ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.mulIndicator_symmDiff
@[to_additive]
theorem mulIndicator_symmDiff (s t : Set α) (f : α → M) :
    mulIndicator (s ∆ t) f = mulIndicator (s \ t) f * mulIndicator (t \ s) f 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] (s t : Set α) (f : α → M)
Goal: (symmDiff s t).mulIndicator f = (s \ t).mulIndicator f * (t \ s).mulIndicator f
-- Set.indicator_symmDiff
@[to_additive]
theorem mulIndicator_symmDiff (s t : Set α) (f : α → M) :
    mulIndicator (s ∆ t) f = mulIndicator (s \ t) f * mulIndicator (t \ s) f 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] (s t : Set α) (f : α → M)
Goal: (symmDiff s t).indicator f = (s \ t).indicator f + (t \ s).indicator f
-- Set.indicator_add
@[to_additive]
theorem mulIndicator_mul (s : Set α) (f g : α → M) :
    (mulIndicator s fun a => f a * g a) = fun a => mulIndicator s f a * mulIndicator s g a 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] (s : Set α) (f g : α → M)
Goal: (s.indicator fun a => f a + g a) = fun a => s.indicator f a + s.indicator g a
-- Set.mulIndicator_mul
@[to_additive]
theorem mulIndicator_mul (s : Set α) (f g : α → M) :
    (mulIndicator s fun a => f a * g a) = fun a => mulIndicator s f a * mulIndicator s g a 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] (s : Set α) (f g : α → M)
Goal: (s.mulIndicator fun a => f a * g a) = fun a => s.mulIndicator f a * s.mulIndicator g a
-- Set.mulIndicator_mul'
@[to_additive]
theorem mulIndicator_mul' (s : Set α) (f g : α → M) :
    mulIndicator s (f * g) = mulIndicator s f * mulIndicator s g 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] (s : Set α) (f g : α → M)
Goal: s.mulIndicator (f * g) = s.mulIndicator f * s.mulIndicator g
-- Set.indicator_add'
@[to_additive]
theorem mulIndicator_mul' (s : Set α) (f g : α → M) :
    mulIndicator s (f * g) = mulIndicator s f * mulIndicator s g 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] (s : Set α) (f g : α → M)
Goal: s.indicator (f + g) = s.indicator f + s.indicator g
-- Set.mulIndicator_compl_mul_self_apply
@[to_additive (attr := simp)]
theorem mulIndicator_compl_mul_self_apply (s : Set α) (f : α → M) (a : α) :
    mulIndicator sᶜ f a * mulIndicator s f a = f a 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] (s : Set α) (f : α → M) (a : α)
Goal: sᶜ.mulIndicator f a * s.mulIndicator f a = f a
-- Set.indicator_compl_add_self_apply
@[to_additive (attr := simp)]
theorem mulIndicator_compl_mul_self_apply (s : Set α) (f : α → M) (a : α) :
    mulIndicator sᶜ f a * mulIndicator s f a = f a 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] (s : Set α) (f : α → M) (a : α)
Goal: sᶜ.indicator f a + s.indicator f a = f a
-- Set.indicator_compl_add_self
@[to_additive (attr := simp)]
theorem mulIndicator_compl_mul_self (s : Set α) (f : α → M) :
    mulIndicator sᶜ f * mulIndicator s f = f 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] (s : Set α) (f : α → M)
Goal: sᶜ.indicator f + s.indicator f = f
-- Set.mulIndicator_compl_mul_self
@[to_additive (attr := simp)]
theorem mulIndicator_compl_mul_self (s : Set α) (f : α → M) :
    mulIndicator sᶜ f * mulIndicator s f = f 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] (s : Set α) (f : α → M)
Goal: sᶜ.mulIndicator f * s.mulIndicator f = f
-- Set.mulIndicator_self_mul_compl_apply
@[to_additive (attr := simp)]
theorem mulIndicator_self_mul_compl_apply (s : Set α) (f : α → M) (a : α) :
