ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.proj_apply
@[simp]
theorem proj_apply (i : ι) (b : ∀ i, φ i) : (proj i : (∀ i, φ i) →L[R] φ i) b = b i 
Arguments: {R : Type u_1} [Semiring R] {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)] (i : ι) (b : (i : ι) → φ i)
Goal: (ContinuousLinearMap.proj i) b = b i
-- ContinuousLinearMap.proj_pi
theorem proj_pi (f : ∀ i, M₂ →L[R] φ i) (i : ι) : (proj i).comp (pi f) = f i 
Arguments: {R : Type u_1} [Semiring R] {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) → M₂ →L[R] φ i) (i : ι)
Goal: (ContinuousLinearMap.proj i).comp (ContinuousLinearMap.pi f) = f i
-- ContinuousLinearMap.iInf_ker_proj
theorem iInf_ker_proj : (⨅ i, ker (proj i : (∀ i, φ i) →L[R] φ i) : Submodule R (∀ i, φ i)) = ⊥ 
Arguments: {R : Type u_1} [Semiring R] {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: ⨅ i, LinearMap.ker (ContinuousLinearMap.proj i) = ⊥
-- Pi.compRightL_apply
@[simp] lemma _root_.Pi.compRightL_apply {α : Type*} (f : α → ι) (v : (i : ι) → φ i) (i : α) :
    Pi.compRightL R φ f v i = v (f i) 
Arguments: (R : Type u_1) [Semiring R] {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)] {α : Type u_6} (f : α → ι) (v : (i : ι) → φ i) (i : α)
Goal: (Pi.compRightL R φ f) v i = v (f i)
-- ContinuousLinearMap.map_neg
protected theorem map_neg (f : M →SL[σ₁₂] M₂) (x : M) : f (-x) = -f x 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →SL[σ₁₂] M₂) (x : M)
Goal: f (-x) = -f x
-- ContinuousLinearMap.map_sub
protected theorem map_sub (f : M →SL[σ₁₂] M₂) (x y : M) : f (x - y) = f x - f y 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →SL[σ₁₂] M₂) (x y : M)
Goal: f (x - y) = f x - f y
-- ContinuousLinearMap.sub_apply'
@[simp]
theorem sub_apply' (f g : M →SL[σ₁₂] M₂) (x : M) : ((f : M →ₛₗ[σ₁₂] M₂) - g) x = f x - g x 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f g : M →SL[σ₁₂] M₂) (x : M)
Goal: (↑f - ↑g) x = f x - g x
-- ContinuousLinearMap.range_prod_eq
theorem range_prod_eq {f : M →L[R] M₂} {g : M →L[R] M₃} (h : ker f ⊔ ker g = ⊤) :
    range (f.prod g) = (range f).prod (range g) 
Arguments: {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R M₂] [Module R M₃] {f : M →L[R] M₂} {g : M →L[R] M₃} (h : LinearMap.ker f ⊔ LinearMap.ker g = ⊤)
Goal: LinearMap.range (f.prod g) = (LinearMap.range f).prod (LinearMap.range g)
-- ContinuousLinearMap.ker_prod_ker_le_ker_coprod
theorem ker_prod_ker_le_ker_coprod [ContinuousAdd M₃] (f : M →L[R] M₃) (g : M₂ →L[R] M₃) :
    (LinearMap.ker f).prod (LinearMap.ker g) ≤ LinearMap.ker (f.coprod g) 
Arguments: {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R M₂] [Module R M₃] [ContinuousAdd M₃] (f : M →L[R] M₃) (g : M₂ →L[R] M₃)
Goal: (LinearMap.ker f).prod (LinearMap.ker g) ≤ LinearMap.ker (f.coprod g)
-- ContinuousLinearMap.ker_coprod_of_disjoint_range
theorem ker_coprod_of_disjoint_range [ContinuousAdd M₃] (f : M →L[R] M₃) (g : M₂ →L[R] M₃)
