statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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extension_of.ext_iff {a b : extension_of i f} :
a = b ↔ ∃ (domain_eq : a.domain = b.domain),
∀ ⦃x : a.domain⦄ ⦃y : b.domain⦄, (x : N) = y → a.to_linear_pmap x = b.to_linear_pmap y | ⟨λ r, r ▸ ⟨rfl, λ x y h, congr_arg a.to_fun $ by exact_mod_cast h⟩,
λ ⟨h1, h2⟩, extension_of.ext h1 h2⟩ | lemma | module.Baer.extension_of.ext_iff | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
chain_linear_pmap_of_chain_extension_of
{c : set (extension_of i f)} (hchain : is_chain (≤) c) :
(is_chain (≤) $ (λ x : extension_of i f, x.to_linear_pmap) '' c) | begin
rintro _ ⟨a, a_mem, rfl⟩ _ ⟨b, b_mem, rfl⟩ neq,
exact hchain a_mem b_mem (ne_of_apply_ne _ neq),
end | lemma | module.Baer.chain_linear_pmap_of_chain_extension_of | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"is_chain",
"ne_of_apply_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of.max {c : set (extension_of i f)} (hchain : is_chain (≤) c)
(hnonempty : c.nonempty) :
extension_of i f | { le := le_trans hnonempty.some.le $ (linear_pmap.le_Sup _ $ (set.mem_image _ _ _).mpr
⟨hnonempty.some, hnonempty.some_spec, rfl⟩).1,
is_extension := λ m, begin
refine eq.trans (hnonempty.some.is_extension m) _,
symmetry,
generalize_proofs _ h0 h1,
exact linear_pmap.Sup_apply
(is_chain.direc... | def | module.Baer.extension_of.max | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"is_chain",
"is_chain.directed_on",
"linear_pmap.Sup",
"linear_pmap.Sup_apply",
"linear_pmap.le_Sup",
"set.mem_image"
] | The maximal element of every nonempty chain of `extension_of i f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of.le_max {c : set (extension_of i f)} (hchain : is_chain (≤) c)
(hnonempty : c.nonempty) (a : extension_of i f) (ha : a ∈ c) :
a ≤ extension_of.max hchain hnonempty | linear_pmap.le_Sup (is_chain.directed_on $ chain_linear_pmap_of_chain_extension_of hchain) $
(set.mem_image _ _ _).mpr ⟨a, ha, rfl⟩ | lemma | module.Baer.extension_of.le_max | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"is_chain",
"is_chain.directed_on",
"linear_pmap.le_Sup",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of.inhabited : inhabited (extension_of i f) | { default :=
{ domain := i.range,
to_fun :=
{ to_fun := λ x, f x.2.some,
map_add' := λ x y, begin
have eq1 : _ + _ = (x + y).1 := congr_arg2 (+) x.2.some_spec y.2.some_spec,
rw [← map_add, ← (x + y).2.some_spec] at eq1,
rw [← fact.out (function.injective i) eq1, map_add],
... | instance | module.Baer.extension_of.inhabited | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"congr_arg2",
"linear_map.coe_mk",
"linear_map.map_smul",
"linear_pmap.mk_apply",
"ring_hom.id_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max : extension_of i f | (@zorn_nonempty_partial_order (extension_of i f) _ ⟨inhabited.default⟩
(λ c hchain hnonempty,
⟨extension_of.max hchain hnonempty, extension_of.le_max hchain hnonempty⟩)).some | def | module.Baer.extension_of_max | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"zorn_nonempty_partial_order"
] | Since every nonempty chain has a maximal element, by Zorn's lemma, there is a maximal
`extension_of i f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of_max_is_max :
∀ (a : extension_of i f), extension_of_max i f ≤ a → a = extension_of_max i f | (@zorn_nonempty_partial_order (extension_of i f) _ ⟨inhabited.default⟩
((λ c hchain hnonempty,
⟨extension_of.max hchain hnonempty, extension_of.le_max hchain hnonempty⟩))).some_spec | lemma | module.Baer.extension_of_max_is_max | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"zorn_nonempty_partial_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max_adjoin.aux1
{y : N}
(x : (extension_of_max i f).domain ⊔ submodule.span R {y}) :
∃ (a : (extension_of_max i f).domain) (b : R), x.1 = a.1 + b • y | begin
have mem1 : x.1 ∈ (_ : set _) := x.2,
rw submodule.coe_sup at mem1,
rcases mem1 with ⟨a, b, a_mem, (b_mem : b ∈ (submodule.span R _ : submodule R N)), eq1⟩,
