ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Submodule.map_iInf_comap_of_surjective
theorem map_iInf_comap_of_surjective {ι : Sort*} (S : ι → Submodule R₂ M₂) :
    (⨅ i, (S i).comap f).map f = iInf 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)) = iInf S
-- Submodule.comap_le_comap_iff_of_surjective
theorem comap_le_comap_iff_of_surjective (p q : Submodule R₂ M₂) : p.comap f ≤ q.comap 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.comap f p ≤ Submodule.comap f q ↔ p ≤ q
-- Submodule.comap_strictMono_of_surjective
theorem comap_strictMono_of_surjective : StrictMono (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: StrictMono (Submodule.comap f)
-- Submodule.comap_map_eq_of_injective
theorem comap_map_eq_of_injective (p : Submodule R M) : (p.map f).comap 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.Injective ⇑f) (p : Submodule R M)
Goal: Submodule.comap f (Submodule.map f p) = p
-- Submodule.comap_surjective_of_injective
theorem comap_surjective_of_injective : Function.Surjective (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.Injective ⇑f)
Goal: Function.Surjective (Submodule.comap f)
-- Submodule.map_injective_of_injective
theorem map_injective_of_injective : Function.Injective (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.Injective ⇑f)
Goal: Function.Injective (Submodule.map f)
-- Submodule.comap_inf_map_of_injective
theorem comap_inf_map_of_injective (p q : Submodule R M) : (p.map f ⊓ q.map f).comap 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.Injective ⇑f) (p q : Submodule R M)
Goal: Submodule.comap f (Submodule.map f p ⊓ Submodule.map f q) = p ⊓ q
-- Submodule.comap_iInf_map_of_injective
theorem comap_iInf_map_of_injective {ι : Sort*} (S : ι → Submodule R M) :
    (⨅ i, (S i).map f).comap f = iInf 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.Injective ⇑f) {ι : Sort u_10} (S : ι → Submodule R M)
Goal: Submodule.comap f (⨅ i, Submodule.map f (S i)) = iInf S
-- Submodule.comap_sup_map_of_injective
theorem comap_sup_map_of_injective (p q : Submodule R M) : (p.map f ⊔ q.map f).comap 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.Injective ⇑f) (p q : Submodule R M)
Goal: Submodule.comap f (Submodule.map f p ⊔ Submodule.map f q) = p ⊔ q
-- Submodule.comap_iSup_map_of_injective
theorem comap_iSup_map_of_injective {ι : Sort*} (S : ι → Submodule R M) :
    (⨆ i, (S i).map f).comap 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.Injective ⇑f) {ι : Sort u_10} (S : ι → Submodule R M)
Goal: Submodule.comap f (⨆ i, Submodule.map f (S i)) = iSup S
-- Submodule.map_le_map_iff_of_injective
theorem map_le_map_iff_of_injective (p q : Submodule R M) : p.map f ≤ q.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.Injective ⇑f) (p q : Submodule R M)
Goal: Submodule.map f p ≤ Submodule.map f q ↔ p ≤ q
-- Submodule.map_strictMono_of_injective
theorem map_strictMono_of_injective : StrictMono (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.Injective ⇑f)
Goal: StrictMono (Submodule.map f)
