ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearMap.range_smul
theorem range_smul (f : V →ₗ[K] V₂) (a : K) (h : a ≠ 0) : range (a • f) = range f 
Arguments: {K : Type u_4} {V : Type u_9} {V₂ : Type u_10} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (a : K) (h : a ≠ 0)
Goal: LinearMap.range (a • f) = LinearMap.range f
-- LinearMap.range_smul'
theorem range_smul' (f : V →ₗ[K] V₂) (a : K) :
    range (a • f) = ⨆ _ : a ≠ 0, range f 
Arguments: {K : Type u_4} {V : Type u_9} {V₂ : Type u_10} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (a : K)
Goal: LinearMap.range (a • f) = ⨆ (_ : a ≠ 0), LinearMap.range f
-- Submodule.map_top
@[simp]
theorem map_top [RingHomSurjective τ₁₂] (f : F) : map f ⊤ = range f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {M : Type u_6} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_11} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] [RingHomSurjective τ₁₂] (f : F)
Goal: Submodule.map f ⊤ = LinearMap.range f
-- Submodule.range_subtype
@[simp]
theorem range_subtype : range p.subtype = p 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M)
Goal: LinearMap.range p.subtype = p
-- Submodule.map_subtype_le
theorem map_subtype_le (p' : Submodule R p) : map p.subtype p' ≤ p 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (p' : Submodule R ↥p)
Goal: Submodule.map p.subtype p' ≤ p
-- Submodule.map_subtype_top
-- @[simp] -- Porting note (#10618): simp can prove this
theorem map_subtype_top : map p.subtype (⊤ : Submodule R p) = p 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M)
Goal: Submodule.map p.subtype ⊤ = p
-- Submodule.comap_subtype_eq_top
@[simp]
theorem comap_subtype_eq_top {p p' : Submodule R M} : comap p.subtype p' = ⊤ ↔ p ≤ p' 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {p p' : Submodule R M}
Goal: Submodule.comap p.subtype p' = ⊤ ↔ p ≤ p'
-- Submodule.comap_subtype_self
@[simp]
theorem comap_subtype_self : comap p.subtype p = ⊤ 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M)
Goal: Submodule.comap p.subtype p = ⊤
-- Submodule.range_inclusion
theorem range_inclusion (p q : Submodule R M) (h : p ≤ q) :
    range (inclusion h) = comap q.subtype p 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (p q : Submodule R M) (h : p ≤ q)
Goal: LinearMap.range (Submodule.inclusion h) = Submodule.comap q.subtype p
-- Submodule.map_subtype_range_inclusion
@[simp]
theorem map_subtype_range_inclusion {p p' : Submodule R M} (h : p ≤ p') :
    map p'.subtype (range <| inclusion h) = p 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {p p' : Submodule R M} (h : p ≤ p')
Goal: Submodule.map p'.subtype (LinearMap.range (Submodule.inclusion h)) = p
-- Submodule.map_subtype_embedding_eq
@[simp]
theorem map_subtype_embedding_eq (p' : Submodule R p) :
    MapSubtype.orderEmbedding p p' = map p.subtype p' 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (p' : Submodule R ↥p)
Goal: (Submodule.MapSubtype.orderEmbedding p) p' = Submodule.map p.subtype p'
-- Submodule.coe_mapIic_apply
@[simp] lemma coe_mapIic_apply [Semiring R] [AddCommMonoid M] [Module R M]
    (p : Submodule R M) (q : Submodule R p) :
    (p.mapIic q : Submodule R M) = q.map p.subtype 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (q : Submodule R ↥p)