    (hd : Disjoint (range f) (range g)) :
    LinearMap.ker (f.coprod g) = (LinearMap.ker f).prod (LinearMap.ker g) 
Arguments: {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R M₂] [Module R M₃] [ContinuousAdd M₃] (f : M →L[R] M₃) (g : M₂ →L[R] M₃) (hd : Disjoint (LinearMap.range f) (LinearMap.range g))
Goal: LinearMap.ker (f.coprod g) = (LinearMap.ker f).prod (LinearMap.ker g)
-- ContinuousLinearMap.neg_apply
@[simp]
theorem neg_apply (f : M →SL[σ₁₂] M₂) (x : M) : (-f) x = -f x 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) (x : M)
Goal: (-f) x = -f x
-- ContinuousLinearMap.coe_neg
@[simp, norm_cast]
theorem coe_neg (f : M →SL[σ₁₂] M₂) : (↑(-f) : M →ₛₗ[σ₁₂] M₂) = -f 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂)
Goal: ↑(-f) = -↑f
-- ContinuousLinearMap.coe_neg'
@[norm_cast]
theorem coe_neg' (f : M →SL[σ₁₂] M₂) : ⇑(-f) = -f 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂)
Goal: ⇑(-f) = -⇑f
-- ContinuousLinearMap.sub_apply
theorem sub_apply (f g : M →SL[σ₁₂] M₂) (x : M) : (f - g) x = f x - g x 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f g : M →SL[σ₁₂] M₂) (x : M)
Goal: (f - g) x = f x - g x
-- ContinuousLinearMap.coe_sub
@[simp, norm_cast]
theorem coe_sub (f g : M →SL[σ₁₂] M₂) : (↑(f - g) : M →ₛₗ[σ₁₂] M₂) = f - g 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f g : M →SL[σ₁₂] M₂)
Goal: ↑(f - g) = ↑f - ↑g
-- ContinuousLinearMap.coe_sub'
@[simp, norm_cast]
theorem coe_sub' (f g : M →SL[σ₁₂] M₂) : ⇑(f - g) = f - g 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f g : M →SL[σ₁₂] M₂)
Goal: ⇑(f - g) = ⇑f - ⇑g
-- ContinuousLinearMap.comp_neg
@[simp]
theorem comp_neg [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃]
    (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : g.comp (-f) = -g.comp f 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂)
Goal: g.comp (-f) = -g.comp f
-- ContinuousLinearMap.neg_comp
@[simp]
theorem neg_comp [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃)
    (f : M →SL[σ₁₂] M₂) : (-g).comp f = -g.comp f 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂)
Goal: (-g).comp f = -g.comp f
-- ContinuousLinearMap.comp_sub
@[simp]
theorem comp_sub [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃]
    (g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M →SL[σ₁₂] M₂) : g.comp (f₁ - f₂) = g.comp f₁ - g.comp f₂ 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M →SL[σ₁₂] M₂)
Goal: g.comp (f₁ - f₂) = g.comp f₁ - g.comp f₂
-- ContinuousLinearMap.sub_comp
@[simp]
theorem sub_comp [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g₁ g₂ : M₂ →SL[σ₂₃] M₃)
    (f : M →SL[σ₁₂] M₂) : (g₁ - g₂).comp f = g₁.comp f - g₂.comp f 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂)
Goal: (g₁ - g₂).comp f = g₁.comp f - g₂.comp f
-- ContinuousLinearMap.intCast_apply
@[simp]
theorem intCast_apply [TopologicalAddGroup M] (z : ℤ) (m : M) : (↑z : M →L[R] M) m = z • m 