-- Submodule.orderIsoMapComapOfBijective_symm_apply
@[simps symm_apply apply]
def orderIsoMapComapOfBijective [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂]
    (f : F) (hf : Bijective f) : Submodule R M ≃o Submodule R₂ 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 : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) (hf : Function.Bijective ⇑f) (p : Submodule R₂ M₂)
Goal: (RelIso.symm (Submodule.orderIsoMapComapOfBijective f hf)) p = Submodule.comap f p
-- Submodule.orderIsoMapComapOfBijective_apply
@[simps symm_apply apply]
def orderIsoMapComapOfBijective [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂]
    (f : F) (hf : Bijective f) : Submodule R M ≃o Submodule R₂ 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 : Type u_9} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) (hf : Function.Bijective ⇑f) (p : Submodule R M)
Goal: (Submodule.orderIsoMapComapOfBijective f hf) p = Submodule.map f p
-- Submodule.orderIsoMapComap_symm_apply
@[simps! symm_apply apply]
def orderIsoMapComap [EquivLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) :
    Submodule R M ≃o Submodule R₂ 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 : Type u_9} [EquivLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) (p : Submodule R₂ M₂)
Goal: (RelIso.symm (Submodule.orderIsoMapComap f)) p = Submodule.comap f p
-- Submodule.orderIsoMapComap_apply
@[simps! symm_apply apply]
def orderIsoMapComap [EquivLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) :
    Submodule R M ≃o Submodule R₂ 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 : Type u_9} [EquivLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂] (f : F) (p : Submodule R M)
Goal: (Submodule.orderIsoMapComap f) p = Submodule.map f p
-- Submodule.map_inf_eq_map_inf_comap
theorem map_inf_eq_map_inf_comap [RingHomSurjective σ₁₂] {f : F} {p : Submodule R M}
    {p' : Submodule R₂ M₂} : map f p ⊓ p' = map f (p ⊓ comap 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} {p' : Submodule R₂ M₂}
Goal: Submodule.map f p ⊓ p' = Submodule.map f (p ⊓ Submodule.comap f p')
-- Submodule.map_comap_subtype
@[simp]
theorem map_comap_subtype : map p.subtype (comap p.subtype p') = p ⊓ p' 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p p' : Submodule R M)
Goal: Submodule.map p.subtype (Submodule.comap p.subtype p') = p ⊓ p'
-- Submodule.eq_zero_of_bot_submodule
theorem eq_zero_of_bot_submodule : ∀ b : (⊥ : Submodule R M), b = 0
  
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (b : ↥⊥)
Goal: b = 0
-- LinearMap.iInf_invariant
theorem _root_.LinearMap.iInf_invariant {σ : R →+* R} [RingHomSurjective σ] {ι : Sort*}
    (f : M →ₛₗ[σ] M) {p : ι → Submodule R M} (hf : ∀ i, ∀ v ∈ p i, f v ∈ p i) :
    ∀ v ∈ iInf p, f v ∈ iInf p 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {σ : R →+* R} [RingHomSurjective σ] {ι : Sort u_10} (f : M →ₛₗ[σ] M) {p : ι → Submodule R M} (hf : ∀ (i : ι), ∀ v ∈ p i, f v ∈ p i) (v : M)
Goal: v ∈ iInf p → f v ∈ iInf p
-- Submodule.disjoint_iff_comap_eq_bot