rw submodule.mem_span_singleton at b_mem,
rcases b_mem with ⟨z, eq2⟩,
exact ⟨⟨a, a_mem⟩, z, by rw [← eq1, ← eq2]⟩,
end | lemma | module.Baer.extension_of_max_adjoin.aux1 | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"submodule",
"submodule.coe_sup",
"submodule.mem_span_singleton",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max_adjoin.fst
{y : N} (x : (extension_of_max i f).domain ⊔ submodule.span R {y}) :
(extension_of_max i f).domain | (extension_of_max_adjoin.aux1 i x).some | def | module.Baer.extension_of_max_adjoin.fst | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"submodule.span"
] | If `x ∈ M ⊔ ⟨y⟩`, then `x = m + r • y`, `fst` pick an arbitrary such `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of_max_adjoin.snd
{y : N} (x : (extension_of_max i f).domain ⊔ submodule.span R {y}) : R | (extension_of_max_adjoin.aux1 i x).some_spec.some | def | module.Baer.extension_of_max_adjoin.snd | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"submodule.span"
] | If `x ∈ M ⊔ ⟨y⟩`, then `x = m + r • y`, `snd` pick an arbitrary such `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of_max_adjoin.eqn
{y : N} (x : (extension_of_max i f).domain ⊔ submodule.span R {y}) :
↑x = ↑(extension_of_max_adjoin.fst i x) + (extension_of_max_adjoin.snd i x) • y | (extension_of_max_adjoin.aux1 i x).some_spec.some_spec | lemma | module.Baer.extension_of_max_adjoin.eqn | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max_adjoin.ideal (y : N) :
ideal R | (extension_of_max i f).domain.comap ((linear_map.id : R →ₗ[R] R).smul_right y) | def | module.Baer.extension_of_max_adjoin.ideal | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"ideal",
"linear_map.id"
] | the ideal `I = {r | r • y ∈ N}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of_max_adjoin.ideal_to (y : N) :
extension_of_max_adjoin.ideal i f y →ₗ[R] Q | { to_fun := λ z, (extension_of_max i f).to_linear_pmap ⟨(↑z : R) • y, z.prop⟩,
map_add' := λ z1 z2, by simp [← (extension_of_max i f).to_linear_pmap.map_add, add_smul],
map_smul' := λ z1 z2, by simp [← (extension_of_max i f).to_linear_pmap.map_smul, mul_smul]; refl } | def | module.Baer.extension_of_max_adjoin.ideal_to | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"add_smul"
] | A linear map `I ⟶ Q` by `x ↦ f' (x • y)` where `f'` is the maximal extension | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of_max_adjoin.extend_ideal_to (h : module.Baer R Q) (y : N) : R →ₗ[R] Q | (h (extension_of_max_adjoin.ideal i f y) (extension_of_max_adjoin.ideal_to i f y)).some | def | module.Baer.extension_of_max_adjoin.extend_ideal_to | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"module.Baer"
] | Since we assumed `Q` being Baer, the linear map `x ↦ f' (x • y) : I ⟶ Q` extends to `R ⟶ Q`,
call this extended map `φ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of_max_adjoin.extend_ideal_to_is_extension (h : module.Baer R Q) (y : N) :
∀ (x : R) (mem : x ∈ extension_of_max_adjoin.ideal i f y),
extension_of_max_adjoin.extend_ideal_to i f h y x =
extension_of_max_adjoin.ideal_to i f y ⟨x, mem⟩ | (h (extension_of_max_adjoin.ideal i f y) (extension_of_max_adjoin.ideal_to i f y)).some_spec | lemma | module.Baer.extension_of_max_adjoin.extend_ideal_to_is_extension | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"module.Baer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max_adjoin.extend_ideal_to_wd' (h : module.Baer R Q) {y : N} (r : R)
(eq1 : r • y = 0) :
extension_of_max_adjoin.extend_ideal_to i f h y r = 0 | begin
rw extension_of_max_adjoin.extend_ideal_to_is_extension i f h y r
(by rw eq1; exact submodule.zero_mem _ : r • y ∈ _),
simp only [extension_of_max_adjoin.ideal_to, linear_map.coe_mk, eq1, subtype.coe_mk,
← zero_mem_class.zero_def, (extension_of_max i f).to_linear_pmap.map_zero]
end | lemma | module.Baer.extension_of_max_adjoin.extend_ideal_to_wd' | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"linear_map.coe_mk",