Arguments: {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] [Module R M] [TopologicalAddGroup M] (z : ℤ) (m : M)
Goal: ↑z m = z • m
-- ContinuousLinearMap.smulRight_one_pow
theorem smulRight_one_pow [TopologicalSpace R] [TopologicalRing R] (c : R) (n : ℕ) :
    smulRight (1 : R →L[R] R) c ^ n = smulRight (1 : R →L[R] R) (c ^ n) 
Arguments: {R : Type u_1} [Ring R] [TopologicalSpace R] [TopologicalRing R] (c : R) (n : ℕ)
Goal: ContinuousLinearMap.smulRight 1 c ^ n = ContinuousLinearMap.smulRight 1 (c ^ n)
-- ContinuousLinearMap.coe_projKerOfRightInverse_apply
@[simp]
theorem coe_projKerOfRightInverse_apply [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂)
    (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (x : M) :
    (f₁.projKerOfRightInverse f₂ h x : M) = x - f₂ (f₁ x) 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (x : M)
Goal: ↑((f₁.projKerOfRightInverse f₂ h) x) = x - f₂ (f₁ x)
-- ContinuousLinearMap.projKerOfRightInverse_apply_idem
@[simp]
theorem projKerOfRightInverse_apply_idem [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂)
    (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (x : LinearMap.ker f₁) :
    f₁.projKerOfRightInverse f₂ h x = x 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (x : ↥(LinearMap.ker f₁))
Goal: (f₁.projKerOfRightInverse f₂ h) ↑x = x
-- ContinuousLinearMap.projKerOfRightInverse_comp_inv
@[simp]
theorem projKerOfRightInverse_comp_inv [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂)
    (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (y : M₂) :
    f₁.projKerOfRightInverse f₂ h (f₂ y) = 0 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (y : M₂)
Goal: (f₁.projKerOfRightInverse f₂ h) (f₂ y) = 0
-- ContinuousLinearMap.isOpenMap_of_ne_zero
protected theorem isOpenMap_of_ne_zero [TopologicalSpace R] [DivisionRing R] [ContinuousSub R]
    [AddCommGroup M] [TopologicalSpace M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M]
    (f : M →L[R] R) (hf : f ≠ 0) : IsOpenMap f 
Arguments: {R : Type u_1} {M : Type u_2} [TopologicalSpace R] [DivisionRing R] [ContinuousSub R] [AddCommGroup M] [TopologicalSpace M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] (f : M →L[R] R) (hf : f ≠ 0)
Goal: IsOpenMap ⇑f
-- ContinuousLinearMap.smul_comp
@[simp]
theorem smul_comp (c : S₃) (h : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
    (c • h).comp f = c • h.comp f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {S₃ : Type u_5} [Semiring R] [Semiring R₂] [Semiring R₃] [Monoid S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_7} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] [DistribMulAction S₃ M₃] [SMulCommClass R₃ S₃ M₃] [ContinuousConstSMul S₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (c : S₃) (h : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂)
Goal: (c • h).comp f = c • h.comp f
-- ContinuousLinearMap.comp_smul
@[simp]