    mulIndicator s f a * mulIndicator sᶜ f a = f a 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] (s : Set α) (f : α → M) (a : α)
Goal: s.mulIndicator f a * sᶜ.mulIndicator f a = f a
-- Set.indicator_self_add_compl_apply
@[to_additive (attr := simp)]
theorem mulIndicator_self_mul_compl_apply (s : Set α) (f : α → M) (a : α) :
    mulIndicator s f a * mulIndicator sᶜ f a = f a 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] (s : Set α) (f : α → M) (a : α)
Goal: s.indicator f a + sᶜ.indicator f a = f a
-- Set.indicator_self_add_compl
@[to_additive (attr := simp)]
theorem mulIndicator_self_mul_compl (s : Set α) (f : α → M) :
    mulIndicator s f * mulIndicator sᶜ f = f 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] (s : Set α) (f : α → M)
Goal: s.indicator f + sᶜ.indicator f = f
-- Set.mulIndicator_self_mul_compl
@[to_additive (attr := simp)]
theorem mulIndicator_self_mul_compl (s : Set α) (f : α → M) :
    mulIndicator s f * mulIndicator sᶜ f = f 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] (s : Set α) (f : α → M)
Goal: s.mulIndicator f * sᶜ.mulIndicator f = f
-- Set.mulIndicator_mul_eq_left
@[to_additive]
theorem mulIndicator_mul_eq_left {f g : α → M} (h : Disjoint (mulSupport f) (mulSupport g)) :
    (mulSupport f).mulIndicator (f * g) = f 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] {f g : α → M} (h : Disjoint (Function.mulSupport f) (Function.mulSupport g))
Goal: (Function.mulSupport f).mulIndicator (f * g) = f
-- Set.indicator_add_eq_left
@[to_additive]
theorem mulIndicator_mul_eq_left {f g : α → M} (h : Disjoint (mulSupport f) (mulSupport g)) :
    (mulSupport f).mulIndicator (f * g) = f 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] {f g : α → M} (h : Disjoint (Function.support f) (Function.support g))
Goal: (Function.support f).indicator (f + g) = f
-- Set.mulIndicator_mul_eq_right
@[to_additive]
theorem mulIndicator_mul_eq_right {f g : α → M} (h : Disjoint (mulSupport f) (mulSupport g)) :
    (mulSupport g).mulIndicator (f * g) = g 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] {f g : α → M} (h : Disjoint (Function.mulSupport f) (Function.mulSupport g))
Goal: (Function.mulSupport g).mulIndicator (f * g) = g
-- Set.indicator_add_eq_right
@[to_additive]
theorem mulIndicator_mul_eq_right {f g : α → M} (h : Disjoint (mulSupport f) (mulSupport g)) :
    (mulSupport g).mulIndicator (f * g) = g 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] {f g : α → M} (h : Disjoint (Function.support f) (Function.support g))
Goal: (Function.support g).indicator (f + g) = g
-- Set.indicator_add_compl_eq_piecewise
@[to_additive]
theorem mulIndicator_mul_compl_eq_piecewise [DecidablePred (· ∈ s)] (f g : α → M) :
    s.mulIndicator f * sᶜ.mulIndicator g = s.piecewise f g 
Arguments: {α : Type u_1} {M : Type u_4} [AddZeroClass M] {s : Set α} [DecidablePred fun x => x ∈ s] (f g : α → M)
Goal: s.indicator f + sᶜ.indicator g = s.piecewise f g
-- Set.mulIndicator_mul_compl_eq_piecewise
@[to_additive]
theorem mulIndicator_mul_compl_eq_piecewise [DecidablePred (· ∈ s)] (f g : α → M) :
    s.mulIndicator f * sᶜ.mulIndicator g = s.piecewise f g 
Arguments: {α : Type u_1} {M : Type u_4} [MulOneClass M] {s : Set α} [DecidablePred fun x => x ∈ s] (f g : α → M)
Goal: s.mulIndicator f * sᶜ.mulIndicator g = s.piecewise f g
-- Set.mulIndicator_inv'
@[to_additive]
theorem mulIndicator_inv' (s : Set α) (f : α → G) : mulIndicator s f⁻¹ = (mulIndicator s f)⁻¹ 
