ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearEquiv.smulOfNeZero_symm_apply
@[simps!]
def smulOfNeZero (a : K) (ha : a ≠ 0) : M ≃ₗ[K] M 
Arguments: (K : Type u_6) (M : Type u_8) [Field K] [AddCommGroup M] [Module K M] (a : K) (ha : a ≠ 0)
Goal: ∀ (a_1 : M), (LinearEquiv.smulOfNeZero K M a ha).symm a_1 = (Units.mk0 a ha)⁻¹ • a_1
-- LinearEquiv.smulOfNeZero_apply
@[simps!]
def smulOfNeZero (a : K) (ha : a ≠ 0) : M ≃ₗ[K] M 
Arguments: (K : Type u_6) (M : Type u_8) [Field K] [AddCommGroup M] [Module K M] (a : K) (ha : a ≠ 0)
Goal: ∀ (a_1 : M), (LinearEquiv.smulOfNeZero K M a ha) a_1 = Units.mk0 a ha • a_1
-- LinearMap.funLeft_apply
@[simp]
theorem funLeft_apply (f : m → n) (g : n → M) (i : m) : funLeft R M f g i = g (f i) 
Arguments: (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] {m : Type u_17} {n : Type u_18} (f : m → n) (g : n → M) (i : m)
Goal: (LinearMap.funLeft R M f) g i = g (f i)
-- LinearMap.funLeft_id
@[simp]
theorem funLeft_id (g : n → M) : funLeft R M _root_.id g = g 
Arguments: (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] {n : Type u_18} (g : n → M)
Goal: (LinearMap.funLeft R M id) g = g
-- LinearMap.funLeft_comp
theorem funLeft_comp (f₁ : n → p) (f₂ : m → n) :
    funLeft R M (f₁ ∘ f₂) = (funLeft R M f₂).comp (funLeft R M f₁) 
Arguments: (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] {m : Type u_17} {n : Type u_18} {p : Type u_19} (f₁ : n → p) (f₂ : m → n)
Goal: LinearMap.funLeft R M (f₁ ∘ f₂) = LinearMap.funLeft R M f₂ ∘ₗ LinearMap.funLeft R M f₁
-- LinearMap.funLeft_surjective_of_injective
theorem funLeft_surjective_of_injective (f : m → n) (hf : Injective f) :
    Surjective (funLeft R M f) 
Arguments: (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] {m : Type u_17} {n : Type u_18} (f : m → n) (hf : Function.Injective f)
Goal: Function.Surjective ⇑(LinearMap.funLeft R M f)
-- LinearMap.funLeft_injective_of_surjective
theorem funLeft_injective_of_surjective (f : m → n) (hf : Surjective f) :
    Injective (funLeft R M f) 
Arguments: (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] {m : Type u_17} {n : Type u_18} (f : m → n) (hf : Function.Surjective f)
Goal: Function.Injective ⇑(LinearMap.funLeft R M f)
-- LinearEquiv.funCongrLeft_apply
@[simp]
theorem funCongrLeft_apply (e : m ≃ n) (x : n → M) : funCongrLeft R M e x = funLeft R M e x 
Arguments: (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] {m : Type u_17} {n : Type u_18} (e : m ≃ n) (x : n → M)
Goal: (LinearEquiv.funCongrLeft R M e) x = (LinearMap.funLeft R M ⇑e) x
-- LinearEquiv.funCongrLeft_id
@[simp]
theorem funCongrLeft_id : funCongrLeft R M (Equiv.refl n) = LinearEquiv.refl R (n → M) 
Arguments: (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] {n : Type u_18}
Goal: LinearEquiv.funCongrLeft R M (Equiv.refl n) = LinearEquiv.refl R (n → M)
-- LinearEquiv.funCongrLeft_comp
@[simp]
theorem funCongrLeft_comp (e₁ : m ≃ n) (e₂ : n ≃ p) :
    funCongrLeft R M (Equiv.trans e₁ e₂) =
      LinearEquiv.trans (funCongrLeft R M e₂) (funCongrLeft R M e₁) 
Arguments: (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] {m : Type u_17} {n : Type u_18} {p : Type u_19} (e₁ : m ≃ n) (e₂ : n ≃ p)
Goal: LinearEquiv.funCongrLeft R M (e₁.trans e₂) = LinearEquiv.funCongrLeft R M e₂ ≪≫ₗ LinearEquiv.funCongrLeft R M e₁
-- LinearEquiv.funCongrLeft_symm
@[simp]