theorem comp_smul [LinearMap.CompatibleSMul N₂ N₃ S R] (hₗ : N₂ →L[R] N₃) (c : S)
    (fₗ : M →L[R] N₂) : hₗ.comp (c • fₗ) = c • hₗ.comp fₗ 
Arguments: {R : Type u_1} {S : Type u_4} [Semiring R] [Monoid S] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] [DistribMulAction S N₃] [SMulCommClass R S N₃] [ContinuousConstSMul S N₃] [DistribMulAction S N₂] [ContinuousConstSMul S N₂] [SMulCommClass R S N₂] [LinearMap.CompatibleSMul N₂ N₃ S R] (hₗ : N₂ →L[R] N₃) (c : S) (fₗ : M →L[R] N₂)
Goal: hₗ.comp (c • fₗ) = c • hₗ.comp fₗ
-- ContinuousLinearMap.comp_smulₛₗ
@[simp]
theorem comp_smulₛₗ [SMulCommClass R₂ R₂ M₂] [SMulCommClass R₃ R₃ M₃] [ContinuousConstSMul R₂ M₂]
    [ContinuousConstSMul R₃ M₃] (h : M₂ →SL[σ₂₃] M₃) (c : R₂) (f : M →SL[σ₁₂] M₂) :
    h.comp (c • f) = σ₂₃ c • h.comp f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R] [Semiring R₂] [Semiring R₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_7} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [SMulCommClass R₂ R₂ M₂] [SMulCommClass R₃ R₃ M₃] [ContinuousConstSMul R₂ M₂] [ContinuousConstSMul R₃ M₃] (h : M₂ →SL[σ₂₃] M₃) (c : R₂) (f : M →SL[σ₁₂] M₂)
Goal: h.comp (c • f) = σ₂₃ c • h.comp f
-- ContinuousLinearMap.prodEquiv_apply
@[simps apply]
def prodEquiv : (M →L[R] N₂) × (M →L[R] N₃) ≃ (M →L[R] N₂ × N₃) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] (f : (M →L[R] N₂) × (M →L[R] N₃))
Goal: ContinuousLinearMap.prodEquiv f = f.1.prod f.2
-- ContinuousLinearMap.prod_ext_iff
theorem prod_ext_iff {f g : M × N₂ →L[R] N₃} :
    f = g ↔ f.comp (inl _ _ _) = g.comp (inl _ _ _) ∧ f.comp (inr _ _ _) = g.comp (inr _ _ _) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] {f g : M × N₂ →L[R] N₃}
Goal: f = g ↔
  f.comp (ContinuousLinearMap.inl R M N₂) = g.comp (ContinuousLinearMap.inl R M N₂) ∧
    f.comp (ContinuousLinearMap.inr R M N₂) = g.comp (ContinuousLinearMap.inr R M N₂)
-- ContinuousLinearMap.prod_ext
@[ext]
theorem prod_ext {f g : M × N₂ →L[R] N₃} (hl : f.comp (inl _ _ _) = g.comp (inl _ _ _))
    (hr : f.comp (inr _ _ _) = g.comp (inr _ _ _)) : f = g 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] {f g : M × N₂ →L[R] N₃} (hl : f.comp (ContinuousLinearMap.inl R M N₂) = g.comp (ContinuousLinearMap.inl R M N₂)) (hr : f.comp (ContinuousLinearMap.inr R M N₂) = g.comp (ContinuousLinearMap.inr R M N₂))
Goal: f = g
-- ContinuousLinearMap.prodₗ_apply
@[simps apply]
def prodₗ : ((M →L[R] N₂) × (M →L[R] N₃)) ≃ₗ[S] M →L[R] N₂ × N₃ 
Arguments: {R : Type u_1} (S : Type u_4) [Semiring R] [Semiring S] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] [Module S N₂] [ContinuousConstSMul S N₂] [SMulCommClass R S N₂] [Module S N₃] [SMulCommClass R S N₃] [ContinuousConstSMul S N₃] [ContinuousAdd N₂] [ContinuousAdd N₃]
Goal: ∀ (a : (M →L[R] N₂) × (M →L[R] N₃)), (ContinuousLinearMap.prodₗ S) a = ContinuousLinearMap.prodEquiv.toFun a
-- ContinuousLinearMap.coeLM_apply
@[simps]