Arguments: {α : Type u_1} {G : Type u_6} [Group G] (s : Set α) (f : α → G)
Goal: s.mulIndicator f⁻¹ = (s.mulIndicator f)⁻¹
-- Set.indicator_neg'
@[to_additive]
theorem mulIndicator_inv' (s : Set α) (f : α → G) : mulIndicator s f⁻¹ = (mulIndicator s f)⁻¹ 
Arguments: {α : Type u_1} {G : Type u_6} [AddGroup G] (s : Set α) (f : α → G)
Goal: s.indicator (-f) = -s.indicator f
-- Set.indicator_neg
@[to_additive]
theorem mulIndicator_inv (s : Set α) (f : α → G) :
    (mulIndicator s fun a => (f a)⁻¹) = fun a => (mulIndicator s f a)⁻¹ 
Arguments: {α : Type u_1} {G : Type u_6} [AddGroup G] (s : Set α) (f : α → G)
Goal: (s.indicator fun a => -f a) = fun a => -s.indicator f a
-- Set.mulIndicator_inv
@[to_additive]
theorem mulIndicator_inv (s : Set α) (f : α → G) :
    (mulIndicator s fun a => (f a)⁻¹) = fun a => (mulIndicator s f a)⁻¹ 
Arguments: {α : Type u_1} {G : Type u_6} [Group G] (s : Set α) (f : α → G)
Goal: (s.mulIndicator fun a => (f a)⁻¹) = fun a => (s.mulIndicator f a)⁻¹
-- Set.indicator_sub
@[to_additive]
theorem mulIndicator_div (s : Set α) (f g : α → G) :
    (mulIndicator s fun a => f a / g a) = fun a => mulIndicator s f a / mulIndicator s g a 
Arguments: {α : Type u_1} {G : Type u_6} [AddGroup G] (s : Set α) (f g : α → G)
Goal: (s.indicator fun a => f a - g a) = fun a => s.indicator f a - s.indicator g a
-- Set.mulIndicator_div
@[to_additive]
theorem mulIndicator_div (s : Set α) (f g : α → G) :
    (mulIndicator s fun a => f a / g a) = fun a => mulIndicator s f a / mulIndicator s g a 
Arguments: {α : Type u_1} {G : Type u_6} [Group G] (s : Set α) (f g : α → G)
Goal: (s.mulIndicator fun a => f a / g a) = fun a => s.mulIndicator f a / s.mulIndicator g a
-- Set.indicator_sub'
@[to_additive]
theorem mulIndicator_div' (s : Set α) (f g : α → G) :
    mulIndicator s (f / g) = mulIndicator s f / mulIndicator s g 
Arguments: {α : Type u_1} {G : Type u_6} [AddGroup G] (s : Set α) (f g : α → G)
Goal: s.indicator (f - g) = s.indicator f - s.indicator g
-- Set.mulIndicator_div'
@[to_additive]
theorem mulIndicator_div' (s : Set α) (f g : α → G) :
    mulIndicator s (f / g) = mulIndicator s f / mulIndicator s g 
Arguments: {α : Type u_1} {G : Type u_6} [Group G] (s : Set α) (f g : α → G)
Goal: s.mulIndicator (f / g) = s.mulIndicator f / s.mulIndicator g
-- Set.mulIndicator_compl
@[to_additive indicator_compl']
theorem mulIndicator_compl (s : Set α) (f : α → G) :
    mulIndicator sᶜ f = f * (mulIndicator s f)⁻¹ 
Arguments: {α : Type u_1} {G : Type u_6} [Group G] (s : Set α) (f : α → G)
Goal: sᶜ.mulIndicator f = f * (s.mulIndicator f)⁻¹
-- Set.indicator_compl'
@[to_additive indicator_compl']
theorem mulIndicator_compl (s : Set α) (f : α → G) :
    mulIndicator sᶜ f = f * (mulIndicator s f)⁻¹ 
Arguments: {α : Type u_1} {G : Type u_6} [AddGroup G] (s : Set α) (f : α → G)
Goal: sᶜ.indicator f = f + -s.indicator f
-- Set.indicator_compl
@[to_additive indicator_compl]
theorem mulIndicator_compl' (s : Set α) (f : α → G) :
    mulIndicator sᶜ f = f / mulIndicator s f 
Arguments: {α : Type u_1} {G : Type u_6} [AddGroup G] (s : Set α) (f : α → G)
Goal: sᶜ.indicator f = f - s.indicator f
-- Set.mulIndicator_compl'
@[to_additive indicator_compl]
theorem mulIndicator_compl' (s : Set α) (f : α → G) :
    mulIndicator sᶜ f = f / mulIndicator s f 
Arguments: {α : Type u_1} {G : Type u_6} [Group G] (s : Set α) (f : α → G)
