File: Mathlib/Topology/Instances/TrivSqZeroExt.lean

full_name: _aux_Mathlib_Topology_Instances_TrivSqZeroExt___unexpand_TrivSqZeroExt_1
code: local notation "tsze" => TrivSqZeroExt
goal: Lean.PrettyPrinter.Unexpander

full_name: _aux_Mathlib_Topology_Instances_TrivSqZeroExt___macroRules_termTsze_1
code: local notation "tsze" => TrivSqZeroExt
goal: Lean.Macro

full_name: TrivSqZeroExt.instTopologicalSpace
code: instance instTopologicalSpace : TopologicalSpace (tsze R M) 
goal: TopologicalSpace (tsze R M)

full_name: TrivSqZeroExt.instT2Space
code: instance [T2Space R] [T2Space M] : T2Space (tsze R M) 
goal: T2Space (tsze R M)

full_name: TrivSqZeroExt.nhds_def
code: theorem nhds_def (x : tsze R M) : 𝓝 x = (𝓝 x.fst).prod (𝓝 x.snd) 
goal: 𝓝 x = (𝓝 x.fst).prod (𝓝 x.snd)

full_name: TrivSqZeroExt.nhds_inl
code: theorem nhds_inl [Zero M] (x : R) : 𝓝 (inl x : tsze R M) = (𝓝 x).prod (𝓝 0) 
goal: 𝓝 (TrivSqZeroExt.inl x) = (𝓝 x).prod (𝓝 0)

full_name: TrivSqZeroExt.nhds_inr
code: theorem nhds_inr [Zero R] (m : M) : 𝓝 (inr m : tsze R M) = (𝓝 0).prod (𝓝 m) 
goal: 𝓝 (TrivSqZeroExt.inr m) = (𝓝 0).prod (𝓝 m)

full_name: TrivSqZeroExt.continuous_fst
code: nonrec theorem continuous_fst : Continuous (fst : tsze R M → R) 
goal: Continuous TrivSqZeroExt.fst

full_name: TrivSqZeroExt.continuous_snd
code: nonrec theorem continuous_snd : Continuous (snd : tsze R M → M) 
goal: Continuous TrivSqZeroExt.snd

full_name: TrivSqZeroExt.continuous_inl
code: theorem continuous_inl [Zero M] : Continuous (inl : R → tsze R M) 
goal: Continuous TrivSqZeroExt.inl

full_name: TrivSqZeroExt.continuous_inr
code: theorem continuous_inr [Zero R] : Continuous (inr : M → tsze R M) 
goal: Continuous TrivSqZeroExt.inr

full_name: TrivSqZeroExt.embedding_inl
code: theorem embedding_inl [Zero M] : Embedding (inl : R → tsze R M) 
goal: Embedding TrivSqZeroExt.inl

full_name: TrivSqZeroExt.embedding_inr
code: theorem embedding_inr [Zero R] : Embedding (inr : M → tsze R M) 
goal: Embedding TrivSqZeroExt.inr

full_name: TrivSqZeroExt.fstCLM_apply
code: /-- `TrivSqZeroExt.fst` as a continuous linear map. -/
@[simps]
def fstCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] R 
goal: (TrivSqZeroExt.fstCLM R M) x = x.fst

full_name: TrivSqZeroExt.fstCLM_toFun
code: /-- `TrivSqZeroExt.fst` as a continuous linear map. -/
@[simps]
def fstCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] R 
goal: (TrivSqZeroExt.fstCLM R M) x = x.fst

full_name: TrivSqZeroExt.fstCLM
code: /-- `TrivSqZeroExt.fst` as a continuous linear map. -/
@[simps]
def fstCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] R 
goal: tsze R M →L[R] R

full_name: TrivSqZeroExt.sndCLM
code: /-- `TrivSqZeroExt.snd` as a continuous linear map. -/
@[simps]
def sndCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] M 
goal: tsze R M →L[R] M