def coeLM : (M →L[R] N₃) →ₗ[S] M →ₗ[R] N₃ 
Arguments: {R : Type u_1} (S : Type u_4) [Semiring R] [Semiring S] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] [Module S N₃] [SMulCommClass R S N₃] [ContinuousConstSMul S N₃] [ContinuousAdd N₃] (self : M →L[R] N₃)
Goal: (ContinuousLinearMap.coeLM S) self = ↑self
-- ContinuousLinearMap.coeLMₛₗ_apply
@[simps]
def coeLMₛₗ : (M →SL[σ₁₃] M₃) →ₗ[S₃] M →ₛₗ[σ₁₃] M₃ 
Arguments: {R : Type u_1} {R₃ : Type u_3} {S₃ : Type u_5} [Semiring R] [Semiring R₃] [Semiring S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] [Module S₃ M₃] [SMulCommClass R₃ S₃ M₃] [ContinuousConstSMul S₃ M₃] (σ₁₃ : R →+* R₃) [ContinuousAdd M₃] (self : M →SL[σ₁₃] M₃)
Goal: (ContinuousLinearMap.coeLMₛₗ σ₁₃) self = ↑self
-- ContinuousLinearMap.coe_smulRightₗ
@[simp]
theorem coe_smulRightₗ (c : M →L[R] S) : ⇑(smulRightₗ c : M₂ →ₗ[T] M →L[R] M₂) = c.smulRight 
Arguments: {R : Type u_1} {S : Type u_2} {T : Type u_3} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring S] [Semiring T] [Module R S] [AddCommMonoid M₂] [Module R M₂] [Module S M₂] [IsScalarTower R S M₂] [TopologicalSpace S] [TopologicalSpace M₂] [ContinuousSMul S M₂] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousAdd M₂] [Module T M₂] [ContinuousConstSMul T M₂] [SMulCommClass R T M₂] [SMulCommClass S T M₂] (c : M →L[R] S)
Goal: ⇑c.smulRightₗ = c.smulRight
-- ContinuousLinearMap.algebraMap_apply
@[simp] theorem algebraMap_apply (r : R) (m : M₂) : algebraMap R (M₂ →L[R] M₂) r m = r • m 
Arguments: {R : Type u_1} [CommRing R] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M₂] [TopologicalAddGroup M₂] [ContinuousConstSMul R M₂] (r : R) (m : M₂)
Goal: ((algebraMap R (M₂ →L[R] M₂)) r) m = r • m
-- ContinuousLinearMap.coe_restrictScalars
@[simp] -- @[norm_cast] -- Porting note: This theorem can't be a `norm_cast` theorem.
theorem coe_restrictScalars (f : M →L[A] M₂) :
    (f.restrictScalars R : M →ₗ[R] M₂) = (f : M →ₗ[A] M₂).restrictScalars R 
Arguments: {A : Type u_1} {M : Type u_2} {M₂ : Type u_3} [Ring A] [AddCommGroup M] [AddCommGroup M₂] [Module A M] [Module A M₂] [TopologicalSpace M] [TopologicalSpace M₂] {R : Type u_4} [Ring R] [Module R M] [Module R M₂] [LinearMap.CompatibleSMul M M₂ R A] (f : M →L[A] M₂)
Goal: ↑(ContinuousLinearMap.restrictScalars R f) = ↑R ↑f
-- ContinuousLinearMap.coe_restrictScalars'
@[simp]
theorem coe_restrictScalars' (f : M →L[A] M₂) : ⇑(f.restrictScalars R) = f 
Arguments: {A : Type u_1} {M : Type u_2} {M₂ : Type u_3} [Ring A] [AddCommGroup M] [AddCommGroup M₂] [Module A M] [Module A M₂] [TopologicalSpace M] [TopologicalSpace M₂] {R : Type u_4} [Ring R] [Module R M] [Module R M₂] [LinearMap.CompatibleSMul M M₂ R A] (f : M →L[A] M₂)
Goal: ⇑(ContinuousLinearMap.restrictScalars R f) = ⇑f
-- ContinuousLinearMap.restrictScalars_zero
@[simp]
theorem restrictScalars_zero : (0 : M →L[A] M₂).restrictScalars R = 0 
