ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.mul_def
theorem mul_def (f g : M₁ →L[R₁] M₁) : f * g = f.comp g 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f g : M₁ →L[R₁] M₁)
Goal: f * g = f.comp g
-- ContinuousLinearMap.coe_mul
@[simp]
theorem coe_mul (f g : M₁ →L[R₁] M₁) : ⇑(f * g) = f ∘ g 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f g : M₁ →L[R₁] M₁)
Goal: ⇑(f * g) = ⇑f ∘ ⇑g
-- ContinuousLinearMap.mul_apply
theorem mul_apply (f g : M₁ →L[R₁] M₁) (x : M₁) : (f * g) x = f (g x) 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f g : M₁ →L[R₁] M₁) (x : M₁)
Goal: (f * g) x = f (g x)
-- ContinuousLinearMap.coe_pow
theorem coe_pow (f : M₁ →L[R₁] M₁) (n : ℕ) : ⇑(f ^ n) = f^[n] 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f : M₁ →L[R₁] M₁) (n : ℕ)
Goal: ⇑(f ^ n) = (⇑f)^[n]
-- ContinuousLinearMap.toLinearMapRingHom_apply
@[simps]
def toLinearMapRingHom [ContinuousAdd M₁] : (M₁ →L[R₁] M₁) →+* M₁ →ₗ[R₁] M₁ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (self : M₁ →L[R₁] M₁)
Goal: ContinuousLinearMap.toLinearMapRingHom self = ↑self
-- ContinuousLinearMap.natCast_apply
@[simp]
theorem natCast_apply [ContinuousAdd M₁] (n : ℕ) (m : M₁) : (↑n : M₁ →L[R₁] M₁) m = n • m 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (n : ℕ) (m : M₁)
Goal: ↑n m = n • m
-- ContinuousLinearMap.ofNat_apply
@[simp]
theorem ofNat_apply [ContinuousAdd M₁] (n : ℕ) [n.AtLeastTwo] (m : M₁) :
    ((no_index (OfNat.ofNat n) : M₁ →L[R₁] M₁)) m = OfNat.ofNat n • m 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (n : ℕ) [n.AtLeastTwo] (m : M₁)
Goal: (OfNat.ofNat n) m = OfNat.ofNat n • m
-- ContinuousLinearMap.smul_def
@[simp]
protected theorem smul_def (f : M₁ →L[R₁] M₁) (a : M₁) : f • a = f a 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (f : M₁ →L[R₁] M₁) (a : M₁)
Goal: f • a = f a
-- ContinuousLinearMap.coe_prod
@[simp, norm_cast]
theorem coe_prod [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) :
    (f₁.prod f₂ : M₁ →ₗ[R₁] M₂ × M₃) = LinearMap.prod f₁ f₂ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃)
Goal: ↑(f₁.prod f₂) = (↑f₁).prod ↑f₂
-- ContinuousLinearMap.prod_apply
@[simp, norm_cast]
theorem prod_apply [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) (x : M₁) :
    f₁.prod f₂ x = (f₁ x, f₂ x) 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) (x : M₁)
Goal: (f₁.prod f₂) x = (f₁ x, f₂ x)
-- ContinuousLinearMap.inl_apply
@[simp]
theorem inl_apply [Module R₁ M₂] (x : M₁) : inl R₁ M₁ M₂ x = (x, 0) 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (x : M₁)
Goal: (ContinuousLinearMap.inl R₁ M₁ M₂) x = (x, 0)
-- ContinuousLinearMap.inr_apply
@[simp]
theorem inr_apply [Module R₁ M₂] (x : M₂) : inr R₁ M₁ M₂ x = (0, x) 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (x : M₂)
Goal: (ContinuousLinearMap.inr R₁ M₁ M₂) x = (0, x)
-- ContinuousLinearMap.coe_inl
@[simp, norm_cast]
theorem coe_inl [Module R₁ M₂] : (inl R₁ M₁ M₂ : M₁ →ₗ[R₁] M₁ × M₂) = LinearMap.inl R₁ M₁ M₂ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂]
Goal: ↑(ContinuousLinearMap.inl R₁ M₁ M₂) = LinearMap.inl R₁ M₁ M₂
-- ContinuousLinearMap.coe_inr
@[simp, norm_cast]