full_name: TrivSqZeroExt.sndCLM_toFun
code: /-- `TrivSqZeroExt.snd` as a continuous linear map. -/
@[simps]
def sndCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] M 
goal: (TrivSqZeroExt.sndCLM R M) x = x.snd

full_name: TrivSqZeroExt.sndCLM_apply
code: /-- `TrivSqZeroExt.snd` as a continuous linear map. -/
@[simps]
def sndCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] M 
goal: (TrivSqZeroExt.sndCLM R M) x = x.snd

full_name: TrivSqZeroExt.inlCLM_toFun
code: /-- `TrivSqZeroExt.inl` as a continuous linear map. -/
@[simps]
def inlCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : R →L[R] tsze R M 
goal: (TrivSqZeroExt.inlCLM R M) r = TrivSqZeroExt.inl r

full_name: TrivSqZeroExt.inlCLM
code: /-- `TrivSqZeroExt.inl` as a continuous linear map. -/
@[simps]
def inlCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : R →L[R] tsze R M 
goal: R →L[R] tsze R M

full_name: TrivSqZeroExt.inlCLM_apply
code: /-- `TrivSqZeroExt.inl` as a continuous linear map. -/
@[simps]
def inlCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : R →L[R] tsze R M 
goal: (TrivSqZeroExt.inlCLM R M) r = TrivSqZeroExt.inl r

full_name: TrivSqZeroExt.inrCLM_apply
code: /-- `TrivSqZeroExt.inr` as a continuous linear map. -/
@[simps]
def inrCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : M →L[R] tsze R M 
goal: (TrivSqZeroExt.inrCLM R M) m = TrivSqZeroExt.inr m

full_name: TrivSqZeroExt.inrCLM_toFun
code: /-- `TrivSqZeroExt.inr` as a continuous linear map. -/
@[simps]
def inrCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : M →L[R] tsze R M 
goal: (TrivSqZeroExt.inrCLM R M) m = TrivSqZeroExt.inr m

full_name: TrivSqZeroExt.inrCLM
code: /-- `TrivSqZeroExt.inr` as a continuous linear map. -/
@[simps]
def inrCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : M →L[R] tsze R M 
goal: M →L[R] tsze R M

full_name: TrivSqZeroExt.instContinuousAdd
code: instance [Add R] [Add M] [ContinuousAdd R] [ContinuousAdd M] : ContinuousAdd (tsze R M) 
goal: ContinuousAdd (tsze R M)

full_name: TrivSqZeroExt.instContinuousMulOfContinuousSMulMulOppositeOfContinuousAdd
code: instance [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] [ContinuousMul R] [ContinuousSMul R M]
    [ContinuousSMul Rᵐᵒᵖ M] [ContinuousAdd M] : ContinuousMul (tsze R M) 
goal: ContinuousMul (tsze R M)

full_name: TrivSqZeroExt.instContinuousNeg
code: instance [Neg R] [Neg M] [ContinuousNeg R] [ContinuousNeg M] : ContinuousNeg (tsze R M) 
goal: ContinuousNeg (tsze R M)

full_name: TrivSqZeroExt.topologicalSemiring
code: /-- This is not an instance due to complaints by the `fails_quickly` linter. At any rate, we only
really care about the `TopologicalRing` instance below. -/
theorem topologicalSemiring [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M]
    [TopologicalSemiring R] [ContinuousAdd M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] :
    TopologicalSemiring (tsze R M) 
goal: TopologicalSemiring (tsze R M)

full_name: TrivSqZeroExt.instTopologicalRingOfTopologicalAddGroupOfContinuousSMulMulOpposite
code: instance [Ring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [TopologicalRing R]
    [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] :
    TopologicalRing (tsze R M) 
goal: TopologicalRing (tsze R M)

full_name: TrivSqZeroExt.instContinuousConstSMul
code: instance [SMul S R] [SMul S M] [ContinuousConstSMul S R] [ContinuousConstSMul S M] :
    ContinuousConstSMul S (tsze R M) 
goal: ContinuousConstSMul S (tsze R M)