"module.Baer",
"submodule.zero_mem",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max_adjoin.extend_ideal_to_wd (h : module.Baer R Q) {y : N} (r r' : R)
(eq1 : r • y = r' • y) :
extension_of_max_adjoin.extend_ideal_to i f h y r =
extension_of_max_adjoin.extend_ideal_to i f h y r' | begin
rw [← sub_eq_zero, ← map_sub],
convert extension_of_max_adjoin.extend_ideal_to_wd' i f h (r - r') _,
rw [sub_smul, sub_eq_zero, eq1],
end | lemma | module.Baer.extension_of_max_adjoin.extend_ideal_to_wd | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"module.Baer",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max_adjoin.extend_ideal_to_eq (h : module.Baer R Q) {y : N} (r : R)
(hr : r • y ∈ (extension_of_max i f).domain) :
extension_of_max_adjoin.extend_ideal_to i f h y r =
(extension_of_max i f).to_linear_pmap ⟨r • y, hr⟩ | by simp only [extension_of_max_adjoin.extend_ideal_to_is_extension i f h _ _ hr,
extension_of_max_adjoin.ideal_to, linear_map.coe_mk, subtype.coe_mk] | lemma | module.Baer.extension_of_max_adjoin.extend_ideal_to_eq | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"linear_map.coe_mk",
"module.Baer",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max_adjoin.extension_to_fun (h : module.Baer R Q)
{y : N} :
(extension_of_max i f).domain ⊔ submodule.span R {y} → Q | λ x, (extension_of_max i f).to_linear_pmap (extension_of_max_adjoin.fst i x) +
extension_of_max_adjoin.extend_ideal_to i f h y (extension_of_max_adjoin.snd i x) | def | module.Baer.extension_of_max_adjoin.extension_to_fun | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"module.Baer",
"submodule.span"
] | We can finally define a linear map `M ⊔ ⟨y⟩ ⟶ Q` by `x + r • y ↦ f x + φ r` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of_max_adjoin.extension_to_fun_wd (h : module.Baer R Q)
{y : N} (x : (extension_of_max i f).domain ⊔ submodule.span R {y})
(a : (extension_of_max i f).domain) (r : R)
(eq1 : ↑x = ↑a + r • y) :
extension_of_max_adjoin.extension_to_fun i f h x =
(extension_of_max i f).to_linear_pmap a +
extens... | begin
cases a with a ha,
rw subtype.coe_mk at eq1,
have eq2 : (extension_of_max_adjoin.fst i x - a : N) = (r - extension_of_max_adjoin.snd i x) • y,
{ rwa [extension_of_max_adjoin.eqn, ← sub_eq_zero, ←sub_sub_sub_eq,
sub_eq_zero, ← sub_smul] at eq1 },
have eq3 := extension_of_max_adjoin.extend_ideal_to_... | lemma | module.Baer.extension_of_max_adjoin.extension_to_fun_wd | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"module.Baer",
"sub_smul",
"submodule.span",
"submodule.sub_mem",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max_adjoin (h : module.Baer R Q) (y : N) :
extension_of i f | { domain := (extension_of_max i f).domain ⊔ submodule.span R {y},
le := le_trans (extension_of_max i f).le le_sup_left,
to_fun :=
{ to_fun := extension_of_max_adjoin.extension_to_fun i f h,
map_add' := λ a b, begin
have eq1 : ↑a + ↑b =
↑((extension_of_max_adjoin.fst i a) + (extension_of_... | def | module.Baer.extension_of_max_adjoin | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"add_smul",
"le_sup_left",
"linear_map.coe_mk",
"linear_map.map_smul",
"linear_pmap.map_add",
"linear_pmap.map_smul",
"linear_pmap.mk_apply",
"module.Baer",
"ring_hom.id_apply",
"smul_add",
"smul_eq_mul",
"submodule.span"
] | The linear map `M ⊔ ⟨y⟩ ⟶ Q` by `x + r • y ↦ f x + φ r` is an extension of `f` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of_max_le (h : module.Baer R Q) {y : N} :
extension_of_max i f ≤ extension_of_max_adjoin i f h y | ⟨le_sup_left, λ x x' EQ, begin
symmetry,
change extension_of_max_adjoin.extension_to_fun i f h _ = _,
rw [extension_of_max_adjoin.extension_to_fun_wd i f h x' x 0 (by simp [EQ]), map_zero, add_zero],
end⟩ | lemma | module.Baer.extension_of_max_le | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"module.Baer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_of_max_to_submodule_eq_top (h : module.Baer R Q) :
(extension_of_max i f).domain = ⊤ | begin