Goal: sᶜ.mulIndicator f = f / s.mulIndicator f
-- Set.mulIndicator_diff
@[to_additive indicator_diff']
theorem mulIndicator_diff (h : s ⊆ t) (f : α → G) :
    mulIndicator (t \ s) f = mulIndicator t f * (mulIndicator s f)⁻¹ 
Arguments: {α : Type u_1} {G : Type u_6} [Group G] {s t : Set α} (h : s ⊆ t) (f : α → G)
Goal: (t \ s).mulIndicator f = t.mulIndicator f * (s.mulIndicator f)⁻¹
-- Set.indicator_diff'
@[to_additive indicator_diff']
theorem mulIndicator_diff (h : s ⊆ t) (f : α → G) :
    mulIndicator (t \ s) f = mulIndicator t f * (mulIndicator s f)⁻¹ 
Arguments: {α : Type u_1} {G : Type u_6} [AddGroup G] {s t : Set α} (h : s ⊆ t) (f : α → G)
Goal: (t \ s).indicator f = t.indicator f + -s.indicator f
-- Set.mulIndicator_diff'
@[to_additive indicator_diff]
theorem mulIndicator_diff' (h : s ⊆ t) (f : α → G) :
    mulIndicator (t \ s) f = mulIndicator t f / mulIndicator s f 
Arguments: {α : Type u_1} {G : Type u_6} [Group G] {s t : Set α} (h : s ⊆ t) (f : α → G)
Goal: (t \ s).mulIndicator f = t.mulIndicator f / s.mulIndicator f
-- Set.indicator_diff
@[to_additive indicator_diff]
theorem mulIndicator_diff' (h : s ⊆ t) (f : α → G) :
    mulIndicator (t \ s) f = mulIndicator t f / mulIndicator s f 
Arguments: {α : Type u_1} {G : Type u_6} [AddGroup G] {s t : Set α} (h : s ⊆ t) (f : α → G)
Goal: (t \ s).indicator f = t.indicator f - s.indicator f
-- Set.apply_indicator_symmDiff
@[to_additive]
theorem apply_mulIndicator_symmDiff {g : G → β} (hg : ∀ x, g x⁻¹ = g x)
    (s t : Set α) (f : α → G) (x : α) :
    g (mulIndicator (s ∆ t) f x) = g (mulIndicator s f x / mulIndicator t f x) 
Arguments: {α : Type u_1} {β : Type u_2} {G : Type u_6} [AddGroup G] {g : G → β} (hg : ∀ (x : G), g (-x) = g x) (s t : Set α) (f : α → G) (x : α)
Goal: g ((symmDiff s t).indicator f x) = g (s.indicator f x - t.indicator f x)
-- Set.apply_mulIndicator_symmDiff
@[to_additive]
theorem apply_mulIndicator_symmDiff {g : G → β} (hg : ∀ x, g x⁻¹ = g x)
    (s t : Set α) (f : α → G) (x : α) :
    g (mulIndicator (s ∆ t) f x) = g (mulIndicator s f x / mulIndicator t f x) 
Arguments: {α : Type u_1} {β : Type u_2} {G : Type u_6} [Group G] {g : G → β} (hg : ∀ (x : G), g x⁻¹ = g x) (s t : Set α) (f : α → G) (x : α)
Goal: g ((symmDiff s t).mulIndicator f x) = g (s.mulIndicator f x / t.mulIndicator f x)
-- AddMonoidHom.map_indicator
@[to_additive]
theorem MonoidHom.map_mulIndicator {M N : Type*} [MulOneClass M] [MulOneClass N] (f : M →* N)
    (s : Set α) (g : α → M) (x : α) : f (s.mulIndicator g x) = s.mulIndicator (f ∘ g) x 
Arguments: {α : Type u_1} {M : Type u_6} {N : Type u_7} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (s : Set α) (g : α → M) (x : α)
Goal: f (s.indicator g x) = s.indicator (⇑f ∘ g) x
-- MonoidHom.map_mulIndicator
@[to_additive]
theorem MonoidHom.map_mulIndicator {M N : Type*} [MulOneClass M] [MulOneClass N] (f : M →* N)
    (s : Set α) (g : α → M) (x : α) : f (s.mulIndicator g x) = s.mulIndicator (f ∘ g) x 
Arguments: {α : Type u_1} {M : Type u_6} {N : Type u_7} [MulOneClass M] [MulOneClass N] (f : M →* N) (s : Set α) (g : α → M) (x : α)
Goal: f (s.mulIndicator g x) = s.mulIndicator (⇑f ∘ g) x
-- MulOpposite.algebraMap_apply
@[simp]
theorem algebraMap_apply (c : R) : algebraMap R Aᵐᵒᵖ c = op (algebraMap R A c) 
Arguments: {R : Type u_1} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (c : R)
Goal: (algebraMap R Aᵐᵒᵖ) c = MulOpposite.op ((algebraMap R A) c)
-- AlgEquiv.opOp_apply
@[simps!]
