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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from_localized_module' : localized_module S M → M' | λ p, p.lift_on (λ x, (is_localized_module.map_units f x.2).unit⁻¹ (f x.1))
begin
rintros ⟨a, b⟩ ⟨a', b'⟩ ⟨c, eq1⟩,
dsimp,
generalize_proofs h1 h2,
erw [module.End_algebra_map_is_unit_inv_apply_eq_iff, ←h2.unit⁻¹.1.map_smul,
module.End_algebra_map_is_unit_inv_apply_eq_iff', ←linear_map.map_smul, ←linear_map.... | def | is_localized_module.from_localized_module' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"module.End_algebra_map_is_unit_inv_apply_eq_iff'"
] | If `(M', f : M ⟶ M')` satisfies universal property of localized module, there is a canonical map
`localized_module S M ⟶ M'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_localized_module'_mk (m : M) (s : S) :
from_localized_module' S f (localized_module.mk m s) =
(is_localized_module.map_units f s).unit⁻¹ (f m) | rfl | lemma | is_localized_module.from_localized_module'_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_localized_module'_add (x y : localized_module S M) :
from_localized_module' S f (x + y) =
from_localized_module' S f x + from_localized_module' S f y | localized_module.induction_on₂ begin
intros a a' b b',
simp only [localized_module.mk_add_mk, from_localized_module'_mk],
generalize_proofs h1 h2 h3,
erw [module.End_algebra_map_is_unit_inv_apply_eq_iff, smul_add, ←h2.unit⁻¹.1.map_smul,
←h3.unit⁻¹.1.map_smul, map_add],
congr' 1,
all_goals { erw [module.... | lemma | is_localized_module.from_localized_module'_add | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"localized_module.induction_on₂",
"localized_module.mk_add_mk",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"module.End_algebra_map_is_unit_inv_apply_eq_iff'",
"mul_comm",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_localized_module'_smul (r : R) (x : localized_module S M) :
r • from_localized_module' S f x = from_localized_module' S f (r • x) | localized_module.induction_on begin
intros a b,
rw [from_localized_module'_mk, localized_module.smul'_mk, from_localized_module'_mk],
generalize_proofs h1, erw [f.map_smul, h1.unit⁻¹.1.map_smul], refl, | lemma | from_localized_module'_smul | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"localized_module.induction_on",
"localized_module.smul'_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_localized_module : localized_module S M →ₗ[R] M' | { to_fun := from_localized_module' S f,
map_add' := from_localized_module'_add S f,
map_smul' := λ r x, by rw [from_localized_module'_smul, ring_hom.id_apply] } | def | from_localized_module | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"from_localized_module'_smul",
"localized_module",
"ring_hom.id_apply"
] | If `(M', f : M ⟶ M')` satisfies universal property of localized module, there is a canonical map
`localized_module S M ⟶ M'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_localized_module_mk (m : M) (s : S) :
from_localized_module S f (localized_module.mk m s) =
(is_localized_module.map_units f s).unit⁻¹ (f m) | rfl | lemma | from_localized_module_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"from_localized_module",
"localized_module.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_localized_module.inj : function.injective $ from_localized_module S f | λ x y eq1,
begin
induction x using localized_module.induction_on with a b,
induction y using localized_module.induction_on with a' b',
simp only [from_localized_module_mk] at eq1,
generalize_proofs h1 h2 at eq1,
erw [module.End_algebra_map_is_unit_inv_apply_eq_iff, ←linear_map.map_smul,
module.End_algebra... | lemma | from_localized_module.inj | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"from_localized_module",
"from_localized_module_mk",
"localized_module.induction_on",
"localized_module.mk_eq",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"module.End_algebra_map_is_unit_inv_apply_eq_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_localized_module.surj : function.surjective $ from_localized_module S f | λ x, let ⟨⟨m, s⟩, eq1⟩ := is_localized_module.surj S f x in ⟨localized_module.mk m s,