refine submodule.eq_top_iff'.mpr (λ y, _),
rw [← extension_of_max_is_max i f _ (extension_of_max_le i f h), extension_of_max_adjoin,
submodule.mem_sup],
exact ⟨0, submodule.zero_mem _, y, submodule.mem_span_singleton_self _, zero_add _⟩
end | lemma | module.Baer.extension_of_max_to_submodule_eq_top | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"module.Baer",
"submodule.mem_span_singleton_self",
"submodule.mem_sup",
"submodule.zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (h : module.Baer R Q) :
module.injective R Q | { out := λ X Y ins1 ins2 ins3 ins4 i hi f, begin
haveI : fact (function.injective i) := ⟨hi⟩,
exact ⟨{ to_fun := λ y, (extension_of_max i f).to_linear_pmap
⟨y, (extension_of_max_to_submodule_eq_top i f h).symm ▸ trivial⟩,
map_add' := λ x y, by { rw ← linear_pmap.map_add, congr, },
map_smul' ... | theorem | module.Baer.injective | algebra.module | src/algebra/module/injective.lean | [
"category_theory.preadditive.injective",
"algebra.category.Module.epi_mono",
"ring_theory.ideal.basic",
"linear_algebra.linear_pmap"
] | [
"fact",
"linear_pmap.map_add",
"linear_pmap.map_smul",
"module.Baer",
"module.injective"
] | **Baer's criterion** for injective module : a Baer module is an injective module, i.e. if every
linear map from an ideal can be extended, then the module is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_linear_map (R : Type u) {M : Type v} {M₂ : Type w}
[semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂]
(f : M → M₂) : Prop | (map_add : ∀ x y, f (x + y) = f x + f y)
(map_smul : ∀ (c : R) x, f (c • x) = c • f x) | structure | is_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"module",
"semiring"
] | A map `f` between modules over a semiring is linear if it satisfies the two properties
`f (x + y) = f x + f y` and `f (c • x) = c • f x`. The predicate `is_linear_map R f` asserts this
property. A bundled version is available with `linear_map`, and should be favored over
`is_linear_map` most of the time. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map {R : Type*} {S : Type*} [semiring R] [semiring S] (σ : R →+* S)
(M : Type*) (M₂ : Type*)
[add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module S M₂]
extends add_hom M M₂ | (map_smul' : ∀ (r : R) (x : M), to_fun (r • x) = (σ r) • to_fun x) | structure | linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"add_hom",
"module",
"semiring"
] | A map `f` between an `R`-module and an `S`-module over a ring homomorphism `σ : R →+* S`
is semilinear if it satisfies the two properties `f (x + y) = f x + f y` and
`f (c • x) = (σ c) • f x`. Elements of `linear_map σ M M₂` (available under the notation
`M →ₛₗ[σ] M₂`) are bundled versions of such maps. For plain linea... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semilinear_map_class (F : Type*) {R S : out_param Type*} [semiring R] [semiring S]
(σ : out_param $ R →+* S) (M M₂ : out_param Type*)
[add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module S M₂]
extends add_hom_class F M M₂ | (map_smulₛₗ : ∀ (f : F) (r : R) (x : M), f (r • x) = (σ r) • f x) | class | semilinear_map_class | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"add_hom_class",
"module",
"semiring"
] | `semilinear_map_class F σ M M₂` asserts `F` is a type of bundled `σ`-semilinear maps `M → M₂`.
See also `linear_map_class F R M M₂` for the case where `σ` is the identity map on `R`.
A map `f` between an `R`-module and an `S`-module over a ring homomorphism `σ : R →+* S`
is semilinear if it satisfies the two properti... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map_class (F : Type*) (R M M₂ : out_param Type*)
[semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] | semilinear_map_class F (ring_hom.id R) M M₂ | abbreviation | linear_map_class | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"module",
"ring_hom.id",
"semilinear_map_class",
"semiring"
] | `linear_map_class F R M M₂` asserts `F` is a type of bundled `R`-linear maps `M → M₂`.
