File: Mathlib/Topology/Algebra/Constructions/DomMulAct.lean

full_name: DomMulAct.instTopologicalSpace
code: @[to_additive]
instance instTopologicalSpace : TopologicalSpace Mᵈᵐᵃ 
goal: TopologicalSpace Mᵈᵐᵃ

full_name: DomAddAct.instTopologicalSpace
code: @[to_additive]
instance instTopologicalSpace : TopologicalSpace Mᵈᵐᵃ 
goal: TopologicalSpace Mᵈᵃᵃ

full_name: DomMulAct.continuous_mk
code: @[to_additive (attr := continuity, fun_prop)]
theorem continuous_mk : Continuous (@mk M) 
goal: Continuous ⇑DomMulAct.mk

full_name: DomAddAct.continuous_mk
code: @[to_additive (attr := continuity, fun_prop)]
theorem continuous_mk : Continuous (@mk M) 
goal: Continuous ⇑DomAddAct.mk

full_name: DomMulAct.continuous_mk_symm
code: @[to_additive (attr := continuity, fun_prop)]
theorem continuous_mk_symm : Continuous (@mk M).symm 
goal: Continuous ⇑DomMulAct.mk.symm

full_name: DomAddAct.continuous_mk_symm
code: @[to_additive (attr := continuity, fun_prop)]
theorem continuous_mk_symm : Continuous (@mk M).symm 
goal: Continuous ⇑DomAddAct.mk.symm

full_name: DomAddAct.mkHomeomorph_toEquiv
code: /-- `DomMulAct.mk` as a homeomorphism. -/
@[to_additive (attr := simps toEquiv) "`DomAddAct.mk` as a homeomorphism"]
def mkHomeomorph : M ≃ₜ Mᵈᵐᵃ 
goal: DomAddAct.mkHomeomorph.toEquiv = DomAddAct.mk

full_name: DomMulAct.mkHomeomorph
code: /-- `DomMulAct.mk` as a homeomorphism. -/
@[to_additive (attr := simps toEquiv) "`DomAddAct.mk` as a homeomorphism"]
def mkHomeomorph : M ≃ₜ Mᵈᵐᵃ 
goal: M ≃ₜ Mᵈᵐᵃ

full_name: DomAddAct.mkHomeomorph
code: /-- `DomMulAct.mk` as a homeomorphism. -/
@[to_additive (attr := simps toEquiv) "`DomAddAct.mk` as a homeomorphism"]
def mkHomeomorph : M ≃ₜ Mᵈᵐᵃ 
goal: M ≃ₜ Mᵈᵃᵃ

full_name: DomMulAct.mkHomeomorph_toEquiv
code: /-- `DomMulAct.mk` as a homeomorphism. -/
@[to_additive (attr := simps toEquiv) "`DomAddAct.mk` as a homeomorphism"]
def mkHomeomorph : M ≃ₜ Mᵈᵐᵃ 
goal: DomMulAct.mkHomeomorph.toEquiv = DomMulAct.mk

full_name: DomAddAct.coe_mkHomeomorph
code: @[to_additive (attr := simp)] theorem coe_mkHomeomorph : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ) = mk 
goal: ⇑DomAddAct.mkHomeomorph = ⇑DomAddAct.mk

full_name: DomMulAct.coe_mkHomeomorph
code: @[to_additive (attr := simp)] theorem coe_mkHomeomorph : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ) = mk 
goal: ⇑DomMulAct.mkHomeomorph = ⇑DomMulAct.mk

full_name: DomMulAct.coe_mkHomeomorph_symm
code: @[to_additive (attr := simp)]
theorem coe_mkHomeomorph_symm : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ).symm = mk.symm 
goal: ⇑DomMulAct.mkHomeomorph.symm = ⇑DomMulAct.mk.symm

full_name: DomAddAct.coe_mkHomeomorph_symm
code: @[to_additive (attr := simp)]
theorem coe_mkHomeomorph_symm : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ).symm = mk.symm 
goal: ⇑DomAddAct.mkHomeomorph.symm = ⇑DomAddAct.mk.symm

full_name: DomAddAct.inducing_mk
code: @[to_additive] theorem inducing_mk : Inducing (@mk M) 
goal: Inducing ⇑DomAddAct.mk

full_name: DomMulAct.inducing_mk
code: @[to_additive] theorem inducing_mk : Inducing (@mk M) 
goal: Inducing ⇑DomMulAct.mk