by { rw [from_localized_module_mk, module.End_algebra_map_is_unit_inv_apply_eq_iff, ←eq1], refl }⟩ | lemma | from_localized_module.surj | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"from_localized_module",
"from_localized_module_mk",
"module.End_algebra_map_is_unit_inv_apply_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_localized_module.bij : function.bijective $ from_localized_module S f | ⟨from_localized_module.inj _ _, from_localized_module.surj _ _⟩ | lemma | from_localized_module.bij | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"from_localized_module",
"from_localized_module.surj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso : localized_module S M ≃ₗ[R] M' | { ..from_localized_module S f,
..equiv.of_bijective (from_localized_module S f) $ from_localized_module.bij _ _} | def | iso | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"equiv.of_bijective",
"from_localized_module",
"from_localized_module.bij",
"localized_module"
] | If `(M', f : M ⟶ M')` satisfies universal property of localized module, then `M'` is isomorphic to
`localized_module S M` as an `R`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_apply_mk (m : M) (s : S) :
iso S f (localized_module.mk m s) = (is_localized_module.map_units f s).unit⁻¹ (f m) | rfl | lemma | iso_apply_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"iso",
"localized_module.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_symm_apply_aux (m : M') :
(iso S f).symm m = localized_module.mk (is_localized_module.surj S f m).some.1
(is_localized_module.surj S f m).some.2 | begin
generalize_proofs _ h2,
apply_fun (iso S f) using linear_equiv.injective _,
rw [linear_equiv.apply_symm_apply],
simp only [iso_apply, linear_map.to_fun_eq_coe, from_localized_module_mk],
erw [module.End_algebra_map_is_unit_inv_apply_eq_iff', h2.some_spec],
end | lemma | iso_symm_apply_aux | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"from_localized_module_mk",
"iso",
"linear_equiv.apply_symm_apply",
"linear_equiv.injective",
"linear_map.to_fun_eq_coe",
"localized_module.mk",
"module.End_algebra_map_is_unit_inv_apply_eq_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_symm_apply' (m : M') (a : M) (b : S) (eq1 : b • m = f a) :
(iso S f).symm m = localized_module.mk a b | (iso_symm_apply_aux S f m).trans $ localized_module.mk_eq.mpr $
begin
generalize_proofs h1,
erw [←is_localized_module.eq_iff_exists S f, f.map_smul, f.map_smul, ←h1.some_spec, ←mul_smul,
mul_comm, mul_smul, eq1],
end | lemma | iso_symm_apply' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"iso",
"iso_symm_apply_aux",
"localized_module.mk",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_symm_comp : (iso S f).symm.to_linear_map.comp f = localized_module.mk_linear_map S M | begin
ext m, rw [linear_map.comp_apply, localized_module.mk_linear_map_apply],
change (iso S f).symm _ = _, rw [iso_symm_apply'], exact one_smul _ _,
end | lemma | iso_symm_comp | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"iso",
"iso_symm_apply'",
"linear_map.comp_apply",
"localized_module.mk_linear_map",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x)) :
M' →ₗ[R] M'' | (localized_module.lift S g h).comp (iso S f).symm.to_linear_map | def | lift | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"iso",
"module.End"
] | If `M'` is a localized module and `g` is a linear map `M' → M''` such that all scalar multiplication
by `s : S` is invertible, then there is a linear map `M' → M''`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_comp (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x)) :
(lift S f g h).comp f = g | begin
dunfold is_localized_module.lift,
rw [linear_map.comp_assoc],
convert localized_module.lift_comp S g h,
exact iso_symm_comp _ _,
end | lemma | lift_comp | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"iso_symm_comp",
"lift",
"linear_map.comp_assoc",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x))
(l : M' →ₗ[R] M'') (hl : l.comp f = g) :
lift S f g h = l | begin
dunfold is_localized_module.lift,
rw [localized_module.lift_unique S g h (l.comp (iso S f).to_linear_map), linear_map.comp_assoc,