Goal: ↑(p.mapIic q) = Submodule.map p.subtype q
-- LinearMap.ker_eq_bot_of_cancel
theorem ker_eq_bot_of_cancel {f : M →ₛₗ[τ₁₂] M₂}
    (h : ∀ u v : ker f →ₗ[R] M, f.comp u = f.comp v → u = v) : ker f = ⊥ 
Arguments: {R : Type u_1} {R₂ : Type u_2} {M : Type u_6} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} (h : ∀ (u v : ↥(LinearMap.ker f) →ₗ[R] M), f.comp u = f.comp v → u = v)
Goal: LinearMap.ker f = ⊥
-- LinearMap.range_comp_of_range_eq_top
theorem range_comp_of_range_eq_top [RingHomSurjective τ₁₂] [RingHomSurjective τ₂₃]
    [RingHomSurjective τ₁₃] {f : M →ₛₗ[τ₁₂] M₂} (g : M₂ →ₛₗ[τ₂₃] M₃) (hf : range f = ⊤) :
    range (g.comp f : M →ₛₗ[τ₁₃] M₃) = range g 
Arguments: {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {M : Type u_6} {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] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [RingHomCompTriple τ₁₂ τ₂₃ τ₁₃] [RingHomSurjective τ₁₂] [RingHomSurjective τ₂₃] [RingHomSurjective τ₁₃] {f : M →ₛₗ[τ₁₂] M₂} (g : M₂ →ₛₗ[τ₂₃] M₃) (hf : LinearMap.range f = ⊤)
Goal: LinearMap.range (g.comp f) = LinearMap.range g
-- LinearMap.mem_submoduleImage
@[simp]
theorem mem_submoduleImage {M' : Type*} [AddCommMonoid M'] [Module R M'] {O : Submodule R M}
    {ϕ : O →ₗ[R] M'} {N : Submodule R M} {x : M'} :
    x ∈ ϕ.submoduleImage N ↔ ∃ (y : _) (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {M' : Type u_11} [AddCommMonoid M'] [Module R M'] {O : Submodule R M} {ϕ : ↥O →ₗ[R] M'} {N : Submodule R M} {x : M'}
Goal: x ∈ ϕ.submoduleImage N ↔ ∃ y, ∃ (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x
-- LinearMap.mem_submoduleImage_of_le
theorem mem_submoduleImage_of_le {M' : Type*} [AddCommMonoid M'] [Module R M'] {O : Submodule R M}
    {ϕ : O →ₗ[R] M'} {N : Submodule R M} (hNO : N ≤ O) {x : M'} :
    x ∈ ϕ.submoduleImage N ↔ ∃ (y : _) (yN : y ∈ N), ϕ ⟨y, hNO yN⟩ = x 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {M' : Type u_11} [AddCommMonoid M'] [Module R M'] {O : Submodule R M} {ϕ : ↥O →ₗ[R] M'} {N : Submodule R M} (hNO : N ≤ O) {x : M'}
Goal: x ∈ ϕ.submoduleImage N ↔ ∃ y, ∃ (yN : y ∈ N), ϕ ⟨y, ⋯⟩ = x
-- LinearMap.submoduleImage_apply_of_le
theorem submoduleImage_apply_of_le {M' : Type*} [AddCommGroup M'] [Module R M']
    {O : Submodule R M} (ϕ : O →ₗ[R] M') (N : Submodule R M) (hNO : N ≤ O) :
    ϕ.submoduleImage N = range (ϕ.comp (Submodule.inclusion hNO)) 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {M' : Type u_11} [AddCommGroup M'] [Module R M'] {O : Submodule R M} (ϕ : ↥O →ₗ[R] M') (N : Submodule R M) (hNO : N ≤ O)
Goal: ϕ.submoduleImage N = LinearMap.range (ϕ ∘ₗ Submodule.inclusion hNO)
-- LinearMap.range_rangeRestrict
@[simp] theorem range_rangeRestrict : range f.rangeRestrict = ⊤ 
Arguments: {R : Type u_1} {R₂ : Type u_2} {M : Type u_6} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} [RingHomSurjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂)
Goal: LinearMap.range f.rangeRestrict = ⊤
-- LinearMap.surjective_rangeRestrict
theorem surjective_rangeRestrict : Surjective f.rangeRestrict 
Arguments: {R : Type u_1} {R₂ : Type u_2} {M : Type u_6} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} [RingHomSurjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂)