This is an abbreviation for `semilinear_map_class F (ring_hom.id R) M M₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_smul_inv {σ' : S →+* R} [ring_hom_inv_pair σ σ'] (c : S) (x : M) :
c • f x = f (σ' c • x) | by simp | lemma | semilinear_map_class.map_smul_inv | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"ring_hom_inv_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_distrib_mul_action_hom (f : M →ₗ[R] M₂) : distrib_mul_action_hom R M M₂ | { map_zero' := show f 0 = 0, from map_zero f, ..f } | def | linear_map.to_distrib_mul_action_hom | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"distrib_mul_action_hom"
] | The `distrib_mul_action_hom` underlying a `linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : M →ₛₗ[σ] M₃} : f.to_fun = (f : M → M₃) | rfl | lemma | linear_map.to_fun_eq_coe | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : M →ₛₗ[σ] M₃} (h : ∀ x, f x = g x) : f = g | fun_like.ext f g h | theorem | linear_map.ext | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : M →ₛₗ[σ] M₃) (f' : M → M₃) (h : f' = ⇑f) : M →ₛₗ[σ] M₃ | { to_fun := f',
map_add' := h.symm ▸ f.map_add',
map_smul' := h.symm ▸ f.map_smul' } | def | linear_map.copy | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | Copy of a `linear_map` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : M →ₛₗ[σ] M₃) (f' : M → M₃) (h : f' = ⇑f) : ⇑(f.copy f' h) = f' | rfl | lemma | linear_map.coe_copy | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : M →ₛₗ[σ] M₃) (f' : M → M₃) (h : f' = ⇑f) : f.copy f' h = f | fun_like.ext' h | lemma | linear_map.copy_eq | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply {R S : Type*} [semiring R] [semiring S] (σ : R →+* S)
(M M₃ : Type*) [add_comm_monoid M] [add_comm_monoid M₃] [module R M] [module S M₃]
(f : M →ₛₗ[σ] M₃) : M → M₃ | f
initialize_simps_projections linear_map (to_fun → apply) | def | linear_map.simps.apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"linear_map",
"module",
"semiring"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk {σ : R →+* S} (f : M → M₃) (h₁ h₂) :
((linear_map.mk f h₁ h₂ : M →ₛₗ[σ] M₃) : M → M₃) = f | rfl | lemma | linear_map.coe_mk | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : M →ₗ[R] M | { to_fun := id, ..distrib_mul_action_hom.id R } | def | linear_map.id | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"distrib_mul_action_hom.id"
] | Identity map as a `linear_map` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply (x : M) :
@id R M _ _ _ x = x | rfl | lemma | linear_map.id_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_coe : ((linear_map.id : M →ₗ[R] M) : M → M) = _root_.id | rfl | lemma | linear_map.id_coe | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_linear : is_linear_map R fₗ | ⟨fₗ.map_add', fₗ.map_smul'⟩ | theorem | linear_map.is_linear | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"is_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @injective (M →ₛₗ[σ] M₃) (M → M₃) coe_fn | fun_like.coe_injective | theorem | linear_map.coe_injective | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg {x x' : M} : x = x' → f x = f x' | fun_like.congr_arg f | lemma | linear_map.congr_arg | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"fun_like.congr_arg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun (h : f = g) (x : M) : f x = g x | fun_like.congr_fun h x | lemma | linear_map.congr_fun | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"fun_like.congr_fun"
] | If two linear maps are equal, they are equal at each point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_iff : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | theorem | linear_map.ext_iff | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : M →ₛₗ[σ] M₃) (h₁ h₂) :
(linear_map.mk f h₁ h₂ : M →ₛₗ[σ] M₃) = f | ext $ λ _, rfl | lemma | linear_map.mk_coe | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (x y : M) : f (x + y) = f x + f y | map_add f x y | lemma | linear_map.map_add | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : f 0 = 0 | map_zero f | lemma | linear_map.map_zero | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smulₛₗ (c : R) (x : M) : f (c • x) = (σ c) • f x | map_smulₛₗ f c x | lemma | linear_map.map_smulₛₗ | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul (c : R) (x : M) : fₗ (c • x) = c • fₗ x | map_smul fₗ c x | lemma | linear_map.map_smul | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff (h : function.injective f) {x : M} : f x = 0 ↔ x = 0 | ⟨λ w, by { apply h, simp [w], }, λ w, by { subst w, simp, }⟩ | lemma | linear_map.map_eq_zero_iff | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.image_smul_setₛₗ [semilinear_map_class F σ M M₃] (c : R) (s : set M) :
h '' (c • s) = (σ c) • h '' s | begin
apply set.subset.antisymm,
{ rintros x ⟨y, ⟨z, zs, rfl⟩, rfl⟩,
exact ⟨h z, set.mem_image_of_mem _ zs, (map_smulₛₗ _ _ _).symm ⟩ },
{ rintros x ⟨y, ⟨z, hz, rfl⟩, rfl⟩,
exact (set.mem_image _ _ _).2 ⟨c • z, set.smul_mem_smul_set hz, map_smulₛₗ _ _ _⟩ }
end | lemma | image_smul_setₛₗ | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"semilinear_map_class",
"set.mem_image",
"set.mem_image_of_mem",
"set.smul_mem_smul_set",
"set.subset.antisymm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.preimage_smul_setₛₗ [semilinear_map_class F σ M M₃] {c : R} (hc : is_unit c)
(s : set M₃) : h ⁻¹' (σ c • s) = c • h ⁻¹' s | begin
apply set.subset.antisymm,
{ rintros x ⟨y, ys, hy⟩,
refine ⟨(hc.unit.inv : R) • x, _, _⟩,
{ simp only [←hy, smul_smul, set.mem_preimage, units.inv_eq_coe_inv, map_smulₛₗ h, ← map_mul,
is_unit.coe_inv_mul, one_smul, map_one, ys] },
{ simp only [smul_smul, is_unit.mul_coe_inv, one_smul, unit... | lemma | preimage_smul_setₛₗ | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"is_unit",
"is_unit.coe_inv_mul",
"is_unit.mul_coe_inv",
"map_mul",
"map_one",
"one_smul",
"ring_hom.id_apply",
"semilinear_map_class",
"set.mem_preimage",
"set.subset.antisymm",
"smul_smul",
"units.inv_eq_coe_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.image_smul_set [linear_map_class F R M M₂] (c : R) (s : set M) :
h '' (c • s) = c • h '' s | image_smul_setₛₗ _ _ _ h c s | lemma | image_smul_set | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"image_smul_setₛₗ",
"linear_map_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.preimage_smul_set [linear_map_class F R M M₂] {c : R} (hc : is_unit c) (s : set M₂) :
h ⁻¹' (c • s) = c • h ⁻¹' s | preimage_smul_setₛₗ _ _ _ h hc s | lemma | preimage_smul_set | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"is_unit",
"linear_map_class",
"preimage_smul_setₛₗ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compatible_smul (R S : Type*) [semiring S] [has_smul R M]
[module S M] [has_smul R M₂] [module S M₂] | (map_smul : ∀ (fₗ : M →ₗ[S] M₂) (c : R) (x : M), fₗ (c • x) = c • fₗ x) | class | linear_map.compatible_smul | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"has_smul",
"module",
"semiring"
] | A typeclass for `has_smul` structures which can be moved through a `linear_map`.