theorem funCongrLeft_symm (e : m ≃ n) : (funCongrLeft R M e).symm = funCongrLeft R M e.symm 
Arguments: (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] {m : Type u_17} {n : Type u_18} (e : m ≃ n)
Goal: (LinearEquiv.funCongrLeft R M e).symm = LinearEquiv.funCongrLeft R M e.symm
-- Submodule.map_coe
@[simp]
theorem map_coe (f : F) (p : Submodule R M) : (map f p : Set M₂) = f '' p 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) (p : Submodule R M)
Goal: ↑(Submodule.map f p) = ⇑f '' ↑p
-- Submodule.map_toAddSubmonoid
theorem map_toAddSubmonoid (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M) :
    (p.map f).toAddSubmonoid = p.toAddSubmonoid.map (f : M →+ M₂) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M)
Goal: (Submodule.map f p).toAddSubmonoid = AddSubmonoid.map (↑f) p.toAddSubmonoid
-- Submodule.map_toAddSubmonoid'
theorem map_toAddSubmonoid' (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M) :
    (p.map f).toAddSubmonoid = p.toAddSubmonoid.map f 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M)
Goal: (Submodule.map f p).toAddSubmonoid = AddSubmonoid.map f p.toAddSubmonoid
-- AddMonoidHom.coe_toIntLinearMap_map
@[simp]
theorem _root_.AddMonoidHom.coe_toIntLinearMap_map {A A₂ : Type*} [AddCommGroup A] [AddCommGroup A₂]
    (f : A →+ A₂) (s : AddSubgroup A) :
    (AddSubgroup.toIntSubmodule s).map f.toIntLinearMap =
      AddSubgroup.toIntSubmodule (s.map f) 
Arguments: {A : Type u_10} {A₂ : Type u_11} [AddCommGroup A] [AddCommGroup A₂] (f : A →+ A₂) (s : AddSubgroup A)
Goal: Submodule.map f.toIntLinearMap (AddSubgroup.toIntSubmodule s) = AddSubgroup.toIntSubmodule (AddSubgroup.map f s)
-- MonoidHom.coe_toAdditive_map
@[simp]
theorem _root_.MonoidHom.coe_toAdditive_map {G G₂ : Type*} [Group G] [Group G₂] (f : G →* G₂)
    (s : Subgroup G) :
    s.toAddSubgroup.map (MonoidHom.toAdditive f) = Subgroup.toAddSubgroup (s.map f) 
Arguments: {G : Type u_10} {G₂ : Type u_11} [Group G] [Group G₂] (f : G →* G₂) (s : Subgroup G)
Goal: AddSubgroup.map (MonoidHom.toAdditive f) (Subgroup.toAddSubgroup s) = Subgroup.toAddSubgroup (Subgroup.map f s)
-- AddMonoidHom.coe_toMultiplicative_map
@[simp]
theorem _root_.AddMonoidHom.coe_toMultiplicative_map {G G₂ : Type*} [AddGroup G] [AddGroup G₂]
    (f : G →+ G₂) (s : AddSubgroup G) :
    s.toSubgroup.map (AddMonoidHom.toMultiplicative f) = AddSubgroup.toSubgroup (s.map f) 
Arguments: {G : Type u_10} {G₂ : Type u_11} [AddGroup G] [AddGroup G₂] (f : G →+ G₂) (s : AddSubgroup G)
Goal: Subgroup.map (AddMonoidHom.toMultiplicative f) (AddSubgroup.toSubgroup s) = AddSubgroup.toSubgroup (AddSubgroup.map f s)
-- Submodule.mem_map
@[simp]
theorem mem_map {f : F} {p : Submodule R M} {x : M₂} : x ∈ map f p ↔ ∃ y, y ∈ p ∧ f y = x 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] {f : F} {p : Submodule R M} {x : M₂}
Goal: x ∈ Submodule.map f p ↔ ∃ y ∈ p, f y = x
-- Submodule.mem_map_of_mem
theorem mem_map_of_mem {f : F} {p : Submodule R M} {r} (h : r ∈ p) : f r ∈ map f p 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] {f : F} {p : Submodule R M} {r : M} (h : r ∈ p)
Goal: f r ∈ Submodule.map f p
-- Submodule.apply_coe_mem_map
theorem apply_coe_mem_map (f : F) {p : Submodule R M} (r : p) : f r ∈ map f p 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) {p : Submodule R M} (r : ↥p)
Goal: f ↑r ∈ Submodule.map f p