theorem disjoint_iff_comap_eq_bot {p q : Submodule R M} : Disjoint p q ↔ comap p.subtype q = ⊥ 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M}
Goal: Disjoint p q ↔ Submodule.comap p.subtype q = ⊥
-- Submodule.map_neg
@[simp]
protected theorem map_neg (f : M →ₗ[R] M₂) : map (-f) p = map f p 
Arguments: {R : Type u_1} {M : Type u_5} {M₂ : Type u_7} [Ring R] [AddCommGroup M] [Module R M] (p : Submodule R M) [AddCommGroup M₂] [Module R M₂] (f : M →ₗ[R] M₂)
Goal: Submodule.map (-f) p = Submodule.map f p
-- Submodule.comap_neg
@[simp]
lemma comap_neg {f : M →ₗ[R] M₂} {p : Submodule R M₂} :
    p.comap (-f) = p.comap f 
Arguments: {R : Type u_1} {M : Type u_5} {M₂ : Type u_7} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup M₂] [Module R M₂] {f : M →ₗ[R] M₂} {p : Submodule R M₂}
Goal: Submodule.comap (-f) p = Submodule.comap f p
-- Submodule.comap_smul
theorem comap_smul (f : V →ₗ[K] V₂) (p : Submodule K V₂) (a : K) (h : a ≠ 0) :
    p.comap (a • f) = p.comap f 
Arguments: {K : Type u_9} {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) (p : Submodule K V₂) (a : K) (h : a ≠ 0)
Goal: Submodule.comap (a • f) p = Submodule.comap f p
-- Submodule.map_smul
protected theorem map_smul (f : V →ₗ[K] V₂) (p : Submodule K V) (a : K) (h : a ≠ 0) :
    p.map (a • f) = p.map f 
Arguments: {K : Type u_9} {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) (p : Submodule K V) (a : K) (h : a ≠ 0)
Goal: Submodule.map (a • f) p = Submodule.map f p
-- Submodule.comap_smul'
theorem comap_smul' (f : V →ₗ[K] V₂) (p : Submodule K V₂) (a : K) :
    p.comap (a • f) = ⨅ _ : a ≠ 0, p.comap f 
Arguments: {K : Type u_9} {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) (p : Submodule K V₂) (a : K)
Goal: Submodule.comap (a • f) p = ⨅ (_ : a ≠ 0), Submodule.comap f p
-- Submodule.map_smul'
theorem map_smul' (f : V →ₗ[K] V₂) (p : Submodule K V) (a : K) :
    p.map (a • f) = ⨆ _ : a ≠ 0, map f p 
Arguments: {K : Type u_9} {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) (p : Submodule K V) (a : K)
Goal: Submodule.map (a • f) p = ⨆ (_ : a ≠ 0), Submodule.map f p
-- Submodule.comapSubtypeEquivOfLe_symm_apply
@[simps symm_apply]
def comapSubtypeEquivOfLe {p q : Submodule R M} (hpq : p ≤ q) : comap q.subtype p ≃ₗ[R] p 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} (hpq : p ≤ q) (x : ↥p)
Goal: (Submodule.comapSubtypeEquivOfLe hpq).symm x = ⟨⟨↑x, ⋯⟩, ⋯⟩
-- Submodule.comapSubtypeEquivOfLe_apply_coe
@[simp]
theorem comapSubtypeEquivOfLe_apply_coe {p q : Submodule R M} (hpq : p ≤ q)
    (x : comap q.subtype p) :
    (comapSubtypeEquivOfLe hpq x : M) = (x : M) 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} (hpq : p ≤ q) (x : ↥(Submodule.comap q.subtype p))
Goal: ↑((Submodule.comapSubtypeEquivOfLe hpq) x) = ↑↑x
-- Submodule.mem_map_equiv
@[simp high]
theorem mem_map_equiv {e : M ≃ₛₗ[τ₁₂] M₂} {x : M₂} :
    x ∈ p.map (e : M →ₛₗ[τ₁₂] M₂) ↔ e.symm 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] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [RingHomInvPair τ₁₂ τ₂₁] [RingHomInvPair τ₂₁ τ₁₂] (p : Submodule R M) {e : M ≃ₛₗ[τ₁₂] M₂} {x : M₂}