This typeclass is generated automatically from a `is_scalar_tower` instance, but exists so that
we can also add an instance for `add_comm_group.int_module`, allowing `z •` to be moved even if
`R` does not support negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_scalar_tower.compatible_smul
{R S : Type*} [semiring S] [has_smul R S]
[has_smul R M] [module S M] [is_scalar_tower R S M]
[has_smul R M₂] [module S M₂] [is_scalar_tower R S M₂] : compatible_smul M M₂ R S | ⟨λ fₗ c x, by rw [← smul_one_smul S c x, ← smul_one_smul S c (fₗ x), map_smul]⟩ | instance | linear_map.is_scalar_tower.compatible_smul | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"has_smul",
"is_scalar_tower",
"module",
"semiring",
"smul_one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul_of_tower {R S : Type*} [semiring S] [has_smul R M]
[module S M] [has_smul R M₂] [module S M₂]
[compatible_smul M M₂ R S] (fₗ : M →ₗ[S] M₂) (c : R) (x : M) :
fₗ (c • x) = c • fₗ x | compatible_smul.map_smul fₗ c x | lemma | linear_map.map_smul_of_tower | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"has_smul",
"module",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_hom : M →+ M₃ | { to_fun := f,
map_zero' := f.map_zero,
map_add' := f.map_add } | def | linear_map.to_add_monoid_hom | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | convert a linear map to an additive map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add_monoid_hom_coe : ⇑f.to_add_monoid_hom = f | rfl | lemma | linear_map.to_add_monoid_hom_coe | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars (fₗ : M →ₗ[S] M₂) : M →ₗ[R] M₂ | { to_fun := fₗ,
map_add' := fₗ.map_add,
map_smul' := fₗ.map_smul_of_tower } | def | linear_map.restrict_scalars | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"restrict_scalars"
] | If `M` and `M₂` are both `R`-modules and `S`-modules and `R`-module structures
are defined by an action of `R` on `S` (formally, we have two scalar towers), then any `S`-linear
map from `M` to `M₂` is `R`-linear.
See also `linear_map.map_smul_of_tower`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_scalars (fₗ : M →ₗ[S] M₂) : ⇑(restrict_scalars R fₗ) = fₗ | rfl | lemma | linear_map.coe_restrict_scalars | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_apply (fₗ : M →ₗ[S] M₂) (x) : restrict_scalars R fₗ x = fₗ x | rfl | lemma | linear_map.restrict_scalars_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_injective :
function.injective (restrict_scalars R : (M →ₗ[S] M₂) → (M →ₗ[R] M₂)) | λ fₗ gₗ h, ext (linear_map.congr_fun h : _) | lemma | linear_map.restrict_scalars_injective | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"linear_map.congr_fun",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_inj (fₗ gₗ : M →ₗ[S] M₂) :
fₗ.restrict_scalars R = gₗ.restrict_scalars R ↔ fₗ = gₗ | (restrict_scalars_injective R).eq_iff | lemma | linear_map.restrict_scalars_inj | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_hom_injective :
function.injective (to_add_monoid_hom : (M →ₛₗ[σ] M₃) → (M →+ M₃)) | λ f g h, ext $ add_monoid_hom.congr_fun h | theorem | linear_map.to_add_monoid_hom_injective | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_ring {f g : R →ₛₗ[σ] M₃} (h : f 1 = g 1) : f = g | ext $ λ x, by rw [← mul_one x, ← smul_eq_mul, f.map_smulₛₗ, g.map_smulₛₗ, h] | theorem | linear_map.ext_ring | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"mul_one",
"smul_eq_mul"
] | If two `σ`-linear maps from `R` are equal on `1`, then they are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_ring_iff {σ : R →+* R} {f g : R →ₛₗ[σ] M} : f = g ↔ f 1 = g 1 | ⟨λ h, h ▸ rfl, ext_ring⟩ | theorem | linear_map.ext_ring_iff | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_ring_op {σ : Rᵐᵒᵖ →+* S} {f g : R →ₛₗ[σ] M₃} (h : f 1 = g 1) : f = g | ext $ λ x, by rw [← one_mul x, ← op_smul_eq_mul, f.map_smulₛₗ, g.map_smulₛₗ, h] | theorem | linear_map.ext_ring_op | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"one_mul",
"op_smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ring_hom.to_semilinear_map (f : R →+* S) : R →ₛₗ[f] S | { to_fun := f,
map_smul' := f.map_mul,
.. f} | def | ring_hom.to_semilinear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | Interpret a `ring_hom` `f` as an `f`-semilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp : M₁ →ₛₗ[σ₁₃] M₃ | { to_fun := f ∘ g,