-- Submodule.map_id
@[simp]
theorem map_id : map (LinearMap.id : M →ₗ[R] M) p = p 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M)
Goal: Submodule.map LinearMap.id p = p
-- Submodule.map_comp
theorem map_comp [RingHomSurjective σ₂₃] [RingHomSurjective σ₁₃] (f : M →ₛₗ[σ₁₂] M₂)
    (g : M₂ →ₛₗ[σ₂₃] M₃) (p : Submodule R M) : map (g.comp f : M →ₛₗ[σ₁₃] M₃) p = map g (map f p) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_5} {M₂ : Type u_7} {M₃ : Type u_8} [Semiring R] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomSurjective σ₁₂] [RingHomSurjective σ₂₃] [RingHomSurjective σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (p : Submodule R M)
Goal: Submodule.map (g.comp f) p = Submodule.map g (Submodule.map f p)
-- Submodule.map_mono
theorem map_mono {f : F} {p p' : Submodule R M} : p ≤ p' → map f p ≤ map f p' 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] {f : F} {p p' : Submodule R M}
Goal: p ≤ p' → Submodule.map f p ≤ Submodule.map f p'
-- Submodule.map_zero
@[simp]
theorem map_zero : map (0 : M →ₛₗ[σ₁₂] M₂) p = ⊥ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (p : Submodule R M) [RingHomSurjective σ₁₂]
Goal: Submodule.map 0 p = ⊥
-- Submodule.map_add_le
theorem map_add_le (f g : M →ₛₗ[σ₁₂] M₂) : map (f + g) p ≤ map f p ⊔ map g p 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (p : Submodule R M) [RingHomSurjective σ₁₂] (f g : M →ₛₗ[σ₁₂] M₂)
Goal: Submodule.map (f + g) p ≤ Submodule.map f p ⊔ Submodule.map g p
-- Submodule.map_inf_le
theorem map_inf_le (f : F) {p q : Submodule R M} :
    (p ⊓ q).map f ≤ p.map f ⊓ q.map f 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) {p q : Submodule R M}
Goal: Submodule.map f (p ⊓ q) ≤ Submodule.map f p ⊓ Submodule.map f q
-- Submodule.map_inf
theorem map_inf (f : F) {p q : Submodule R M} (hf : Injective f) :
    (p ⊓ q).map f = p.map f ⊓ q.map f 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) {p q : Submodule R M} (hf : Function.Injective ⇑f)
Goal: Submodule.map f (p ⊓ q) = Submodule.map f p ⊓ Submodule.map f q
-- Submodule.range_map_nonempty
theorem range_map_nonempty (N : Submodule R M) :
    (Set.range (fun ϕ => Submodule.map ϕ N : (M →ₛₗ[σ₁₂] M₂) → Submodule R₂ M₂)).Nonempty 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] (N : Submodule R M)
Goal: (Set.range fun ϕ => Submodule.map ϕ N).Nonempty
-- Submodule.coe_equivMapOfInjective_apply
@[simp]
theorem coe_equivMapOfInjective_apply (f : F) (i : Injective f) (p : Submodule R M) (x : p) :
    (equivMapOfInjective f i p x : M₂) = f x 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) (i : Function.Injective ⇑f) (p : Submodule R M) (x : ↥p)
Goal: ↑((Submodule.equivMapOfInjective f i p) x) = f ↑x
-- Submodule.map_equivMapOfInjective_symm_apply
@[simp]
theorem map_equivMapOfInjective_symm_apply (f : F) (i : Injective f) (p : Submodule R M)
    (x : p.map f) : f ((equivMapOfInjective f i p).symm x) = x 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) (i : Function.Injective ⇑f) (p : Submodule R M) (x : ↥(Submodule.map f p))
Goal: f ↑((Submodule.equivMapOfInjective f i p).symm x) = ↑x
-- Submodule.comap_coe
@[simp]
theorem comap_coe (f : F) (p : Submodule R₂ M₂) : (comap f p : Set M) = f ⁻¹' p 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) (p : Submodule R₂ M₂)
Goal: ↑(Submodule.comap f p) = ⇑f ⁻¹' ↑p
