File: Mathlib/RingTheory/Regular/IsSMulRegular.lean

full_name: LinearEquiv.isSMulRegular_congr'
code: lemma LinearEquiv.isSMulRegular_congr' [AddCommMonoid N] [Module S N]
    (e : M ≃ₛₗ[σ] N) (r : R) : IsSMulRegular M r ↔ IsSMulRegular N (σ r) 
goal: IsSMulRegular M r ↔ IsSMulRegular N (σ r)

full_name: LinearEquiv.isSMulRegular_congr
code: lemma LinearEquiv.isSMulRegular_congr [AddCommMonoid N] [Module R N]
    (e : M ≃ₗ[R] N) (r : R) : IsSMulRegular M r ↔ IsSMulRegular N r 
goal: IsSMulRegular M r ↔ IsSMulRegular N r

full_name: IsSMulRegular.submodule
code: lemma IsSMulRegular.submodule [Semiring R] [AddCommMonoid M] [Module R M]
    (N : Submodule R M) (r : R) (h : IsSMulRegular M r) : IsSMulRegular N r 
goal: IsSMulRegular (↥N) r

full_name: IsSMulRegular.lTensor
code: lemma IsSMulRegular.lTensor : IsSMulRegular (M ⊗[R] M') r 
goal: IsSMulRegular (TensorProduct R M M') r

full_name: IsSMulRegular.rTensor
code: lemma IsSMulRegular.rTensor : IsSMulRegular (M' ⊗[R] M) r 
goal: IsSMulRegular (TensorProduct R M' M) r

full_name: isSMulRegular_algebraMap_iff
code: lemma isSMulRegular_algebraMap_iff [CommSemiring R] [Semiring S] [Algebra R S]
    [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] (r : R) :
    IsSMulRegular M (algebraMap R S r) ↔ IsSMulRegular M r 
goal: IsSMulRegular M ((algebraMap R S) r) ↔ IsSMulRegular M r

full_name: isSMulRegular_iff_smul_eq_zero_imp_eq_zero
code: lemma isSMulRegular_iff_smul_eq_zero_imp_eq_zero :
    IsSMulRegular M r ↔ ∀ x : M, r • x = 0 → x = 0 
goal: IsSMulRegular M r ↔ ∀ (x : M), r • x = 0 → x = 0

full_name: isSMulRegular_iff_mem_nonZeroSMulDivisors
code: lemma isSMulRegular_iff_mem_nonZeroSMulDivisors :
    IsSMulRegular M r ↔ r ∈ nonZeroSMulDivisors R M 
goal: IsSMulRegular M r ↔ r ∈ nonZeroSMulDivisors R M

full_name: isSMulRegular_of_smul_eq_zero_imp_eq_zero
code: lemma isSMulRegular_of_smul_eq_zero_imp_eq_zero
    (h : ∀ x : M, r • x = 0 → x = 0) : IsSMulRegular M r 
goal: IsSMulRegular M r

full_name: isSMulRegular_on_submodule_iff_mem_imp_smul_eq_zero_imp_eq_zero
code: lemma isSMulRegular_on_submodule_iff_mem_imp_smul_eq_zero_imp_eq_zero :
    IsSMulRegular N r ↔ ∀ x ∈ N, r • x = 0 → x = 0 
goal: IsSMulRegular (↥N) r ↔ ∀ x ∈ N, r • x = 0 → x = 0

full_name: isSMulRegular_on_quot_iff_smul_mem_implies_mem
code: lemma isSMulRegular_on_quot_iff_smul_mem_implies_mem :
    IsSMulRegular (M ⧸ N) r ↔ ∀ x : M, r • x ∈ N → x ∈ N 
goal: IsSMulRegular (M ⧸ N) r ↔ ∀ (x : M), r • x ∈ N → x ∈ N

full_name: mem_of_isSMulRegular_on_quot_of_smul_mem
code: lemma mem_of_isSMulRegular_on_quot_of_smul_mem (h1 : IsSMulRegular (M ⧸ N) r)
    {x : M} (h2 : r • x ∈ N) : x ∈ N 
goal: x ∈ N

full_name: isSMulRegular_of_range_eq_ker
code: /-- Given a left exact sequence `0 → M → M' → M''`, if `r` is regular on both
`M` and `M''` it's regular `M'` too. -/
lemma isSMulRegular_of_range_eq_ker {f : M →ₗ[R] M'} {g : M' →ₗ[R] M''}
    (hf : Function.Injective f) (hfg : LinearMap.range f = LinearMap.ker g)
    (h1 : IsSMulRegular M r) (h2 : IsSMulRegular M'' r) :
    IsSMulRegular M' r 
goal: IsSMulRegular M' r