Arguments: {A : Type u_1} {M : Type u_2} {M₂ : Type u_3} [Ring A] [AddCommGroup M] [AddCommGroup M₂] [Module A M] [Module A M₂] [TopologicalSpace M] [TopologicalSpace M₂] {R : Type u_4} [Ring R] [Module R M] [Module R M₂] [LinearMap.CompatibleSMul M M₂ R A]
Goal: ContinuousLinearMap.restrictScalars R 0 = 0
-- ContinuousLinearMap.restrictScalars_add
@[simp]
theorem restrictScalars_add (f g : M →L[A] M₂) :
    (f + g).restrictScalars R = f.restrictScalars R + g.restrictScalars R 
Arguments: {A : Type u_1} {M : Type u_2} {M₂ : Type u_3} [Ring A] [AddCommGroup M] [AddCommGroup M₂] [Module A M] [Module A M₂] [TopologicalSpace M] [TopologicalSpace M₂] {R : Type u_4} [Ring R] [Module R M] [Module R M₂] [LinearMap.CompatibleSMul M M₂ R A] [TopologicalAddGroup M₂] (f g : M →L[A] M₂)
Goal: ContinuousLinearMap.restrictScalars R (f + g) =
  ContinuousLinearMap.restrictScalars R f + ContinuousLinearMap.restrictScalars R g
-- ContinuousLinearMap.restrictScalars_neg
@[simp]
theorem restrictScalars_neg (f : M →L[A] M₂) : (-f).restrictScalars R = -f.restrictScalars R 
Arguments: {A : Type u_1} {M : Type u_2} {M₂ : Type u_3} [Ring A] [AddCommGroup M] [AddCommGroup M₂] [Module A M] [Module A M₂] [TopologicalSpace M] [TopologicalSpace M₂] {R : Type u_4} [Ring R] [Module R M] [Module R M₂] [LinearMap.CompatibleSMul M M₂ R A] [TopologicalAddGroup M₂] (f : M →L[A] M₂)
Goal: ContinuousLinearMap.restrictScalars R (-f) = -ContinuousLinearMap.restrictScalars R f
-- ContinuousLinearMap.restrictScalars_smul
@[simp]
theorem restrictScalars_smul (c : S) (f : M →L[A] M₂) :
    (c • f).restrictScalars R = c • f.restrictScalars R 
Arguments: {A : Type u_1} {M : Type u_2} {M₂ : Type u_3} [Ring A] [AddCommGroup M] [AddCommGroup M₂] [Module A M] [Module A M₂] [TopologicalSpace M] [TopologicalSpace M₂] {R : Type u_4} [Ring R] [Module R M] [Module R M₂] [LinearMap.CompatibleSMul M M₂ R A] {S : Type u_5} [Ring S] [Module S M₂] [ContinuousConstSMul S M₂] [SMulCommClass A S M₂] [SMulCommClass R S M₂] (c : S) (f : M →L[A] M₂)
Goal: ContinuousLinearMap.restrictScalars R (c • f) = c • ContinuousLinearMap.restrictScalars R f
-- ContinuousLinearMap.coe_restrictScalarsₗ
@[simp]
theorem coe_restrictScalarsₗ : ⇑(restrictScalarsₗ A M M₂ R S) = restrictScalars R 
Arguments: {A : Type u_1} {M : Type u_2} {M₂ : Type u_3} [Ring A] [AddCommGroup M] [AddCommGroup M₂] [Module A M] [Module A M₂] [TopologicalSpace M] [TopologicalSpace M₂] {R : Type u_4} [Ring R] [Module R M] [Module R M₂] [LinearMap.CompatibleSMul M M₂ R A] {S : Type u_5} [Ring S] [Module S M₂] [ContinuousConstSMul S M₂] [SMulCommClass A S M₂] [SMulCommClass R S M₂] [TopologicalAddGroup M₂]
Goal: ⇑(ContinuousLinearMap.restrictScalarsₗ A M M₂ R S) = ContinuousLinearMap.restrictScalars R
-- ContinuousLinearEquiv.coe_apply
theorem coe_apply (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : (e : M₁ →SL[σ₁₂] M₂) b = e b 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁)
Goal: ↑e b = e b