Goal: x ∈ Submodule.map (↑e) p ↔ e.symm x ∈ p
-- Submodule.map_equiv_eq_comap_symm
theorem map_equiv_eq_comap_symm (e : M ≃ₛₗ[τ₁₂] M₂) (K : Submodule R M) :
    K.map (e : M →ₛₗ[τ₁₂] M₂) = K.comap (e.symm : M₂ →ₛₗ[τ₂₁] M) 
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] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [RingHomInvPair τ₁₂ τ₂₁] [RingHomInvPair τ₂₁ τ₁₂] (e : M ≃ₛₗ[τ₁₂] M₂) (K : Submodule R M)
Goal: Submodule.map (↑e) K = Submodule.comap (↑e.symm) K
-- Submodule.comap_equiv_eq_map_symm
theorem comap_equiv_eq_map_symm (e : M ≃ₛₗ[τ₁₂] M₂) (K : Submodule R₂ M₂) :
    K.comap (e : M →ₛₗ[τ₁₂] M₂) = K.map (e.symm : M₂ →ₛₗ[τ₂₁] M) 
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] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [RingHomInvPair τ₁₂ τ₂₁] [RingHomInvPair τ₂₁ τ₁₂] (e : M ≃ₛₗ[τ₁₂] M₂) (K : Submodule R₂ M₂)
Goal: Submodule.comap (↑e) K = Submodule.map (↑e.symm) K
-- Submodule.map_symm_eq_iff
theorem map_symm_eq_iff (e : M ≃ₛₗ[τ₁₂] M₂) {K : Submodule R₂ M₂} :
    K.map e.symm = p ↔ p.map e = K 
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] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [RingHomInvPair τ₁₂ τ₂₁] [RingHomInvPair τ₂₁ τ₁₂] {p : Submodule R M} (e : M ≃ₛₗ[τ₁₂] M₂) {K : Submodule R₂ M₂}
Goal: Submodule.map e.symm K = p ↔ Submodule.map e p = K
-- Submodule.orderIsoMapComap_apply'
theorem orderIsoMapComap_apply' (e : M ≃ₛₗ[τ₁₂] M₂) (p : Submodule R M) :
    orderIsoMapComap e p = comap e.symm 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] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [RingHomInvPair τ₁₂ τ₂₁] [RingHomInvPair τ₂₁ τ₁₂] (e : M ≃ₛₗ[τ₁₂] M₂) (p : Submodule R M)
Goal: (Submodule.orderIsoMapComap e) p = Submodule.comap e.symm p
-- Submodule.orderIsoMapComap_symm_apply'
theorem orderIsoMapComap_symm_apply' (e : M ≃ₛₗ[τ₁₂] M₂) (p : Submodule R₂ M₂) :
    (orderIsoMapComap e).symm p = map e.symm 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] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [RingHomInvPair τ₁₂ τ₂₁] [RingHomInvPair τ₂₁ τ₁₂] (e : M ≃ₛₗ[τ₁₂] M₂) (p : Submodule R₂ M₂)
Goal: (Submodule.orderIsoMapComap e).symm p = Submodule.map e.symm p
-- Submodule.inf_comap_le_comap_add
theorem inf_comap_le_comap_add (f₁ f₂ : M →ₛₗ[τ₁₂] M₂) :
    comap f₁ q ⊓ comap f₂ q ≤ comap (f₁ + f₂) 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] {τ₁₂ : R →+* R₂} (q : Submodule R₂ M₂) (f₁ f₂ : M →ₛₗ[τ₁₂] M₂)
Goal: Submodule.comap f₁ q ⊓ Submodule.comap f₂ q ≤ Submodule.comap (f₁ + f₂) q
-- Submodule.comap_le_comap_smul
theorem comap_le_comap_smul (fₗ : N →ₗ[R] N₂) (c : R) : comap fₗ qₗ ≤ comap (c • fₗ) qₗ 
Arguments: {R : Type u_1} {N : Type u_9} {N₂ : Type u_10} [CommSemiring R] [AddCommMonoid N] [AddCommMonoid N₂] [Module R N] [Module R N₂] (qₗ : Submodule R N₂) (fₗ : N →ₗ[R] N₂) (c : R)
Goal: Submodule.comap fₗ qₗ ≤ Submodule.comap (c • fₗ) qₗ
-- LinearMap.submoduleMap_coe_apply
@[simp]
theorem submoduleMap_coe_apply (f : M →ₗ[R] M₁) {p : Submodule R M} (x : p) :