theorem coe_inr [Module R₁ M₂] : (inr R₁ M₁ M₂ : M₂ →ₗ[R₁] M₁ × M₂) = LinearMap.inr R₁ M₁ M₂ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂]
Goal: ↑(ContinuousLinearMap.inr R₁ M₁ M₂) = LinearMap.inr R₁ M₁ M₂
-- ContinuousLinearMap.isClosed_ker
theorem isClosed_ker [T1Space M₂] [FunLike F M₁ M₂] [ContinuousSemilinearMapClass F σ₁₂ M₁ M₂]
    (f : F) :
    IsClosed (ker f : Set M₁) 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {F : Type u_9} [T1Space M₂] [FunLike F M₁ M₂] [ContinuousSemilinearMapClass F σ₁₂ M₁ M₂] (f : F)
Goal: IsClosed ↑(LinearMap.ker f)
-- ContinuousLinearMap.isComplete_ker
theorem isComplete_ker {M' : Type*} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M']
    [Module R₁ M'] [T1Space M₂] [FunLike F M' M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂]
    (f : F) :
    IsComplete (ker f : Set M') 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {F : Type u_9} {M' : Type u_10} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M'] [Module R₁ M'] [T1Space M₂] [FunLike F M' M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂] (f : F)
Goal: IsComplete ↑(LinearMap.ker f)
-- ContinuousLinearMap.ker_prod
@[simp]
theorem ker_prod [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :
    ker (f.prod g) = ker f ⊓ ker g 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃)
Goal: LinearMap.ker (f.prod g) = LinearMap.ker f ⊓ LinearMap.ker g
-- ContinuousLinearMap.coe_codRestrict
@[norm_cast]
theorem coe_codRestrict (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) :
    (f.codRestrict p h : M₁ →ₛₗ[σ₁₂] p) = (f : M₁ →ₛₗ[σ₁₂] M₂).codRestrict p h 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), f x ∈ p)
Goal: ↑(f.codRestrict p h) = LinearMap.codRestrict p (↑f) h
-- ContinuousLinearMap.coe_codRestrict_apply
@[simp]
theorem coe_codRestrict_apply (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) (x) :
    (f.codRestrict p h x : M₂) = f x 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), f x ∈ p) (x : M₁)
Goal: ↑((f.codRestrict p h) x) = f x
-- ContinuousLinearMap.ker_codRestrict
@[simp]
theorem ker_codRestrict (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) :
    ker (f.codRestrict p h) = ker f 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), f x ∈ p)
Goal: LinearMap.ker (f.codRestrict p h) = LinearMap.ker f
-- ContinuousLinearMap.coe_rangeRestrict
@[simp]
theorem coe_rangeRestrict [RingHomSurjective σ₁₂] (f : M₁ →SL[σ₁₂] M₂) :
    (f.rangeRestrict : M₁ →ₛₗ[σ₁₂] LinearMap.range f) = (f : M₁ →ₛₗ[σ₁₂] M₂).rangeRestrict 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [RingHomSurjective σ₁₂] (f : M₁ →SL[σ₁₂] M₂)
Goal: ↑f.rangeRestrict = (↑f).rangeRestrict
-- Submodule.coe_subtypeL
@[simp, norm_cast]
theorem _root_.Submodule.coe_subtypeL (p : Submodule R₁ M₁) :
    (p.subtypeL : p →ₗ[R₁] M₁) = p.subtype 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁)
Goal: ↑p.subtypeL = p.subtype
-- Submodule.coe_subtypeL'
@[simp]
theorem _root_.Submodule.coe_subtypeL' (p : Submodule R₁ M₁) : ⇑p.subtypeL = p.subtype 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁)
Goal: ⇑p.subtypeL = ⇑p.subtype
-- Submodule.subtypeL_apply
@[simp] -- @[norm_cast] -- Porting note: A theorem with this can't have a rhs starting with `↑`.