Goal: Function.Surjective ⇑f.rangeRestrict
-- LinearMap.ker_rangeRestrict
@[simp] theorem ker_rangeRestrict : ker f.rangeRestrict = ker f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {M : Type u_6} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} [RingHomSurjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂)
Goal: LinearMap.ker f.rangeRestrict = LinearMap.ker f
-- LinearMap.restrictScalarsLinearMap_apply
@[simps]
def restrictScalarsLinearMap : (M →ₗ[A] N) →ₗ[k] M →ₗ[k] N 
Arguments: (k : Type u_1) [CommSemiring k] (A : Type u_2) [Semiring A] [Algebra k A] (M : Type u_3) [AddCommMonoid M] [Module k M] [Module A M] [IsScalarTower k A M] (N : Type u_4) [AddCommMonoid N] [Module k N] [Module A N] [IsScalarTower k A N] (fₗ : M →ₗ[A] N)
Goal: (LinearMap.restrictScalarsLinearMap k A M N) fₗ = ↑k fₗ
-- LinearMap.mem_ker
@[simp]
theorem mem_ker {f : F} {y} : y ∈ ker f ↔ f y = 0 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f : F} {y : M}
Goal: y ∈ LinearMap.ker f ↔ f y = 0
-- LinearMap.ker_id
@[simp]
theorem ker_id : ker (LinearMap.id : M →ₗ[R] M) = ⊥ 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: LinearMap.ker LinearMap.id = ⊥
-- LinearMap.map_coe_ker
@[simp]
theorem map_coe_ker (f : F) (x : ker f) : f x = 0 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] (f : F) (x : ↥(LinearMap.ker f))
Goal: f ↑x = 0
-- LinearMap.ker_toAddSubmonoid
theorem ker_toAddSubmonoid (f : M →ₛₗ[τ₁₂] M₂) : f.ker.toAddSubmonoid = (AddMonoidHom.mker f) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂)
Goal: (LinearMap.ker f).toAddSubmonoid = AddMonoidHom.mker f
-- LinearMap.comp_ker_subtype
theorem comp_ker_subtype (f : M →ₛₗ[τ₁₂] M₂) : f.comp f.ker.subtype = 0 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂)
Goal: f.comp (LinearMap.ker f).subtype = 0
-- LinearMap.ker_comp
theorem ker_comp (f : M →ₛₗ[τ₁₂] M₂) (g : M₂ →ₛₗ[τ₂₃] M₃) :
    ker (g.comp f : M →ₛₗ[τ₁₃] M₃) = comap f (ker g) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_6} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [RingHomCompTriple τ₁₂ τ₂₃ τ₁₃] (f : M →ₛₗ[τ₁₂] M₂) (g : M₂ →ₛₗ[τ₂₃] M₃)
Goal: LinearMap.ker (g.comp f) = Submodule.comap f (LinearMap.ker g)
-- LinearMap.ker_le_ker_comp
theorem ker_le_ker_comp (f : M →ₛₗ[τ₁₂] M₂) (g : M₂ →ₛₗ[τ₂₃] M₃) :
    ker f ≤ ker (g.comp f : M →ₛₗ[τ₁₃] M₃) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_6} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [RingHomCompTriple τ₁₂ τ₂₃ τ₁₃] (f : M →ₛₗ[τ₁₂] M₂) (g : M₂ →ₛₗ[τ₂₃] M₃)
Goal: LinearMap.ker f ≤ LinearMap.ker (g.comp f)
-- LinearMap.ker_sup_ker_le_ker_comp_of_commute
theorem ker_sup_ker_le_ker_comp_of_commute {f g : M →ₗ[R] M} (h : Commute f g) :
    ker f ⊔ ker g ≤ ker (f ∘ₗ g) 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {f g : M →ₗ[R] M} (h : Commute f g)
Goal: LinearMap.ker f ⊔ LinearMap.ker g ≤ LinearMap.ker (f ∘ₗ g)
-- LinearMap.ker_le_comap
@[simp]
theorem ker_le_comap {p : Submodule R₂ M₂} (f : M →ₛₗ[τ₁₂] M₂) :
    ker f ≤ p.comap f 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {p : Submodule R₂ M₂} (f : M →ₛₗ[τ₁₂] M₂)
Goal: LinearMap.ker f ≤ Submodule.comap f p
-- LinearMap.disjoint_ker
theorem disjoint_ker {f : F} {p : Submodule R M} :