-- Submodule.AddMonoidHom.coe_toIntLinearMap_comap
@[simp]
theorem AddMonoidHom.coe_toIntLinearMap_comap {A A₂ : Type*} [AddCommGroup A] [AddCommGroup A₂]
    (f : A →+ A₂) (s : AddSubgroup A₂) :
    (AddSubgroup.toIntSubmodule s).comap f.toIntLinearMap =
      AddSubgroup.toIntSubmodule (s.comap f) 
Arguments: {A : Type u_10} {A₂ : Type u_11} [AddCommGroup A] [AddCommGroup A₂] (f : A →+ A₂) (s : AddSubgroup A₂)
Goal: Submodule.comap f.toIntLinearMap (AddSubgroup.toIntSubmodule s) = AddSubgroup.toIntSubmodule (AddSubgroup.comap f s)
-- Submodule.mem_comap
@[simp]
theorem mem_comap {f : F} {p : Submodule R₂ M₂} : x ∈ comap f p ↔ f x ∈ p 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {x : M} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] {f : F} {p : Submodule R₂ M₂}
Goal: x ∈ Submodule.comap f p ↔ f x ∈ p
-- Submodule.comap_id
@[simp]
theorem comap_id : comap (LinearMap.id : M →ₗ[R] M) p = p 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M)
Goal: Submodule.comap LinearMap.id p = p
-- Submodule.comap_comp
theorem comap_comp (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (p : Submodule R₃ M₃) :
    comap (g.comp f : M →ₛₗ[σ₁₃] M₃) p = comap f (comap g p) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_5} {M₂ : Type u_7} {M₃ : Type u_8} [Semiring R] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (p : Submodule R₃ M₃)
Goal: Submodule.comap (g.comp f) p = Submodule.comap f (Submodule.comap g p)
-- Submodule.comap_mono
theorem comap_mono {f : F} {q q' : Submodule R₂ M₂} : q ≤ q' → comap f q ≤ comap f q' 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] {f : F} {q q' : Submodule R₂ M₂}
Goal: q ≤ q' → Submodule.comap f q ≤ Submodule.comap f q'
-- Submodule.le_comap_pow_of_le_comap
theorem le_comap_pow_of_le_comap (p : Submodule R M) {f : M →ₗ[R] M} (h : p ≤ p.comap f) (k : ℕ) :
    p ≤ p.comap (f ^ k) 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) {f : M →ₗ[R] M} (h : p ≤ Submodule.comap f p) (k : ℕ)
Goal: p ≤ Submodule.comap (f ^ k) p
-- Submodule.map_le_iff_le_comap
theorem map_le_iff_le_comap {f : F} {p : Submodule R M} {q : Submodule R₂ M₂} :
    map f p ≤ q ↔ p ≤ comap f q 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] {f : F} {p : Submodule R M} {q : Submodule R₂ M₂}
Goal: Submodule.map f p ≤ q ↔ p ≤ Submodule.comap f q
-- Submodule.gc_map_comap
theorem gc_map_comap (f : F) : GaloisConnection (map f) (comap f)
  
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] (f : F)
Goal: GaloisConnection (Submodule.map f) (Submodule.comap f)
-- Submodule.map_bot
@[simp]
theorem map_bot (f : F) : map f ⊥ = ⊥ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] (f : F)
Goal: Submodule.map f ⊥ = ⊥
-- Submodule.map_sup
@[simp]
theorem map_sup (f : F) : map f (p ⊔ p') = map f p ⊔ map f p' 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (p p' : Submodule R M) {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] (f : F)
Goal: Submodule.map f (p ⊔ p') = Submodule.map f p ⊔ Submodule.map f p'
-- Submodule.map_iSup
@[simp]
theorem map_iSup {ι : Sort*} (f : F) (p : ι → Submodule R M) :
    map f (⨆ i, p i) = ⨆ i, map f (p i) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] {ι : Sort u_10} (f : F) (p : ι → Submodule R M)
Goal: Submodule.map f (⨆ i, p i) = ⨆ i, Submodule.map f (p i)
-- Submodule.comap_top
@[simp]