full_name: isSMulRegular_of_isSMulRegular_on_submodule_on_quotient
code: lemma isSMulRegular_of_isSMulRegular_on_submodule_on_quotient
    (h1 : IsSMulRegular N r) (h2 : IsSMulRegular (M ⧸ N) r) : IsSMulRegular M r 
goal: IsSMulRegular M r

full_name: biUnion_associatedPrimes_eq_compl_regular
code: lemma biUnion_associatedPrimes_eq_compl_regular [IsNoetherianRing R] :
    ⋃ p ∈ associatedPrimes R M, p = { r : R | IsSMulRegular M r }ᶜ 
goal: ⋃ p ∈ associatedPrimes R M, ↑p = {r | IsSMulRegular M r}ᶜ

full_name: isSMulRegular_iff_ker_lsmul_eq_bot
code: lemma isSMulRegular_iff_ker_lsmul_eq_bot :
    IsSMulRegular M r ↔ LinearMap.ker (LinearMap.lsmul R M r) = ⊥ 
goal: IsSMulRegular M r ↔ LinearMap.ker ((LinearMap.lsmul R M) r) = ⊥

full_name: isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule
code: lemma isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule :
    IsSMulRegular N r ↔ Disjoint (LinearMap.ker (LinearMap.lsmul R M r)) N 
goal: IsSMulRegular (↥N) r ↔ Disjoint (LinearMap.ker ((LinearMap.lsmul R M) r)) N

full_name: isSMulRegular_on_quot_iff_lsmul_comap_le
code: lemma isSMulRegular_on_quot_iff_lsmul_comap_le :
    IsSMulRegular (M ⧸ N) r ↔ N.comap (LinearMap.lsmul R M r) ≤ N 
goal: IsSMulRegular (M ⧸ N) r ↔ Submodule.comap ((LinearMap.lsmul R M) r) N ≤ N

full_name: isSMulRegular_on_quot_iff_lsmul_comap_eq
code: lemma isSMulRegular_on_quot_iff_lsmul_comap_eq :
    IsSMulRegular (M ⧸ N) r ↔ N.comap (LinearMap.lsmul R M r) = N 
goal: IsSMulRegular (M ⧸ N) r ↔ Submodule.comap ((LinearMap.lsmul R M) r) N = N

full_name: IsSMulRegular.isSMulRegular_on_quot_iff_smul_top_inf_eq_smul
code: lemma IsSMulRegular.isSMulRegular_on_quot_iff_smul_top_inf_eq_smul :
    IsSMulRegular M r → (IsSMulRegular (M ⧸ N) r ↔ r • ⊤ ⊓ N ≤ r • N) 
goal: IsSMulRegular M r → (IsSMulRegular (M ⧸ N) r ↔ r • ⊤ ⊓ N ≤ r • N)

full_name: isSMulRegular_of_ker_lsmul_eq_bot
code: lemma isSMulRegular_of_ker_lsmul_eq_bot
    (h : LinearMap.ker (LinearMap.lsmul R M r) = ⊥) :
    IsSMulRegular M r 
goal: IsSMulRegular M r

full_name: smul_top_inf_eq_smul_of_isSMulRegular_on_quot
code: lemma smul_top_inf_eq_smul_of_isSMulRegular_on_quot :
    IsSMulRegular (M ⧸ N) r → r • ⊤ ⊓ N ≤ r • N 
goal: IsSMulRegular (M ⧸ N) r → r • ⊤ ⊓ N ≤ r • N

full_name: QuotSMulTop.map_first_exact_on_four_term_exact_of_isSMulRegular_last
code: lemma QuotSMulTop.map_first_exact_on_four_term_exact_of_isSMulRegular_last
    {M'''} [AddCommGroup M'''] [Module R M''']
    {r : R} {f₁ : M →ₗ[R] M'} {f₂ : M' →ₗ[R] M''} {f₃ : M'' →ₗ[R] M'''}
    (h₁₂ : Exact f₁ f₂) (h₂₃ : Exact f₂ f₃) (h : IsSMulRegular M''' r) :
    Exact (map r f₁) (map r f₂) 
goal: Function.Exact ⇑((QuotSMulTop.map r) f₁) ⇑((QuotSMulTop.map r) f₂)