show (iso S f).to_linear_map.comp (iso S f).symm.to_linear_map = linear_map.id, from _,
linear_map.comp_id],
{ rw [linear_equiv.comp_to_linear_map_symm_eq, linear_map.id_... | lemma | lift_unique | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"from_localized_module_mk",
"is_unit",
"iso",
"lift",
"linear_equiv.comp_to_linear_map_symm_eq",
"linear_map.comp_assoc",
"linear_map.comp_id",
"linear_map.id",
"linear_map.id_comp",
"module.End",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_universal :
∀ (g : M →ₗ[R] M'') (map_unit : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x)),
∃! (l : M' →ₗ[R] M''), l.comp f = g | λ g h, ⟨lift S f g h, lift_comp S f g h, λ l hl, (lift_unique S f g h l hl).symm⟩ | lemma | is_universal | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"lift_comp",
"lift_unique",
"module.End"
] | Universal property from localized module:
If `(M', f : M ⟶ M')` is a localized module then it satisfies the following universal property:
For every `R`-module `M''` which every `s : S`-scalar multiplication is invertible and for every
`R`-linear map `g : M ⟶ M''`, there is a unique `R`-linear map `l : M' ⟶ M''` such th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_ext (map_unit : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x))
⦃j k : M' →ₗ[R] M''⦄ (h : j.comp f = k.comp f) : j = k | by { rw [←lift_unique S f (k.comp f) map_unit j h, lift_unique], refl } | lemma | ring_hom_ext | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"lift_unique",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv [is_localized_module S g] : M' ≃ₗ[R] M'' | (iso S f).symm.trans (iso S g) | def | linear_equiv | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"is_localized_module",
"iso"
] | If `(M', f)` and `(M'', g)` both satisfy universal property of localized module, then `M', M''`
are isomorphic as `R`-module | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_injective (s : S) : function.injective (λ m : M', s • m) | ((module.End_is_unit_iff _).mp (is_localized_module.map_units f s)).injective | lemma | smul_injective | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"module.End_is_unit_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_inj (s : S) (m₁ m₂ : M') : s • m₁ = s • m₂ ↔ m₁ = m₂ | (smul_injective f s).eq_iff | lemma | smul_inj | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"smul_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (m : M) (s : S) : M' | from_localized_module S f (localized_module.mk m s) | def | mk' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"from_localized_module",
"localized_module.mk"
] | `mk' f m s` is the fraction `m/s` with respect to the localization map `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk'_smul (r : R) (m : M) (s : S) : mk' f (r • m) s = r • mk' f m s | by { delta mk', rw [← localized_module.smul'_mk, linear_map.map_smul] } | lemma | mk'_smul | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"linear_map.map_smul",
"localized_module.smul'_mk",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_add_mk' (m₁ m₂ : M) (s₁ s₂ : S) :
mk' f m₁ s₁ + mk' f m₂ s₂ = mk' f (s₂ • m₁ + s₁ • m₂) (s₁ * s₂) | by { delta mk', rw [← map_add, localized_module.mk_add_mk] } | lemma | mk'_add_mk' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module.mk_add_mk",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_zero (s : S) :
mk' f 0 s = 0 | by rw [← zero_smul R (0 : M), mk'_smul, zero_smul] | lemma | mk'_zero | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"mk'",
"mk'_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_one (m : M) :
mk' f m (1 : S) = f m | by { delta mk', rw [from_localized_module_mk, module.End_algebra_map_is_unit_inv_apply_eq_iff,
submonoid.coe_one, one_smul] } | lemma | mk'_one | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"from_localized_module_mk",
"mk'",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"one_smul",
"submonoid.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_cancel (m : M) (s : S) :
mk' f (s • m) s = f m | by { delta mk', rw [localized_module.mk_cancel, ← mk'_one S f], refl } | lemma | mk'_cancel | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module.mk_cancel",