    Disjoint p (ker f) ↔ ∀ x ∈ p, f x = 0 → x = 0 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f : F} {p : Submodule R M}
Goal: Disjoint p (LinearMap.ker f) ↔ ∀ x ∈ p, f x = 0 → x = 0
-- LinearMap.ker_eq_bot'
theorem ker_eq_bot' {f : F} : ker f = ⊥ ↔ ∀ m, f m = 0 → m = 0 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f : F}
Goal: LinearMap.ker f = ⊥ ↔ ∀ (m : M), f m = 0 → m = 0
-- LinearMap.ker_eq_bot_of_inverse
theorem ker_eq_bot_of_inverse {τ₂₁ : R₂ →+* R} [RingHomInvPair τ₁₂ τ₂₁] {f : M →ₛₗ[τ₁₂] M₂}
    {g : M₂ →ₛₗ[τ₂₁] M} (h : (g.comp f : M →ₗ[R] M) = id) : ker f = ⊥ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [RingHomInvPair τ₁₂ τ₂₁] {f : M →ₛₗ[τ₁₂] M₂} {g : M₂ →ₛₗ[τ₂₁] M} (h : g.comp f = LinearMap.id)
Goal: LinearMap.ker f = ⊥
-- LinearMap.le_ker_iff_map
theorem le_ker_iff_map [RingHomSurjective τ₁₂] {f : F} {p : Submodule R M} :
    p ≤ ker f ↔ map f p = ⊥ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] [RingHomSurjective τ₁₂] {f : F} {p : Submodule R M}
Goal: p ≤ LinearMap.ker f ↔ Submodule.map f p = ⊥
-- LinearMap.ker_codRestrict
theorem ker_codRestrict {τ₂₁ : R₂ →+* R} (p : Submodule R M) (f : M₂ →ₛₗ[τ₂₁] M) (hf) :
    ker (codRestrict p f hf) = ker f 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₂₁ : R₂ →+* R} (p : Submodule R M) (f : M₂ →ₛₗ[τ₂₁] M) (hf : ∀ (c : M₂), f c ∈ p)
Goal: LinearMap.ker (LinearMap.codRestrict p f hf) = LinearMap.ker f
-- LinearMap.ker_restrict
theorem ker_restrict [AddCommMonoid M₁] [Module R M₁] {p : Submodule R M} {q : Submodule R M₁}
    {f : M →ₗ[R] M₁} (hf : ∀ x : M, x ∈ p → f x ∈ q) :
    ker (f.restrict hf) = LinearMap.ker (f.domRestrict p) 
Arguments: {R : Type u_1} {M : Type u_6} {M₁ : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₁] [Module R M₁] {p : Submodule R M} {q : Submodule R M₁} {f : M →ₗ[R] M₁} (hf : ∀ x ∈ p, f x ∈ q)
Goal: LinearMap.ker (f.restrict hf) = LinearMap.ker (f.domRestrict p)
-- LinearMap.ker_zero
@[simp]
theorem ker_zero : ker (0 : M →ₛₗ[τ₁₂] M₂) = ⊤ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂}
Goal: LinearMap.ker 0 = ⊤
-- LinearMap.ker_eq_top
theorem ker_eq_top {f : M →ₛₗ[τ₁₂] M₂} : ker f = ⊤ ↔ f = 0 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂}
Goal: LinearMap.ker f = ⊤ ↔ f = 0
-- AddMonoidHom.coe_toIntLinearMap_ker
@[simp]
theorem _root_.AddMonoidHom.coe_toIntLinearMap_ker {M M₂ : Type*} [AddCommGroup M] [AddCommGroup M₂]
    (f : M →+ M₂) : LinearMap.ker f.toIntLinearMap = AddSubgroup.toIntSubmodule f.ker 
Arguments: {M : Type u_13} {M₂ : Type u_14} [AddCommGroup M] [AddCommGroup M₂] (f : M →+ M₂)
Goal: LinearMap.ker f.toIntLinearMap = AddSubgroup.toIntSubmodule f.ker
-- LinearMap.ker_eq_bot_of_injective
theorem ker_eq_bot_of_injective {f : F} (hf : Injective f) : ker f = ⊥ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f : F} (hf : Function.Injective ⇑f)
Goal: LinearMap.ker f = ⊥
-- LinearMap.iterateKer_coe
@[simps]
def iterateKer (f : M →ₗ[R] M) : ℕ →o Submodule R M 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (f : M →ₗ[R] M) (n : ℕ)
Goal: f.iterateKer n = LinearMap.ker (f ^ n)
-- LinearMap.ker_toAddSubgroup