full_name: TrivSqZeroExt.instContinuousSMul
code: instance [TopologicalSpace S] [SMul S R] [SMul S M] [ContinuousSMul S R] [ContinuousSMul S M] :
    ContinuousSMul S (tsze R M) 
goal: ContinuousSMul S (tsze R M)

full_name: TrivSqZeroExt.hasSum_inl
code: theorem hasSum_inl [AddCommMonoid R] [AddCommMonoid M] {f : α → R} {a : R} (h : HasSum f a) :
    HasSum (fun x ↦ inl (f x)) (inl a : tsze R M) 
goal: HasSum (fun x => TrivSqZeroExt.inl (f x)) (TrivSqZeroExt.inl a)

full_name: TrivSqZeroExt.hasSum_inr
code: theorem hasSum_inr [AddCommMonoid R] [AddCommMonoid M] {f : α → M} {a : M} (h : HasSum f a) :
    HasSum (fun x ↦ inr (f x)) (inr a : tsze R M) 
goal: HasSum (fun x => TrivSqZeroExt.inr (f x)) (TrivSqZeroExt.inr a)

full_name: TrivSqZeroExt.hasSum_fst
code: theorem hasSum_fst [AddCommMonoid R] [AddCommMonoid M] {f : α → tsze R M} {a : tsze R M}
    (h : HasSum f a) : HasSum (fun x ↦ fst (f x)) (fst a) 
goal: HasSum (fun x => (f x).fst) a.fst

full_name: TrivSqZeroExt.hasSum_snd
code: theorem hasSum_snd [AddCommMonoid R] [AddCommMonoid M] {f : α → tsze R M} {a : tsze R M}
    (h : HasSum f a) : HasSum (fun x ↦ snd (f x)) (snd a) 
goal: HasSum (fun x => (f x).snd) a.snd

full_name: TrivSqZeroExt.instUniformSpace
code: instance instUniformSpace : UniformSpace (tsze R M) 
goal: UniformSpace (tsze R M)

full_name: TrivSqZeroExt.instCompleteSpace
code: instance [CompleteSpace R] [CompleteSpace M] : CompleteSpace (tsze R M) 
goal: CompleteSpace (tsze R M)

full_name: TrivSqZeroExt.instUniformAddGroup
code: instance [AddGroup R] [AddGroup M] [UniformAddGroup R] [UniformAddGroup M] :
    UniformAddGroup (tsze R M) 
goal: UniformAddGroup (tsze R M)

full_name: TrivSqZeroExt.uniformity_def
code: theorem uniformity_def :
    𝓤 (tsze R M) =
      ((𝓤 R).comap fun p => (p.1.fst, p.2.fst)) ⊓ ((𝓤 M).comap fun p => (p.1.snd, p.2.snd)) 
goal: uniformity (tsze R M) =
  Filter.comap (fun p => (p.1.fst, p.2.fst)) (uniformity R) ⊓ Filter.comap (fun p => (p.1.snd, p.2.snd)) (uniformity M)

full_name: TrivSqZeroExt.uniformContinuous_fst
code: nonrec theorem uniformContinuous_fst : UniformContinuous (fst : tsze R M → R) 
goal: UniformContinuous TrivSqZeroExt.fst

full_name: TrivSqZeroExt.uniformContinuous_snd
code: nonrec theorem uniformContinuous_snd : UniformContinuous (snd : tsze R M → M) 
goal: UniformContinuous TrivSqZeroExt.snd

full_name: TrivSqZeroExt.uniformContinuous_inl
code: theorem uniformContinuous_inl [Zero M] : UniformContinuous (inl : R → tsze R M) 
goal: UniformContinuous TrivSqZeroExt.inl

full_name: TrivSqZeroExt.uniformContinuous_inr
code: theorem uniformContinuous_inr [Zero R] : UniformContinuous (inr : M → tsze R M) 
goal: UniformContinuous TrivSqZeroExt.inr