def opOp : A ≃ₐ[R] Aᵐᵒᵖᵐᵒᵖ 
Arguments: (R : Type u_1) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ∀ (a : A), (AlgEquiv.opOp R A) a = MulOpposite.op (MulOpposite.op a)
-- AlgEquiv.opOp_symm_apply
@[simps!]
def opOp : A ≃ₐ[R] Aᵐᵒᵖᵐᵒᵖ 
Arguments: (R : Type u_1) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ∀ (a : Aᵐᵒᵖᵐᵒᵖ), (AlgEquiv.opOp R A).symm a = MulOpposite.unop (MulOpposite.unop a)
-- AlgEquiv.toRingEquiv_opOp
@[simp] theorem toRingEquiv_opOp : (opOp R A : A ≃+* Aᵐᵒᵖᵐᵒᵖ) = RingEquiv.opOp A 
Arguments: (R : Type u_1) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ↑(AlgEquiv.opOp R A) = RingEquiv.opOp A
-- AlgHom.fromOpposite_apply
@[simps (config := .asFn)]
def fromOpposite (f : A →ₐ[R] B) (hf : ∀ x y, Commute (f x) (f y)) : Aᵐᵒᵖ →ₐ[R] B 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y))
Goal: ⇑(f.fromOpposite hf) = ⇑f ∘ MulOpposite.unop
-- AlgHom.toLinearMap_fromOpposite
@[simp]
theorem toLinearMap_fromOpposite (f : A →ₐ[R] B) (hf : ∀ x y, Commute (f x) (f y)) :
    (f.fromOpposite hf).toLinearMap = f.toLinearMap ∘ₗ (opLinearEquiv R (M := A)).symm 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y))
Goal: (f.fromOpposite hf).toLinearMap = f.toLinearMap ∘ₗ ↑(MulOpposite.opLinearEquiv R).symm
-- AlgHom.toRingHom_fromOpposite
@[simp]
theorem toRingHom_fromOpposite (f : A →ₐ[R] B) (hf : ∀ x y, Commute (f x) (f y)) :
    (f.fromOpposite hf : Aᵐᵒᵖ →+* B) = (f : A →+* B).fromOpposite hf 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y))
Goal: ↑(f.fromOpposite hf) = (↑f).fromOpposite hf
-- AlgHom.toOpposite_apply
@[simps (config := .asFn)]
def toOpposite (f : A →ₐ[R] B) (hf : ∀ x y, Commute (f x) (f y)) : A →ₐ[R] Bᵐᵒᵖ 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y))
Goal: ⇑(f.toOpposite hf) = MulOpposite.op ∘ ⇑f
-- AlgHom.toLinearMap_toOpposite
@[simp]
theorem toLinearMap_toOpposite (f : A →ₐ[R] B) (hf : ∀ x y, Commute (f x) (f y)) :
    (f.toOpposite hf).toLinearMap = (opLinearEquiv R : B ≃ₗ[R] Bᵐᵒᵖ) ∘ₗ f.toLinearMap 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y))
Goal: (f.toOpposite hf).toLinearMap = ↑(MulOpposite.opLinearEquiv R) ∘ₗ f.toLinearMap
-- AlgHom.toRingHom_toOpposite
@[simp]
theorem toRingHom_toOpposite (f : A →ₐ[R] B) (hf : ∀ x y, Commute (f x) (f y)) :
    (f.toOpposite hf : A →+* Bᵐᵒᵖ) = (f : A →+* B).toOpposite hf 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y))
Goal: ↑(f.toOpposite hf) = (↑f).toOpposite hf
-- AlgHom.op_apply_apply
@[simps!]