theorem ker_toAddSubgroup (f : M →ₛₗ[τ₁₂] M₂) : (ker f).toAddSubgroup = f.toAddMonoidHom.ker 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Ring R] [Ring R₂] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂)
Goal: (LinearMap.ker f).toAddSubgroup = f.toAddMonoidHom.ker
-- LinearMap.sub_mem_ker_iff
theorem sub_mem_ker_iff {x y} : x - y ∈ ker f ↔ f x = f y 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Ring R] [Ring R₂] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f : F} {x y : M}
Goal: x - y ∈ LinearMap.ker f ↔ f x = f y
-- LinearMap.disjoint_ker'
theorem disjoint_ker' {p : Submodule R M} :
    Disjoint p (ker f) ↔ ∀ x ∈ p, ∀ y ∈ p, f x = f y → x = y 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Ring R] [Ring R₂] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f : F} {p : Submodule R M}
Goal: Disjoint p (LinearMap.ker f) ↔ ∀ x ∈ p, ∀ y ∈ p, f x = f y → x = y
-- LinearMap.injOn_of_disjoint_ker
theorem injOn_of_disjoint_ker {p : Submodule R M} {s : Set M} (h : s ⊆ p)
    (hd : Disjoint p (ker f)) : Set.InjOn f s 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Ring R] [Ring R₂] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f : F} {p : Submodule R M} {s : Set M} (h : s ⊆ ↑p) (hd : Disjoint p (LinearMap.ker f))
Goal: Set.InjOn (⇑f) s
-- LinearMapClass.ker_eq_bot
theorem _root_.LinearMapClass.ker_eq_bot : ker f = ⊥ ↔ Injective f 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Ring R] [Ring R₂] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} (F : Type u_12) [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f : F}
Goal: LinearMap.ker f = ⊥ ↔ Function.Injective ⇑f
-- LinearMap.ker_eq_bot
theorem ker_eq_bot {f : M →ₛₗ[τ₁₂] M₂} : ker f = ⊥ ↔ Injective f 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Ring R] [Ring R₂] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂}
Goal: LinearMap.ker f = ⊥ ↔ Function.Injective ⇑f
-- LinearMap.injective_domRestrict_iff
@[simp] lemma injective_domRestrict_iff {f : M →ₛₗ[τ₁₂] M₂} {S : Submodule R M} :
    Injective (f.domRestrict S) ↔ S ⊓ LinearMap.ker f = ⊥ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Ring R] [Ring R₂] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} {S : Submodule R M}
Goal: Function.Injective ⇑(f.domRestrict S) ↔ S ⊓ LinearMap.ker f = ⊥
-- LinearMap.injective_restrict_iff_disjoint
@[simp] theorem injective_restrict_iff_disjoint {p : Submodule R M} {f : M →ₗ[R] M}
    (hf : ∀ x ∈ p, f x ∈ p) :
    Injective (f.restrict hf) ↔ Disjoint p (ker f) 
Arguments: {R : Type u_1} {M : Type u_6} [Ring R] [AddCommGroup M] [Module R M] {p : Submodule R M} {f : M →ₗ[R] M} (hf : ∀ x ∈ p, f x ∈ p)
Goal: Function.Injective ⇑(f.restrict hf) ↔ Disjoint p (LinearMap.ker f)
-- LinearMap.ker_smul
theorem ker_smul (f : V →ₗ[K] V₂) (a : K) (h : a ≠ 0) : ker (a • f) = ker f 
Arguments: {K : Type u_5} {V : Type u_10} {V₂ : Type u_11} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (a : K) (h : a ≠ 0)
Goal: LinearMap.ker (a • f) = LinearMap.ker f
-- LinearMap.ker_smul'
theorem ker_smul' (f : V →ₗ[K] V₂) (a : K) : ker (a • f) = ⨅ _ : a ≠ 0, ker f 
Arguments: {K : Type u_5} {V : Type u_10} {V₂ : Type u_11} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (a : K)
Goal: LinearMap.ker (a • f) = ⨅ (_ : a ≠ 0), LinearMap.ker f
-- Submodule.comap_bot
@[simp]