    ↑(f.submoduleMap p x) = f x 
Arguments: {R : Type u_1} {M : Type u_5} {M₁ : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] (f : M →ₗ[R] M₁) {p : Submodule R M} (x : ↥p)
Goal: ↑((f.submoduleMap p) x) = f ↑x
-- LinearMap.submoduleMap_surjective
theorem submoduleMap_surjective (f : M →ₗ[R] M₁) (p : Submodule R M) :
    Function.Surjective (f.submoduleMap p) 
Arguments: {R : Type u_1} {M : Type u_5} {M₁ : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] (f : M →ₗ[R] M₁) (p : Submodule R M)
Goal: Function.Surjective ⇑(f.submoduleMap p)
-- LinearMap.map_codRestrict
theorem map_codRestrict [RingHomSurjective σ₂₁] (p : Submodule R M) (f : M₂ →ₛₗ[σ₂₁] M) (h p') :
    Submodule.map (codRestrict p f h) p' = comap p.subtype (p'.map f) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₂₁ : R₂ →+* R} [RingHomSurjective σ₂₁] (p : Submodule R M) (f : M₂ →ₛₗ[σ₂₁] M) (h : ∀ (c : M₂), f c ∈ p) (p' : Submodule R₂ M₂)
Goal: Submodule.map (LinearMap.codRestrict p f h) p' = Submodule.comap p.subtype (Submodule.map f p')
-- LinearMap.comap_codRestrict
theorem comap_codRestrict (p : Submodule R M) (f : M₂ →ₛₗ[σ₂₁] M) (hf p') :
    Submodule.comap (codRestrict p f hf) p' = Submodule.comap f (map p.subtype p') 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₂₁ : R₂ →+* R} (p : Submodule R M) (f : M₂ →ₛₗ[σ₂₁] M) (hf : ∀ (c : M₂), f c ∈ p) (p' : Submodule R ↥p)
Goal: Submodule.comap (LinearMap.codRestrict p f hf) p' = Submodule.comap f (Submodule.map p.subtype p')
-- LinearEquiv.map_eq_comap
theorem map_eq_comap {p : Submodule R M} :
    (p.map (e : M →ₛₗ[σ₁₂] M₂) : Submodule R₂ M₂) = p.comap (e.symm : M₂ →ₛₗ[σ₂₁] M) 
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_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) {p : Submodule R M}
Goal: Submodule.map (↑e) p = Submodule.comap (↑e.symm) p
-- LinearEquiv.submoduleMap_apply
@[simp]
theorem submoduleMap_apply (p : Submodule R M) (x : p) : ↑(e.submoduleMap p x) = e 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_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) (p : Submodule R M) (x : ↥p)
Goal: ↑((e.submoduleMap p) x) = e ↑x
-- LinearEquiv.submoduleMap_symm_apply
@[simp]
theorem submoduleMap_symm_apply (p : Submodule R M)
    (x : (p.map (e : M →ₛₗ[σ₁₂] M₂) : Submodule R₂ M₂)) : ↑((e.submoduleMap p).symm x) = e.symm 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_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) (p : Submodule R M) (x : ↥(Submodule.map (↑e) p))
Goal: ↑((e.submoduleMap p).symm x) = e.symm ↑x
-- MvPolynomial.C_sub
theorem C_sub : (C (a - a') : MvPolynomial σ R) = C a - C a' 
Arguments: {R : Type u} (σ : Type u_1) (a a' : R) [CommRing R]
Goal: MvPolynomial.C (a - a') = MvPolynomial.C a - MvPolynomial.C a'
-- MvPolynomial.C_neg
theorem C_neg : (C (-a) : MvPolynomial σ R) = -C a 
Arguments: {R : Type u} (σ : Type u_1) (a : R) [CommRing R]
Goal: MvPolynomial.C (-a) = -MvPolynomial.C a
-- MvPolynomial.coeff_neg
@[simp]