full_name: DomMulAct.embedding_mk
code: @[to_additive] theorem embedding_mk : Embedding (@mk M) 
goal: Embedding ⇑DomMulAct.mk

full_name: DomAddAct.embedding_mk
code: @[to_additive] theorem embedding_mk : Embedding (@mk M) 
goal: Embedding ⇑DomAddAct.mk

full_name: DomMulAct.openEmbedding_mk
code: @[to_additive] theorem openEmbedding_mk : OpenEmbedding (@mk M) 
goal: OpenEmbedding ⇑DomMulAct.mk

full_name: DomAddAct.openEmbedding_mk
code: @[to_additive] theorem openEmbedding_mk : OpenEmbedding (@mk M) 
goal: OpenEmbedding ⇑DomAddAct.mk

full_name: DomAddAct.closedEmbedding_mk
code: @[to_additive] theorem closedEmbedding_mk : ClosedEmbedding (@mk M) 
goal: ClosedEmbedding ⇑DomAddAct.mk

full_name: DomMulAct.closedEmbedding_mk
code: @[to_additive] theorem closedEmbedding_mk : ClosedEmbedding (@mk M) 
goal: ClosedEmbedding ⇑DomMulAct.mk

full_name: DomAddAct.quotientMap_mk
code: @[to_additive] theorem quotientMap_mk : QuotientMap (@mk M) 
goal: QuotientMap ⇑DomAddAct.mk

full_name: DomMulAct.quotientMap_mk
code: @[to_additive] theorem quotientMap_mk : QuotientMap (@mk M) 
goal: QuotientMap ⇑DomMulAct.mk

full_name: DomAddAct.inducing_mk_symm
code: @[to_additive] theorem inducing_mk_symm : Inducing (@mk M).symm 
goal: Inducing ⇑DomAddAct.mk.symm

full_name: DomMulAct.inducing_mk_symm
code: @[to_additive] theorem inducing_mk_symm : Inducing (@mk M).symm 
goal: Inducing ⇑DomMulAct.mk.symm

full_name: DomMulAct.embedding_mk_symm
code: @[to_additive] theorem embedding_mk_symm : Embedding (@mk M).symm 
goal: Embedding ⇑DomMulAct.mk.symm

full_name: DomAddAct.embedding_mk_symm
code: @[to_additive] theorem embedding_mk_symm : Embedding (@mk M).symm 
goal: Embedding ⇑DomAddAct.mk.symm

full_name: DomAddAct.openEmbedding_mk_symm
code: @[to_additive]
theorem openEmbedding_mk_symm : OpenEmbedding (@mk M).symm 
goal: OpenEmbedding ⇑DomAddAct.mk.symm

full_name: DomMulAct.openEmbedding_mk_symm
code: @[to_additive]
theorem openEmbedding_mk_symm : OpenEmbedding (@mk M).symm 
goal: OpenEmbedding ⇑DomMulAct.mk.symm

full_name: DomMulAct.closedEmbedding_mk_symm
code: @[to_additive]
theorem closedEmbedding_mk_symm : ClosedEmbedding (@mk M).symm 
goal: ClosedEmbedding ⇑DomMulAct.mk.symm

full_name: DomAddAct.closedEmbedding_mk_symm
code: @[to_additive]
theorem closedEmbedding_mk_symm : ClosedEmbedding (@mk M).symm 
goal: ClosedEmbedding ⇑DomAddAct.mk.symm

full_name: DomMulAct.quotientMap_mk_symm
code: @[to_additive]
theorem quotientMap_mk_symm : QuotientMap (@mk M).symm 
goal: QuotientMap ⇑DomMulAct.mk.symm

full_name: DomAddAct.quotientMap_mk_symm
code: @[to_additive]
theorem quotientMap_mk_symm : QuotientMap (@mk M).symm 
goal: QuotientMap ⇑DomAddAct.mk.symm

full_name: DomMulAct.instT0Space
code: @[to_additive] instance instT0Space [T0Space M] : T0Space Mᵈᵐᵃ 
goal: T0Space Mᵈᵐᵃ

full_name: DomAddAct.instT0Space
code: @[to_additive] instance instT0Space [T0Space M] : T0Space Mᵈᵐᵃ 
goal: T0Space Mᵈᵃᵃ

full_name: DomAddAct.instT1Space
code: @[to_additive] instance instT1Space [T1Space M] : T1Space Mᵈᵐᵃ 
goal: T1Space Mᵈᵃᵃ

full_name: DomMulAct.instT1Space
code: @[to_additive] instance instT1Space [T1Space M] : T1Space Mᵈᵐᵃ 
goal: T1Space Mᵈᵐᵃ