map_add' := by simp only [map_add, forall_const, eq_self_iff_true, comp_app],
map_smul' := λ r x, by rw [comp_app, map_smulₛₗ, map_smulₛₗ, ring_hom_comp_triple.comp_apply] } | def | linear_map.comp | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"forall_const",
"ring_hom_comp_triple.comp_apply"
] | Composition of two linear maps is a linear map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_apply (x : M₁) : f.comp g x = f (g x) | rfl | lemma | linear_map.comp_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp : (f.comp g : M₁ → M₃) = f ∘ g | rfl | lemma | linear_map.coe_comp | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id : f.comp id = f | linear_map.ext $ λ x, rfl | theorem | linear_map.comp_id | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"linear_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp : id.comp f = f | linear_map.ext $ λ x, rfl | theorem | linear_map.id_comp | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"linear_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right (hg : function.surjective g) :
f.comp g = f'.comp g ↔ f = f' | ⟨λ h, ext $ hg.forall.2 (ext_iff.1 h), λ h, h ▸ rfl⟩ | theorem | linear_map.cancel_right | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left (hf : function.injective f) :
f.comp g = f.comp g' ↔ g = g' | ⟨λ h, ext $ λ x, hf $ by rw [← comp_apply, h, comp_apply], λ h, h ▸ rfl⟩ | theorem | linear_map.cancel_left | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse [module R M] [module S M₂] {σ : R →+* S} {σ' : S →+* R} [ring_hom_inv_pair σ σ']
(f : M →ₛₗ[σ] M₂) (g : M₂ → M) (h₁ : left_inverse g f) (h₂ : right_inverse g f) :
M₂ →ₛₗ[σ'] M | by dsimp [left_inverse, function.right_inverse] at h₁ h₂; exact
{ to_fun := g,
map_add' := λ x y, by { rw [← h₁ (g (x + y)), ← h₁ (g x + g y)]; simp [h₂] },
map_smul' := λ a b, by { rw [← h₁ (g (a • b)), ← h₁ ((σ' a) • g b)], simp [h₂] } } | def | linear_map.inverse | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module",
"ring_hom_inv_pair"
] | If a function `g` is a left and right inverse of a linear map `f`, then `g` is linear itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_neg (x : M) : f (- x) = - f x | map_neg f x | lemma | linear_map.map_neg | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (x y : M) : f (x - y) = f x - f y | map_sub f x y | lemma | linear_map.map_sub | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compatible_smul.int_module
{S : Type*} [semiring S] [module S M] [module S M₂] : compatible_smul M M₂ ℤ S | ⟨λ fₗ c x, begin
induction c using int.induction_on,
case hz : { simp },
case hp : n ih { simp [add_smul, ih] },
case hn : n ih { simp [sub_smul, ih] }
end⟩ | instance | linear_map.compatible_smul.int_module | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_smul",
"ih",
"int.induction_on",
"module",
"semiring",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compatible_smul.units {R S : Type*}
[monoid R] [mul_action R M] [mul_action R M₂] [semiring S] [module S M] [module S M₂]
[compatible_smul M M₂ R S] :
compatible_smul M M₂ Rˣ S | ⟨λ fₗ c x, (compatible_smul.map_smul fₗ (c : R) x : _)⟩ | instance | linear_map.compatible_smul.units | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module",
"monoid",
"mul_action",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_hom.to_linear_map {R S : Type*} [semiring R] [semiring S] (g : R →+* S) :
(by haveI := comp_hom S g; exact (R →ₗ[R] S)) | by exact
{ to_fun := (g : R → S),
map_add' := g.map_add,
map_smul' := g.map_mul } | def | module.comp_hom.to_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"semiring"
] | `g : R →+* S` is `R`-linear when the module structure on `S` is `module.comp_hom S g` . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_map (fₗ : M →+[R] M₂) : M →ₗ[R] M₂ | { ..fₗ } | def | distrib_mul_action_hom.to_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | A `distrib_mul_action_hom` between two modules is a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_map_eq_coe (f : M →+[R] M₂) :
f.to_linear_map = ↑f | rfl | lemma | distrib_mul_action_hom.to_linear_map_eq_coe | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_linear_map (f : M →+[R] M₂) :
((f : M →ₗ[R] M₂) : M → M₂) = f | rfl | lemma | distrib_mul_action_hom.coe_to_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_injective {f g : M →+[R] M₂} (h : (f : M →ₗ[R] M₂) = (g : M →ₗ[R] M₂)) :