protected def op : (A →ₐ[R] B) ≃ (Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ) 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B)
Goal: ∀ (a : Aᵐᵒᵖ), (AlgHom.op f) a = MulOpposite.op (f (MulOpposite.unop a))
-- AlgHom.op_symm_apply_apply
@[simps!]
protected def op : (A →ₐ[R] B) ≃ (Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ) 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ)
Goal: ∀ (a : A), (AlgHom.op.symm f) a = MulOpposite.unop (f (MulOpposite.op a))
-- AlgHom.toRingHom_op
theorem toRingHom_op (f : A →ₐ[R] B) : f.op.toRingHom = RingHom.op f.toRingHom 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B)
Goal: (AlgHom.op f).toRingHom = RingHom.op f.toRingHom
-- AlgHom.toRingHom_unop
theorem toRingHom_unop (f : Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ) : f.unop.toRingHom = RingHom.unop f.toRingHom 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ)
Goal: (AlgHom.unop f).toRingHom = RingHom.unop f.toRingHom
-- AlgHom.opComm_symm_apply_apply
@[simps!]
def opComm : (A →ₐ[R] Bᵐᵒᵖ) ≃ (Aᵐᵒᵖ →ₐ[R] B) 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
Goal: ∀ (a : Aᵐᵒᵖ →ₐ[R] B) (a_1 : A), (AlgHom.opComm.symm a) a_1 = MulOpposite.op (a (MulOpposite.op a_1))
-- AlgHom.opComm_apply_apply
@[simps!]
def opComm : (A →ₐ[R] Bᵐᵒᵖ) ≃ (Aᵐᵒᵖ →ₐ[R] B) 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
Goal: ∀ (a : A →ₐ[R] Bᵐᵒᵖ) (a_1 : Aᵐᵒᵖ), (AlgHom.opComm a) a_1 = MulOpposite.unop (a (MulOpposite.unop a_1))
-- AlgEquiv.op_apply_apply
@[simps!]
def op : (A ≃ₐ[R] B) ≃ Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B)
Goal: ∀ (a : Aᵐᵒᵖ), (AlgEquiv.op f) a = MulOpposite.op (f (MulOpposite.unop a))
-- AlgEquiv.op_symm_apply_apply
@[simps!]
def op : (A ≃ₐ[R] B) ≃ Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ)
Goal: ∀ (a : A), (AlgEquiv.op.symm f) a = MulOpposite.unop (f (MulOpposite.op a))
-- AlgEquiv.op_apply_symm_apply
@[simps!]
def op : (A ≃ₐ[R] B) ≃ Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B)
Goal: ∀ (a : Bᵐᵒᵖ), (AlgEquiv.op f).symm a = MulOpposite.op ((↑↑f).symm (MulOpposite.unop a))
-- AlgEquiv.op_symm_apply_symm_apply
@[simps!]
def op : (A ≃ₐ[R] B) ≃ Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ)
Goal: ∀ (a : B), (AlgEquiv.op.symm f).symm a = MulOpposite.unop ((↑↑f).symm (MulOpposite.op a))
-- AlgEquiv.toAlgHom_op
theorem toAlgHom_op (f : A ≃ₐ[R] B) :
    (AlgEquiv.op f).toAlgHom = AlgHom.op f.toAlgHom 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B)
Goal: ↑(AlgEquiv.op f) = AlgHom.op ↑f
-- AlgEquiv.toRingEquiv_op
theorem toRingEquiv_op (f : A ≃ₐ[R] B) :
    (AlgEquiv.op f).toRingEquiv = RingEquiv.op f.toRingEquiv 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B)
Goal: (AlgEquiv.op f).toRingEquiv = RingEquiv.op f.toRingEquiv
-- AlgEquiv.toAlgHom_unop
theorem toAlgHom_unop (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ) : f.unop.toAlgHom = AlgHom.unop f.toAlgHom 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ)
Goal: ↑(AlgEquiv.unop f) = AlgHom.unop ↑f
-- AlgEquiv.toRingEquiv_unop
theorem toRingEquiv_unop (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ) :
    (AlgEquiv.unop f).toRingEquiv = RingEquiv.unop f.toRingEquiv 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ)
Goal: (AlgEquiv.unop f).toRingEquiv = RingEquiv.unop f.toRingEquiv
-- AlgEquiv.opComm_apply_apply
@[simps!]