full_name: DomMulAct.instT2Space
code: @[to_additive] instance instT2Space [T2Space M] : T2Space Mᵈᵐᵃ 
goal: T2Space Mᵈᵐᵃ

full_name: DomAddAct.instT2Space
code: @[to_additive] instance instT2Space [T2Space M] : T2Space Mᵈᵐᵃ 
goal: T2Space Mᵈᵃᵃ

full_name: DomAddAct.instT25Space
code: @[to_additive] instance instT25Space [T25Space M] : T25Space Mᵈᵐᵃ 
goal: T25Space Mᵈᵃᵃ

full_name: DomMulAct.instT25Space
code: @[to_additive] instance instT25Space [T25Space M] : T25Space Mᵈᵐᵃ 
goal: T25Space Mᵈᵐᵃ

full_name: DomMulAct.instT3Space
code: @[to_additive] instance instT3Space [T3Space M] : T3Space Mᵈᵐᵃ 
goal: T3Space Mᵈᵐᵃ

full_name: DomAddAct.instT3Space
code: @[to_additive] instance instT3Space [T3Space M] : T3Space Mᵈᵐᵃ 
goal: T3Space Mᵈᵃᵃ

full_name: DomMulAct.instT4Space
code: @[to_additive] instance instT4Space [T4Space M] : T4Space Mᵈᵐᵃ 
goal: T4Space Mᵈᵐᵃ

full_name: DomAddAct.instT4Space
code: @[to_additive] instance instT4Space [T4Space M] : T4Space Mᵈᵐᵃ 
goal: T4Space Mᵈᵃᵃ

full_name: DomAddAct.instT5Space
code: @[to_additive] instance instT5Space [T5Space M] : T5Space Mᵈᵐᵃ 
goal: T5Space Mᵈᵃᵃ

full_name: DomMulAct.instT5Space
code: @[to_additive] instance instT5Space [T5Space M] : T5Space Mᵈᵐᵃ 
goal: T5Space Mᵈᵐᵃ

full_name: DomMulAct.instR0Space
code: @[to_additive] instance instR0Space [R0Space M] : R0Space Mᵈᵐᵃ 
goal: R0Space Mᵈᵐᵃ

full_name: DomAddAct.instR0Space
code: @[to_additive] instance instR0Space [R0Space M] : R0Space Mᵈᵐᵃ 
goal: R0Space Mᵈᵃᵃ

full_name: DomMulAct.instR1Space
code: @[to_additive] instance instR1Space [R1Space M] : R1Space Mᵈᵐᵃ 
goal: R1Space Mᵈᵐᵃ

full_name: DomAddAct.instR1Space
code: @[to_additive] instance instR1Space [R1Space M] : R1Space Mᵈᵐᵃ 
goal: R1Space Mᵈᵃᵃ

full_name: DomAddAct.instRegularSpace
code: @[to_additive]
instance instRegularSpace [RegularSpace M] : RegularSpace Mᵈᵐᵃ 
goal: RegularSpace Mᵈᵃᵃ

full_name: DomMulAct.instRegularSpace
code: @[to_additive]
instance instRegularSpace [RegularSpace M] : RegularSpace Mᵈᵐᵃ 
goal: RegularSpace Mᵈᵐᵃ

full_name: DomAddAct.instNormalSpace
code: @[to_additive]
instance instNormalSpace [NormalSpace M] : NormalSpace Mᵈᵐᵃ 
goal: NormalSpace Mᵈᵃᵃ

full_name: DomMulAct.instNormalSpace
code: @[to_additive]
instance instNormalSpace [NormalSpace M] : NormalSpace Mᵈᵐᵃ 
goal: NormalSpace Mᵈᵐᵃ

full_name: DomAddAct.instCompletelyNormalSpace
code: @[to_additive]
instance instCompletelyNormalSpace [CompletelyNormalSpace M] : CompletelyNormalSpace Mᵈᵐᵃ 
goal: CompletelyNormalSpace Mᵈᵃᵃ

full_name: DomMulAct.instCompletelyNormalSpace
code: @[to_additive]
instance instCompletelyNormalSpace [CompletelyNormalSpace M] : CompletelyNormalSpace Mᵈᵐᵃ 
goal: CompletelyNormalSpace Mᵈᵐᵃ

full_name: DomMulAct.instDiscreteTopology
code: @[to_additive]
instance instDiscreteTopology [DiscreteTopology M] : DiscreteTopology Mᵈᵐᵃ 
goal: DiscreteTopology Mᵈᵐᵃ

