst✝⁷ : MulAction G₀ α
inst✝⁶ : ContinuousConstSMul G₀ α
inst✝⁵ : TopologicalSpace β
f : β → α
b : β
c : G₀
s : Set β
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
⊢ IsClosedMap fun x => 0 • x
state_after: case inl
M : Type u_1
α : Type u_2
β : Type u_3
G₀ : Type u_4
inst✝⁹ : TopologicalSpace α
inst✝⁸ : GroupWithZero G₀
inst✝⁷ : MulAction G₀ α
inst✝⁶ : ContinuousConstSMul G₀ α
inst✝⁵ : TopologicalSpace β
f : β → α
b : β
c : G₀
s : Set β
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
⊢ IsClosedMap fun x => 0

tactic: exact isClosedMap_const
state_before: case inl
M : Type u_1
α : Type u_2
β : Type u_3
G₀ : Type u_4
inst✝⁹ : TopologicalSpace α
inst✝⁸ : GroupWithZero G₀
inst✝⁷ : MulAction G₀ α
inst✝⁶ : ContinuousConstSMul G₀ α
inst✝⁵ : TopologicalSpace β
f : β → α
b : β
c : G₀
s : Set β
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
⊢ IsClosedMap fun x => 0
state_after: no goals

tactic: exact (Homeomorph.smulOfNeZero c hne).isClosedMap
state_before: case inr
M : Type u_1
α : Type u_2
β : Type u_3
G₀ : Type u_4
inst✝⁹ : TopologicalSpace α
inst✝⁸ : GroupWithZero G₀
inst✝⁷ : MulAction G₀ α
inst✝⁶ : ContinuousConstSMul G₀ α
inst✝⁵ : TopologicalSpace β
f : β → α
b : β
c✝ : G₀
s : Set β
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
c : G₀
hne : c ≠ 0
⊢ IsClosedMap fun x => c • x
state_after: no goals