theorem _root_.Submodule.subtypeL_apply (p : Submodule R₁ M₁) (x : p) : p.subtypeL x = x 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) (x : ↥p)
Goal: p.subtypeL x = ↑x
-- Submodule.range_subtypeL
@[simp]
theorem _root_.Submodule.range_subtypeL (p : Submodule R₁ M₁) : range p.subtypeL = p 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁)
Goal: LinearMap.range p.subtypeL = p
-- Submodule.ker_subtypeL
@[simp]
theorem _root_.Submodule.ker_subtypeL (p : Submodule R₁ M₁) : ker p.subtypeL = ⊥ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁)
Goal: LinearMap.ker p.subtypeL = ⊥
-- ContinuousLinearMap.coe_fst
@[simp, norm_cast]
theorem coe_fst [Module R₁ M₂] : ↑(fst R₁ M₁ M₂) = LinearMap.fst R₁ M₁ M₂ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂]
Goal: ↑(ContinuousLinearMap.fst R₁ M₁ M₂) = LinearMap.fst R₁ M₁ M₂
-- ContinuousLinearMap.coe_fst'
@[simp, norm_cast]
theorem coe_fst' [Module R₁ M₂] : ⇑(fst R₁ M₁ M₂) = Prod.fst 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂]
Goal: ⇑(ContinuousLinearMap.fst R₁ M₁ M₂) = Prod.fst
-- ContinuousLinearMap.coe_snd
@[simp, norm_cast]
theorem coe_snd [Module R₁ M₂] : ↑(snd R₁ M₁ M₂) = LinearMap.snd R₁ M₁ M₂ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂]
Goal: ↑(ContinuousLinearMap.snd R₁ M₁ M₂) = LinearMap.snd R₁ M₁ M₂
-- ContinuousLinearMap.coe_snd'
@[simp, norm_cast]
theorem coe_snd' [Module R₁ M₂] : ⇑(snd R₁ M₁ M₂) = Prod.snd 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂]
Goal: ⇑(ContinuousLinearMap.snd R₁ M₁ M₂) = Prod.snd
-- ContinuousLinearMap.fst_prod_snd
@[simp]
theorem fst_prod_snd [Module R₁ M₂] : (fst R₁ M₁ M₂).prod (snd R₁ M₁ M₂) = id R₁ (M₁ × M₂) 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂]
Goal: (ContinuousLinearMap.fst R₁ M₁ M₂).prod (ContinuousLinearMap.snd R₁ M₁ M₂) = ContinuousLinearMap.id R₁ (M₁ × M₂)
-- ContinuousLinearMap.fst_comp_prod
@[simp]
theorem fst_comp_prod [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :
    (fst R₁ M₂ M₃).comp (f.prod g) = f 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃)
Goal: (ContinuousLinearMap.fst R₁ M₂ M₃).comp (f.prod g) = f
-- ContinuousLinearMap.snd_comp_prod
@[simp]
theorem snd_comp_prod [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :
    (snd R₁ M₂ M₃).comp (f.prod g) = g 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃)
Goal: (ContinuousLinearMap.snd R₁ M₂ M₃).comp (f.prod g) = g
-- ContinuousLinearMap.coe_prodMap
@[simp, norm_cast]
theorem coe_prodMap [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂)
    (f₂ : M₃ →L[R₁] M₄) : ↑(f₁.prodMap f₂) = (f₁ : M₁ →ₗ[R₁] M₂).prodMap (f₂ : M₃ →ₗ[R₁] M₄) 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄)
Goal: ↑(f₁.prodMap f₂) = (↑f₁).prodMap ↑f₂
-- ContinuousLinearMap.coe_prodMap'
@[simp, norm_cast]
theorem coe_prodMap' [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂)
    (f₂ : M₃ →L[R₁] M₄) : ⇑(f₁.prodMap f₂) = Prod.map f₁ f₂ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄)
Goal: ⇑(f₁.prodMap f₂) = Prod.map ⇑f₁ ⇑f₂
-- ContinuousLinearMap.coe_coprod
@[norm_cast, simp]
theorem coe_coprod [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃)
    (f₂ : M₂ →L[R₁] M₃) : (f₁.coprod f₂ : M₁ × M₂ →ₗ[R₁] M₃) = LinearMap.coprod f₁ f₂ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃)