full_name: DomAddAct.instDiscreteTopology
code: @[to_additive]
instance instDiscreteTopology [DiscreteTopology M] : DiscreteTopology Mᵈᵐᵃ 
goal: DiscreteTopology Mᵈᵃᵃ

full_name: DomAddAct.instSeparableSpace
code: @[to_additive]
instance instSeparableSpace [SeparableSpace M] : SeparableSpace Mᵈᵐᵃ 
goal: TopologicalSpace.SeparableSpace Mᵈᵃᵃ

full_name: DomMulAct.instSeparableSpace
code: @[to_additive]
instance instSeparableSpace [SeparableSpace M] : SeparableSpace Mᵈᵐᵃ 
goal: TopologicalSpace.SeparableSpace Mᵈᵐᵃ

full_name: DomAddAct.instFirstCountableTopology
code: @[to_additive]
instance instFirstCountableTopology [FirstCountableTopology M] : FirstCountableTopology Mᵈᵐᵃ 
goal: FirstCountableTopology Mᵈᵃᵃ

full_name: DomMulAct.instFirstCountableTopology
code: @[to_additive]
instance instFirstCountableTopology [FirstCountableTopology M] : FirstCountableTopology Mᵈᵐᵃ 
goal: FirstCountableTopology Mᵈᵐᵃ

full_name: DomMulAct.instSecondCountableTopology
code: @[to_additive]
instance instSecondCountableTopology [SecondCountableTopology M] : SecondCountableTopology Mᵈᵐᵃ 
goal: SecondCountableTopology Mᵈᵐᵃ

full_name: DomAddAct.instSecondCountableTopology
code: @[to_additive]
instance instSecondCountableTopology [SecondCountableTopology M] : SecondCountableTopology Mᵈᵐᵃ 
goal: SecondCountableTopology Mᵈᵃᵃ

full_name: DomAddAct.map_mk_nhds
code: @[to_additive (attr := simp)]
theorem map_mk_nhds (x : M) : map (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk x) 
goal: Filter.map (⇑DomAddAct.mk) (𝓝 x) = 𝓝 (DomAddAct.mk x)

full_name: DomMulAct.map_mk_nhds
code: @[to_additive (attr := simp)]
theorem map_mk_nhds (x : M) : map (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk x) 
goal: Filter.map (⇑DomMulAct.mk) (𝓝 x) = 𝓝 (DomMulAct.mk x)

full_name: DomMulAct.map_mk_symm_nhds
code: @[to_additive (attr := simp)]
theorem map_mk_symm_nhds (x : Mᵈᵐᵃ) : map (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk.symm x) 
goal: Filter.map (⇑DomMulAct.mk.symm) (𝓝 x) = 𝓝 (DomMulAct.mk.symm x)

full_name: DomAddAct.map_mk_symm_nhds
code: @[to_additive (attr := simp)]
theorem map_mk_symm_nhds (x : Mᵈᵐᵃ) : map (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk.symm x) 
goal: Filter.map (⇑DomAddAct.mk.symm) (𝓝 x) = 𝓝 (DomAddAct.mk.symm x)

full_name: DomMulAct.comap_mk_nhds
code: @[to_additive (attr := simp)]
theorem comap_mk_nhds (x : Mᵈᵐᵃ) : comap (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk.symm x) 
goal: Filter.comap (⇑DomMulAct.mk) (𝓝 x) = 𝓝 (DomMulAct.mk.symm x)

full_name: DomAddAct.comap_mk_nhds
code: @[to_additive (attr := simp)]
theorem comap_mk_nhds (x : Mᵈᵐᵃ) : comap (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk.symm x) 
goal: Filter.comap (⇑DomAddAct.mk) (𝓝 x) = 𝓝 (DomAddAct.mk.symm x)

full_name: DomAddAct.comap_mk.symm_nhds
code: @[to_additive (attr := simp)]
theorem comap_mk.symm_nhds (x : M) : comap (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk x) 
goal: Filter.comap (⇑DomAddAct.mk.symm) (𝓝 x) = 𝓝 (DomAddAct.mk x)

full_name: DomMulAct.comap_mk.symm_nhds
code: @[to_additive (attr := simp)]
theorem comap_mk.symm_nhds (x : M) : comap (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk x) 
goal: Filter.comap (⇑DomMulAct.mk.symm) (𝓝 x) = 𝓝 (DomMulAct.mk x)