theorem coeff_neg (m : σ →₀ ℕ) (p : MvPolynomial σ R) : coeff m (-p) = -coeff m p 
Arguments: {R : Type u} (σ : Type u_1) [CommRing R] (m : σ →₀ ℕ) (p : MvPolynomial σ R)
Goal: MvPolynomial.coeff m (-p) = -MvPolynomial.coeff m p
-- MvPolynomial.coeff_sub
@[simp]
theorem coeff_sub (m : σ →₀ ℕ) (p q : MvPolynomial σ R) : coeff m (p - q) = coeff m p - coeff m q 
Arguments: {R : Type u} (σ : Type u_1) [CommRing R] (m : σ →₀ ℕ) (p q : MvPolynomial σ R)
Goal: MvPolynomial.coeff m (p - q) = MvPolynomial.coeff m p - MvPolynomial.coeff m q
-- MvPolynomial.support_neg
@[simp]
theorem support_neg : (-p).support = p.support 
Arguments: {R : Type u} (σ : Type u_1) [CommRing R] {p : MvPolynomial σ R}
Goal: (-p).support = p.support
-- MvPolynomial.support_sub
theorem support_sub [DecidableEq σ] (p q : MvPolynomial σ R) :
    (p - q).support ⊆ p.support ∪ q.support 
Arguments: {R : Type u} (σ : Type u_1) [CommRing R] [DecidableEq σ] (p q : MvPolynomial σ R)
Goal: (p - q).support ⊆ p.support ∪ q.support
-- MvPolynomial.degrees_neg
theorem degrees_neg (p : MvPolynomial σ R) : (-p).degrees = p.degrees 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R)
Goal: (-p).degrees = p.degrees
-- MvPolynomial.degrees_sub
theorem degrees_sub [DecidableEq σ] (p q : MvPolynomial σ R) :
    (p - q).degrees ≤ p.degrees ⊔ q.degrees 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] [DecidableEq σ] (p q : MvPolynomial σ R)
Goal: (p - q).degrees ≤ p.degrees ⊔ q.degrees
-- MvPolynomial.vars_neg
@[simp]
theorem vars_neg : (-p).vars = p.vars 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R)
Goal: (-p).vars = p.vars
-- MvPolynomial.vars_sub_subset
theorem vars_sub_subset [DecidableEq σ] : (p - q).vars ⊆ p.vars ∪ q.vars 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R) {q : MvPolynomial σ R} [DecidableEq σ]
Goal: (p - q).vars ⊆ p.vars ∪ q.vars
-- MvPolynomial.vars_sub_of_disjoint
@[simp]
theorem vars_sub_of_disjoint [DecidableEq σ] (hpq : Disjoint p.vars q.vars) :
    (p - q).vars = p.vars ∪ q.vars 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R) {q : MvPolynomial σ R} [DecidableEq σ] (hpq : Disjoint p.vars q.vars)
Goal: (p - q).vars = p.vars ∪ q.vars
-- MvPolynomial.eval₂_sub
@[simp]
theorem eval₂_sub : (p - q).eval₂ f g = p.eval₂ f g - q.eval₂ f g 
Arguments: {R : Type u} {S : Type v} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R) {q : MvPolynomial σ R} [CommRing S] (f : R →+* S) (g : σ → S)
Goal: MvPolynomial.eval₂ f g (p - q) = MvPolynomial.eval₂ f g p - MvPolynomial.eval₂ f g q
-- MvPolynomial.eval_sub
theorem eval_sub (f : σ → R) : eval f (p - q) = eval f p - eval f q 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R) {q : MvPolynomial σ R} (f : σ → R)
Goal: (MvPolynomial.eval f) (p - q) = (MvPolynomial.eval f) p - (MvPolynomial.eval f) q
-- MvPolynomial.eval₂_neg
@[simp]
theorem eval₂_neg : (-p).eval₂ f g = -p.eval₂ f g 
Arguments: {R : Type u} {S : Type v} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R) [CommRing S] (f : R →+* S) (g : σ → S)
Goal: MvPolynomial.eval₂ f g (-p) = -MvPolynomial.eval₂ f g p