def opComm : (A ≃ₐ[R] Bᵐᵒᵖ) ≃ (Aᵐᵒᵖ ≃ₐ[R] B) 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
Goal: ∀ (a : A ≃ₐ[R] Bᵐᵒᵖ) (a_1 : Aᵐᵒᵖ), (AlgEquiv.opComm a) a_1 = MulOpposite.unop (a (MulOpposite.unop a_1))
-- AlgEquiv.opComm_symm_apply_symm_apply
@[simps!]
def opComm : (A ≃ₐ[R] Bᵐᵒᵖ) ≃ (Aᵐᵒᵖ ≃ₐ[R] B) 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
Goal: ∀ (a : Aᵐᵒᵖ ≃ₐ[R] B) (a_1 : Bᵐᵒᵖ),
  (AlgEquiv.opComm.symm a).symm a_1 =
    MulOpposite.unop ((↑↑(AlgEquiv.refl.trans (a.trans (AlgEquiv.opOp R B)))).symm (MulOpposite.op a_1))
-- AlgEquiv.opComm_apply_symm_apply
@[simps!]
def opComm : (A ≃ₐ[R] Bᵐᵒᵖ) ≃ (Aᵐᵒᵖ ≃ₐ[R] B) 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
Goal: ∀ (a : A ≃ₐ[R] Bᵐᵒᵖ) (a_1 : B),
  (AlgEquiv.opComm a).symm a_1 = (↑↑AlgEquiv.refl).symm ((↑↑((AlgEquiv.op a).trans (AlgEquiv.opOp R B).symm)).symm a_1)
-- AlgEquiv.opComm_symm_apply_apply
@[simps!]
def opComm : (A ≃ₐ[R] Bᵐᵒᵖ) ≃ (Aᵐᵒᵖ ≃ₐ[R] B) 
Arguments: {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
Goal: ∀ (a : Aᵐᵒᵖ ≃ₐ[R] B) (a_1 : A), (AlgEquiv.opComm.symm a) a_1 = MulOpposite.op (a (MulOpposite.op a_1))
-- AlgEquiv.toOpposite_apply
@[simps!]
def toOpposite : A ≃ₐ[R] Aᵐᵒᵖ 
Arguments: (R : Type u_1) (A : Type u_3) [CommSemiring R] [CommSemiring A] [Algebra R A]
Goal: ∀ (a : A), (AlgEquiv.toOpposite R A) a = MulOpposite.op a
-- AlgEquiv.toOpposite_symm_apply
@[simps!]
def toOpposite : A ≃ₐ[R] Aᵐᵒᵖ 
Arguments: (R : Type u_1) (A : Type u_3) [CommSemiring R] [CommSemiring A] [Algebra R A]
Goal: ∀ (a : Aᵐᵒᵖ), (AlgEquiv.toOpposite R A).symm a = MulOpposite.unop a
-- AlgEquiv.toRingEquiv_toOpposite
@[simp] lemma toRingEquiv_toOpposite : (toOpposite R A : A ≃+* Aᵐᵒᵖ) = RingEquiv.toOpposite A 
Arguments: (R : Type u_1) (A : Type u_3) [CommSemiring R] [CommSemiring A] [Algebra R A]
Goal: ↑(AlgEquiv.toOpposite R A) = RingEquiv.toOpposite A
-- AlgEquiv.toLinearEquiv_toOpposite
@[simp] lemma toLinearEquiv_toOpposite : toLinearEquiv (toOpposite R A) = opLinearEquiv R 
Arguments: (R : Type u_1) (A : Type u_3) [CommSemiring R] [CommSemiring A] [Algebra R A]
Goal: (AlgEquiv.toOpposite R A).toLinearEquiv = MulOpposite.opLinearEquiv R
-- Algebra.lmul_algebraMap
theorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x 
Arguments: (R : Type u) {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (x : R)
Goal: (Algebra.lmul R A) ((algebraMap R A) x) = (Algebra.lsmul R R A) x