theorem comap_bot (f : F) : comap f ⊥ = ker f 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_12} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] (f : F)
Goal: Submodule.comap f ⊥ = LinearMap.ker f
-- Submodule.ker_subtype
@[simp]
theorem ker_subtype : ker p.subtype = ⊥ 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M)
Goal: LinearMap.ker p.subtype = ⊥
-- Submodule.ker_inclusion
@[simp]
theorem ker_inclusion (p p' : Submodule R M) (h : p ≤ p') : ker (inclusion h) = ⊥ 
Arguments: {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (p p' : Submodule R M) (h : p ≤ p')
Goal: LinearMap.ker (Submodule.inclusion h) = ⊥
-- LinearMap.ker_comp_of_ker_eq_bot
theorem ker_comp_of_ker_eq_bot (f : M →ₛₗ[τ₁₂] M₂) {g : M₂ →ₛₗ[τ₂₃] M₃} (hg : ker g = ⊥) :
    ker (g.comp f : M →ₛₗ[τ₁₃] M₃) = ker f 
Arguments: {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_6} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [RingHomCompTriple τ₁₂ τ₂₃ τ₁₃] (f : M →ₛₗ[τ₁₂] M₂) {g : M₂ →ₛₗ[τ₂₃] M₃} (hg : LinearMap.ker g = ⊥)
Goal: LinearMap.ker (g.comp f) = LinearMap.ker f
-- LinearEquiv.ker
@[simp]
protected theorem ker : LinearMap.ker (e : M →ₛₗ[σ₁₂] M₂) = ⊥ 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_M₂ : Module R₂ M₂} {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} (e : M ≃ₛₗ[σ₁₂] M₂)
Goal: LinearMap.ker ↑e = ⊥
-- LinearEquiv.ker_comp
@[simp]
theorem ker_comp (l : M →ₛₗ[σ₁₂] M₂) :
    LinearMap.ker (((e'' : M₂ →ₛₗ[σ₂₃] M₃).comp l : M →ₛₗ[σ₁₃] M₃) : M →ₛₗ[σ₁₃] M₃) =
    LinearMap.ker l 
Arguments: {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_6} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M : Module R M} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {σ₃₂ : R₃ →+* R₂} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} (e'' : M₂ ≃ₛₗ[σ₂₃] M₃) (l : M →ₛₗ[σ₁₂] M₂)
Goal: LinearMap.ker ((↑e'').comp l) = LinearMap.ker l
-- ULift.vadd_def
@[to_additive (attr := simp)]
theorem smul_def [SMul R M] (s : ULift R) (x : M) : s • x = s.down • x 
Arguments: {R : Type u} {M : Type v} [VAdd R M] (s : ULift.{u_1, u} R) (x : M)
Goal: s +ᵥ x = s.down +ᵥ x
-- ULift.smul_def
@[to_additive (attr := simp)]
theorem smul_def [SMul R M] (s : ULift R) (x : M) : s • x = s.down • x 
Arguments: {R : Type u} {M : Type v} [SMul R M] (s : ULift.{u_1, u} R) (x : M)
Goal: s • x = s.down • x
-- ULift.moduleEquiv_apply
@[simps apply symm_apply]
def moduleEquiv [Semiring R] [AddCommMonoid M] [Module R M] : ULift.{w} M ≃ₗ[R] M 
Arguments: {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (self : ULift.{w, v} M)
Goal: ULift.moduleEquiv self = self.down
-- ULift.moduleEquiv_symm_apply
@[simps apply symm_apply]
def moduleEquiv [Semiring R] [AddCommMonoid M] [Module R M] : ULift.{w} M ≃ₗ[R] M 
Arguments: {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (down : M)
Goal: ULift.moduleEquiv.symm down = { down := down }
-- Module.zero_smul
@[ext]
class Module (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] extends
  DistribMulAction R M where
    protected add_smul : ∀ (r s : R) (x : M), (r + s) • x = r • x + s • x
    protected zero_smul : ∀ x : M, (0 : R) • x = 0
Arguments: {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [self : Module R M] (x : M)
Goal: 0 • x = 0