f = g | by { ext m, exact linear_map.congr_fun h m, } | lemma | distrib_mul_action_hom.to_linear_map_injective | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"linear_map.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (f : M → M₂) (H : is_linear_map R f) : M →ₗ[R] M₂ | { to_fun := f, map_add' := H.1, map_smul' := H.2 } | def | is_linear_map.mk' | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"is_linear_map",
"mk'"
] | Convert an `is_linear_map` predicate to a `linear_map` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk'_apply {f : M → M₂} (H : is_linear_map R f) (x : M) :
mk' f H x = f x | rfl | theorem | is_linear_map.mk'_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"is_linear_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_linear_map_smul {R M : Type*} [comm_semiring R] [add_comm_monoid M] [module R M]
(c : R) :
is_linear_map R (λ (z : M), c • z) | begin
refine is_linear_map.mk (smul_add c) _,
intros _ _,
simp only [smul_smul, mul_comm]
end | lemma | is_linear_map.is_linear_map_smul | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"comm_semiring",
"is_linear_map",
"module",
"mul_comm",
"smul_add",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_linear_map_smul' {R M : Type*} [semiring R] [add_comm_monoid M] [module R M] (a : M) :
is_linear_map R (λ (c : R), c • a) | is_linear_map.mk (λ x y, add_smul x y a) (λ x y, mul_smul x y a) | lemma | is_linear_map.is_linear_map_smul' | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"add_smul",
"is_linear_map",
"module",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : f (0 : M) = (0 : M₂) | (lin.mk' f).map_zero | lemma | is_linear_map.map_zero | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_linear_map_neg :
is_linear_map R (λ (z : M), -z) | is_linear_map.mk neg_add (λ x y, (smul_neg x y).symm) | lemma | is_linear_map.is_linear_map_neg | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"is_linear_map",
"smul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (x : M) : f (- x) = - f x | (lin.mk' f).map_neg x | lemma | is_linear_map.map_neg | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (x y) : f (x - y) = f x - f y | (lin.mk' f).map_sub x y | lemma | is_linear_map.map_sub | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.End (R : Type u) (M : Type v)
[semiring R] [add_comm_monoid M] [module R M] | M →ₗ[R] M | abbreviation | module.End | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"module",
"semiring"
] | Linear endomorphisms of a module, with associated ring structure
`module.End.semiring` and algebra structure `module.End.algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.to_nat_linear_map [add_comm_monoid M] [add_comm_monoid M₂] (f : M →+ M₂) :
M →ₗ[ℕ] M₂ | { to_fun := f, map_add' := f.map_add, map_smul' := map_nsmul f } | def | add_monoid_hom.to_nat_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"map_nsmul"
] | Reinterpret an additive homomorphism as a `ℕ`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.to_nat_linear_map_injective [add_comm_monoid M] [add_comm_monoid M₂] :
function.injective (@add_monoid_hom.to_nat_linear_map M M₂ _ _) | by { intros f g h, ext, exact linear_map.congr_fun h x } | lemma | add_monoid_hom.to_nat_linear_map_injective | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"add_monoid_hom.to_nat_linear_map",
"linear_map.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.to_int_linear_map [add_comm_group M] [add_comm_group M₂] (f : M →+ M₂) :
M →ₗ[ℤ] M₂ | { to_fun := f, map_add' := f.map_add, map_smul' := map_zsmul f } | def | add_monoid_hom.to_int_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_group",
"map_zsmul"
] | Reinterpret an additive homomorphism as a `ℤ`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.to_int_linear_map_injective [add_comm_group M] [add_comm_group M₂] :
function.injective (@add_monoid_hom.to_int_linear_map M M₂ _ _) | by { intros f g h, ext, exact linear_map.congr_fun h x } | lemma | add_monoid_hom.to_int_linear_map_injective | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_group",
"add_monoid_hom.to_int_linear_map",
"linear_map.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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