theorem comap_top (f : F) : comap f ⊤ = ⊤ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F)
Goal: Submodule.comap f ⊤ = ⊤
-- Submodule.comap_inf
@[simp]
theorem comap_inf (f : F) : comap f (q ⊓ q') = comap f q ⊓ comap f q' 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (q q' : Submodule R₂ M₂) {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F)
Goal: Submodule.comap f (q ⊓ q') = Submodule.comap f q ⊓ Submodule.comap f q'
-- Submodule.comap_iInf
@[simp]
theorem comap_iInf [RingHomSurjective σ₁₂] {ι : Sort*} (f : F) (p : ι → Submodule R₂ M₂) :
    comap f (⨅ i, p i) = ⨅ i, comap f (p i) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] {ι : Sort u_10} (f : F) (p : ι → Submodule R₂ M₂)
Goal: Submodule.comap f (⨅ i, p i) = ⨅ i, Submodule.comap f (p i)
-- Submodule.comap_zero
@[simp]
theorem comap_zero : comap (0 : M →ₛₗ[σ₁₂] M₂) q = ⊤ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (q : Submodule R₂ M₂)
Goal: Submodule.comap 0 q = ⊤
-- Submodule.map_comap_le
theorem map_comap_le [RingHomSurjective σ₁₂] (f : F) (q : Submodule R₂ M₂) :
    map f (comap f q) ≤ q 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] (f : F) (q : Submodule R₂ M₂)
Goal: Submodule.map f (Submodule.comap f q) ≤ q
-- Submodule.le_comap_map
theorem le_comap_map [RingHomSurjective σ₁₂] (f : F) (p : Submodule R M) : p ≤ comap f (map f p) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] (f : F) (p : Submodule R M)
Goal: p ≤ Submodule.comap f (Submodule.map f p)
-- Submodule.map_comap_eq_of_surjective
theorem map_comap_eq_of_surjective (p : Submodule R₂ M₂) : (p.comap f).map f = p 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] {f : F} (hf : Function.Surjective ⇑f) (p : Submodule R₂ M₂)
Goal: Submodule.map f (Submodule.comap f p) = p
-- Submodule.map_surjective_of_surjective
theorem map_surjective_of_surjective : Function.Surjective (map f) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] {f : F} (hf : Function.Surjective ⇑f)
Goal: Function.Surjective (Submodule.map f)
-- Submodule.comap_injective_of_surjective
theorem comap_injective_of_surjective : Function.Injective (comap f) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] {f : F} (hf : Function.Surjective ⇑f)
Goal: Function.Injective (Submodule.comap f)
-- Submodule.map_sup_comap_of_surjective
theorem map_sup_comap_of_surjective (p q : Submodule R₂ M₂) :
    (p.comap f ⊔ q.comap f).map f = p ⊔ q 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] {f : F} (hf : Function.Surjective ⇑f) (p q : Submodule R₂ M₂)
Goal: Submodule.map f (Submodule.comap f p ⊔ Submodule.comap f q) = p ⊔ q
-- Submodule.map_iSup_comap_of_sujective
theorem map_iSup_comap_of_sujective {ι : Sort*} (S : ι → Submodule R₂ M₂) :
    (⨆ i, (S i).comap f).map f = iSup S 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] {f : F} (hf : Function.Surjective ⇑f) {ι : Sort u_10} (S : ι → Submodule R₂ M₂)
Goal: Submodule.map f (⨆ i, Submodule.comap f (S i)) = iSup S
-- Submodule.map_inf_comap_of_surjective
theorem map_inf_comap_of_surjective (p q : Submodule R₂ M₂) :
    (p.comap f ⊓ q.comap f).map f = p ⊓ q 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {F : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] [RingHomSurjective σ₁₂] {f : F} (hf : Function.Surjective ⇑f) (p q : Submodule R₂ M₂)
Goal: Submodule.map f (Submodule.comap f p ⊓ Submodule.comap f q) = p ⊓ q