Goal: ↑(f₁.coprod f₂) = (↑f₁).coprod ↑f₂
-- ContinuousLinearMap.coprod_apply
@[simp]
theorem coprod_apply [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃)
    (f₂ : M₂ →L[R₁] M₃) (x) : f₁.coprod f₂ x = f₁ x.1 + f₂ x.2 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) (x : M₁ × M₂)
Goal: (f₁.coprod f₂) x = f₁ x.1 + f₂ x.2
-- ContinuousLinearMap.range_coprod
theorem range_coprod [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃)
    (f₂ : M₂ →L[R₁] M₃) : range (f₁.coprod f₂) = range f₁ ⊔ range f₂ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃)
Goal: LinearMap.range (f₁.coprod f₂) = LinearMap.range f₁ ⊔ LinearMap.range f₂
-- ContinuousLinearMap.comp_fst_add_comp_snd
theorem comp_fst_add_comp_snd [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f : M₁ →L[R₁] M₃)
    (g : M₂ →L[R₁] M₃) :
    f.comp (ContinuousLinearMap.fst R₁ M₁ M₂) + g.comp (ContinuousLinearMap.snd R₁ M₁ M₂) =
      f.coprod g 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f : M₁ →L[R₁] M₃) (g : M₂ →L[R₁] M₃)
Goal: f.comp (ContinuousLinearMap.fst R₁ M₁ M₂) + g.comp (ContinuousLinearMap.snd R₁ M₁ M₂) = f.coprod g
-- ContinuousLinearMap.coprod_inl_inr
theorem coprod_inl_inr [ContinuousAdd M₁] [ContinuousAdd M'₁] :
    (ContinuousLinearMap.inl R₁ M₁ M'₁).coprod (ContinuousLinearMap.inr R₁ M₁ M'₁) =
      ContinuousLinearMap.id R₁ (M₁ × M'₁) 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M'₁ : Type u_5} [TopologicalSpace M'₁] [AddCommMonoid M'₁] [Module R₁ M₁] [Module R₁ M'₁] [ContinuousAdd M₁] [ContinuousAdd M'₁]
Goal: (ContinuousLinearMap.inl R₁ M₁ M'₁).coprod (ContinuousLinearMap.inr R₁ M₁ M'₁) = ContinuousLinearMap.id R₁ (M₁ × M'₁)
-- ContinuousLinearMap.smulRight_apply
@[simp]
theorem smulRight_apply {c : M₁ →L[R] S} {f : M₂} {x : M₁} :
    (smulRight c f : M₁ → M₂) x = c x • f 
Arguments: {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {R : Type u_10} {S : Type u_11} [Semiring R] [Semiring S] [Module R M₁] [Module R M₂] [Module R S] [Module S M₂] [IsScalarTower R S M₂] [TopologicalSpace S] [ContinuousSMul S M₂] {c : M₁ →L[R] S} {f : M₂} {x : M₁}
Goal: (c.smulRight f) x = c x • f
-- ContinuousLinearMap.smulRight_one_one
@[simp]
theorem smulRight_one_one (c : R₁ →L[R₁] M₂) : smulRight (1 : R₁ →L[R₁] R₁) (c 1) = c 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₂] [TopologicalSpace R₁] [ContinuousSMul R₁ M₂] (c : R₁ →L[R₁] M₂)
Goal: ContinuousLinearMap.smulRight 1 (c 1) = c
-- ContinuousLinearMap.smulRight_one_eq_iff
@[simp]
theorem smulRight_one_eq_iff {f f' : M₂} :
    smulRight (1 : R₁ →L[R₁] R₁) f = smulRight (1 : R₁ →L[R₁] R₁) f' ↔ f = f' 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₂] [TopologicalSpace R₁] [ContinuousSMul R₁ M₂] {f f' : M₂}
Goal: ContinuousLinearMap.smulRight 1 f = ContinuousLinearMap.smulRight 1 f' ↔ f = f'
-- ContinuousLinearMap.smulRight_comp
theorem smulRight_comp [ContinuousMul R₁] {x : M₂} {c : R₁} :
    (smulRight (1 : R₁ →L[R₁] R₁) x).comp (smulRight (1 : R₁ →L[R₁] R₁) c) =
      smulRight (1 : R₁ →L[R₁] R₁) (c • x) 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₂] [TopologicalSpace R₁] [ContinuousSMul R₁ M₂] [ContinuousMul R₁] {x : M₂} {c : R₁}
Goal: (ContinuousLinearMap.smulRight 1 x).comp (ContinuousLinearMap.smulRight 1 c) = ContinuousLinearMap.smulRight 1 (c • x)
