File: Mathlib/Topology/Algebra/ProperConstSMul.lean

full_name: ProperConstVAdd
code: /-- A mixin typeclass saying that the `(c +ᵥ ·)` is a proper map for all `c`.

Note that this is **not** the same as a proper additive action (not yet in `Mathlib`). -/
class ProperConstVAdd (M X : Type*) [VAdd M X] [TopologicalSpace X] : Prop where
  /-- `(c +ᵥ ·)` is a proper map. -/
  isProperMap_vadd (c : M) : IsProperMap ((c +ᵥ ·) : X → X)
goal: Prop

full_name: ProperConstVAdd.isProperMap_vadd
code: /-- A mixin typeclass saying that the `(c +ᵥ ·)` is a proper map for all `c`.

Note that this is **not** the same as a proper additive action (not yet in `Mathlib`). -/
class ProperConstVAdd (M X : Type*) [VAdd M X] [TopologicalSpace X] : Prop where
  /-- `(c +ᵥ ·)` is a proper map. -/
  isProperMap_vadd (c : M) : IsProperMap ((c +ᵥ ·) : X → X)
goal: IsProperMap fun x => c +ᵥ x

full_name: ProperConstSMul.isProperMap_smul
code: /-- A mixin typeclass saying that `(c • ·)` is a proper map for all `c`.

Note that this is **not** the same as a proper multiplicative action (not yet in `Mathlib`). -/
@[to_additive]
class ProperConstSMul (M X : Type*) [SMul M X] [TopologicalSpace X] : Prop where
  /-- `(c • ·)` is a proper map. -/
  isProperMap_smul (c : M) : IsProperMap ((c • ·) : X → X)
goal: IsProperMap fun x => c • x

full_name: ProperConstSMul
code: /-- A mixin typeclass saying that `(c • ·)` is a proper map for all `c`.

Note that this is **not** the same as a proper multiplicative action (not yet in `Mathlib`). -/
@[to_additive]
class ProperConstSMul (M X : Type*) [SMul M X] [TopologicalSpace X] : Prop where
  /-- `(c • ·)` is a proper map. -/
  isProperMap_smul (c : M) : IsProperMap ((c • ·) : X → X)
goal: Prop

full_name: isProperMap_smul
code: /-- `(c • ·)` is a proper map. -/
@[to_additive "`(c +ᵥ ·)` is a proper map."]
theorem isProperMap_smul {M : Type*} (c : M) (X : Type*) [SMul M X] [TopologicalSpace X]
    [h : ProperConstSMul M X] : IsProperMap ((c • ·) : X → X) 
goal: IsProperMap fun x => c • x

full_name: isProperMap_vadd
code: /-- `(c • ·)` is a proper map. -/
@[to_additive "`(c +ᵥ ·)` is a proper map."]
theorem isProperMap_smul {M : Type*} (c : M) (X : Type*) [SMul M X] [TopologicalSpace X]
    [h : ProperConstSMul M X] : IsProperMap ((c • ·) : X → X) 
goal: IsProperMap fun x => c +ᵥ x

full_name: IsCompact.preimage_smul
code: /-- The preimage of a compact set under `(c • ·)` is a compact set. -/
@[to_additive "The preimage of a compact set under `(c +ᵥ ·)` is a compact set."]
theorem IsCompact.preimage_smul {M X : Type*} [SMul M X] [TopologicalSpace X]
    [ProperConstSMul M X] {s : Set X} (hs : IsCompact s) (c : M) : IsCompact ((c • ·) ⁻¹' s) 
goal: IsCompact ((fun x => c • x) ⁻¹' s)

full_name: IsCompact.preimage_vadd
code: /-- The preimage of a compact set under `(c • ·)` is a compact set. -/
@[to_additive "The preimage of a compact set under `(c +ᵥ ·)` is a compact set."]
theorem IsCompact.preimage_smul {M X : Type*} [SMul M X] [TopologicalSpace X]
    [ProperConstSMul M X] {s : Set X} (hs : IsCompact s) (c : M) : IsCompact ((c • ·) ⁻¹' s) 
goal: IsCompact ((fun x => c +ᵥ x) ⁻¹' s)

full_name: instProperConstSMulOfContinuousConstSMulOfT2SpaceOfCompactSpace
code: @[to_additive]
instance (priority := 100) {M X : Type*} [SMul M X] [TopologicalSpace X] [ContinuousConstSMul M X]
    [T2Space X] [CompactSpace X] : ProperConstSMul M X 
goal: ProperConstSMul M X

full_name: instProperConstVAddOfContinuousConstVAddOfT2SpaceOfCompactSpace
code: @[to_additive]
instance (priority := 100) {M X : Type*} [SMul M X] [TopologicalSpace X] [ContinuousConstSMul M X]
    [T2Space X] [CompactSpace X] : ProperConstSMul M X 
goal: ProperConstVAdd M X

full_name: instProperConstSMulOfContinuousConstSMul
code: @[to_additive]
instance (priority := 100) {G X : Type*} [Group G] [MulAction G X] [TopologicalSpace X]
    [ContinuousConstSMul G X] : ProperConstSMul G X 
goal: ProperConstSMul G X

full_name: instProperConstVAddOfContinuousConstVAdd
code: @[to_additive]
instance (priority := 100) {G X : Type*} [Group G] [MulAction G X] [TopologicalSpace X]
    [ContinuousConstSMul G X] : ProperConstSMul G X 
goal: ProperConstVAdd G X

full_name: instProperConstSMulProd
code: instance {M X Y : Type*}
    [SMul M X] [TopologicalSpace X] [ProperConstSMul M X]
    [SMul M Y] [TopologicalSpace Y] [ProperConstSMul M Y] :
    ProperConstSMul M (X × Y) 
goal: ProperConstSMul M (X × Y)

full_name: instProperConstSMulForall
code: instance {M ι : Type*} {X : ι → Type*}
    [∀ i, SMul M (X i)] [∀ i, TopologicalSpace (X i)] [∀ i, ProperConstSMul M (X i)] :
    ProperConstSMul M (∀ i, X i) 
goal: ProperConstSMul M ((i : ι) → X i)