-- ContinuousLinearMap.toSpanSingleton_apply
theorem toSpanSingleton_apply (x : M₁) (r : R₁) : toSpanSingleton R₁ x r = r • x 
Arguments: (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] (x : M₁) (r : R₁)
Goal: (ContinuousLinearMap.toSpanSingleton R₁ x) r = r • x
-- ContinuousLinearMap.toSpanSingleton_add
theorem toSpanSingleton_add [ContinuousAdd M₁] (x y : M₁) :
    toSpanSingleton R₁ (x + y) = toSpanSingleton R₁ x + toSpanSingleton R₁ y 
Arguments: (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] (x y : M₁)
Goal: ContinuousLinearMap.toSpanSingleton R₁ (x + y) =
  ContinuousLinearMap.toSpanSingleton R₁ x + ContinuousLinearMap.toSpanSingleton R₁ y
-- ContinuousLinearMap.toSpanSingleton_smul'
theorem toSpanSingleton_smul' {α} [Monoid α] [DistribMulAction α M₁] [ContinuousConstSMul α M₁]
    [SMulCommClass R₁ α M₁] (c : α) (x : M₁) :
    toSpanSingleton R₁ (c • x) = c • toSpanSingleton R₁ x 
Arguments: (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] {α : Type u_10} [Monoid α] [DistribMulAction α M₁] [ContinuousConstSMul α M₁] [SMulCommClass R₁ α M₁] (c : α) (x : M₁)
Goal: ContinuousLinearMap.toSpanSingleton R₁ (c • x) = c • ContinuousLinearMap.toSpanSingleton R₁ x
-- ContinuousLinearMap.toSpanSingleton_smul
theorem toSpanSingleton_smul (R) {M₁} [CommSemiring R] [AddCommMonoid M₁] [Module R M₁]
    [TopologicalSpace R] [TopologicalSpace M₁] [ContinuousSMul R M₁] (c : R) (x : M₁) :
    toSpanSingleton R (c • x) = c • toSpanSingleton R x 
Arguments: (R : Type u_10) {M₁ : Type u_11} [CommSemiring R] [AddCommMonoid M₁] [Module R M₁] [TopologicalSpace R] [TopologicalSpace M₁] [ContinuousSMul R M₁] (c : R) (x : M₁)
Goal: ContinuousLinearMap.toSpanSingleton R (c • x) = c • ContinuousLinearMap.toSpanSingleton R x
-- ContinuousLinearMap.coe_pi'
@[simp]
theorem coe_pi' (f : ∀ i, M →L[R] φ i) : ⇑(pi f) = fun c i => f i c 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i)
Goal: ⇑(ContinuousLinearMap.pi f) = fun c i => (f i) c
-- ContinuousLinearMap.coe_pi
@[simp]
theorem coe_pi (f : ∀ i, M →L[R] φ i) : (pi f : M →ₗ[R] ∀ i, φ i) = LinearMap.pi fun i => f i 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i)
Goal: ↑(ContinuousLinearMap.pi f) = LinearMap.pi fun i => ↑(f i)
-- ContinuousLinearMap.pi_apply
theorem pi_apply (f : ∀ i, M →L[R] φ i) (c : M) (i : ι) : pi f c i = f i c 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) (c : M) (i : ι)
Goal: (ContinuousLinearMap.pi f) c i = (f i) c
-- ContinuousLinearMap.pi_eq_zero
theorem pi_eq_zero (f : ∀ i, M →L[R] φ i) : pi f = 0 ↔ ∀ i, f i = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i)
Goal: ContinuousLinearMap.pi f = 0 ↔ ∀ (i : ι), f i = 0
-- ContinuousLinearMap.pi_zero
theorem pi_zero : pi (fun _ => 0 : ∀ i, M →L[R] φ i) = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)]
Goal: (ContinuousLinearMap.pi fun x => 0) = 0
-- ContinuousLinearMap.pi_comp
theorem pi_comp (f : ∀ i, M →L[R] φ i) (g : M₂ →L[R] M) :
    (pi f).comp g = pi fun i => (f i).comp g 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M₂] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) (g : M₂ →L[R] M)
Goal: (ContinuousLinearMap.pi f).comp g = ContinuousLinearMap.pi fun i => (f i).comp g