"mk'",
"mk'_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_cancel' (m : M) (s : S) :
s • mk' f m s = f m | by rw [submonoid.smul_def, ← mk'_smul, ← submonoid.smul_def, mk'_cancel] | lemma | mk'_cancel' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"mk'",
"mk'_cancel",
"mk'_smul",
"submonoid.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_cancel_left (m : M) (s₁ s₂ : S) :
mk' f (s₁ • m) (s₁ * s₂) = mk' f m s₂ | by { delta mk', rw localized_module.mk_cancel_common_left } | lemma | mk'_cancel_left | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module.mk_cancel_common_left",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_cancel_right (m : M) (s₁ s₂ : S) :
mk' f (s₂ • m) (s₁ * s₂) = mk' f m s₁ | by { delta mk', rw localized_module.mk_cancel_common_right } | lemma | mk'_cancel_right | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module.mk_cancel_common_right",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_add (m₁ m₂ : M) (s : S) : mk' f (m₁ + m₂) s = mk' f m₁ s + mk' f m₂ s | by { rw [mk'_add_mk', ← smul_add, mk'_cancel_left] } | lemma | mk'_add | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"mk'",
"mk'_add_mk'",
"mk'_cancel_left",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_mk'_iff (m₁ m₂ : M) (s₁ s₂ : S) :
mk' f m₁ s₁ = mk' f m₂ s₂ ↔ ∃ s : S, s • s₁ • m₂ = s • s₂ • m₁ | begin
delta mk',
rw [(from_localized_module.inj S f).eq_iff, localized_module.mk_eq],
simp_rw eq_comm
end | lemma | mk'_eq_mk'_iff | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"from_localized_module.inj",
"localized_module.mk_eq",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_neg {M M' : Type*} [add_comm_group M] [add_comm_group M'] [module R M]
[module R M'] (f : M →ₗ[R] M') [is_localized_module S f] (m : M) (s : S) :
mk' f (-m) s = - mk' f m s | by { delta mk', rw [localized_module.mk_neg, map_neg] } | lemma | mk'_neg | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"add_comm_group",
"is_localized_module",
"localized_module.mk_neg",
"mk'",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_sub {M M' : Type*} [add_comm_group M] [add_comm_group M'] [module R M]
[module R M'] (f : M →ₗ[R] M') [is_localized_module S f] (m₁ m₂ : M) (s : S) :
mk' f (m₁ - m₂) s = mk' f m₁ s - mk' f m₂ s | by rw [sub_eq_add_neg, sub_eq_add_neg, mk'_add, mk'_neg] | lemma | mk'_sub | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"add_comm_group",
"is_localized_module",
"mk'",
"mk'_add",
"mk'_neg",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_sub_mk' {M M' : Type*} [add_comm_group M] [add_comm_group M'] [module R M]
[module R M'] (f : M →ₗ[R] M') [is_localized_module S f] (m₁ m₂ : M) (s₁ s₂ : S) :
mk' f m₁ s₁ - mk' f m₂ s₂ = mk' f (s₂ • m₁ - s₁ • m₂) (s₁ * s₂) | by rw [sub_eq_add_neg, ← mk'_neg, mk'_add_mk', smul_neg, ← sub_eq_add_neg] | lemma | mk'_sub_mk' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"add_comm_group",
"is_localized_module",
"mk'",
"mk'_add_mk'",
"mk'_neg",
"module",
"smul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mul_mk'_of_map_mul {M M' : Type*} [semiring M] [semiring M'] [module R M]
[algebra R M'] (f : M →ₗ[R] M') (hf : ∀ m₁ m₂, f (m₁ * m₂) = f m₁ * f m₂)
[is_localized_module S f] (m₁ m₂ : M) (s₁ s₂ : S) :
mk' f m₁ s₁ * mk' f m₂ s₂ = mk' f (m₁ * m₂) (s₁ * s₂) | begin
symmetry,
apply (module.End_algebra_map_is_unit_inv_apply_eq_iff _ _ _).mpr,
simp_rw [submonoid.coe_mul, ← smul_eq_mul],
rw [smul_smul_smul_comm, ← mk'_smul, ← mk'_smul],
simp_rw [← submonoid.smul_def, mk'_cancel, smul_eq_mul, hf],
end | lemma | mk'_mul_mk'_of_map_mul | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra",
"is_localized_module",
"mk'",
"mk'_cancel",
"mk'_smul",
"module",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"semiring",
"smul_eq_mul",
"smul_smul_smul_comm",
"submonoid.coe_mul",
"submonoid.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mul_mk' {M M' : Type*} [semiring M] [semiring M'] [algebra R M]
[algebra R M'] (f : M →ₐ[R] M')
[is_localized_module S f.to_linear_map] (m₁ m₂ : M) (s₁ s₂ : S) :
mk' f.to_linear_map m₁ s₁ * mk' f.to_linear_map m₂ s₂ =
mk' f.to_linear_map (m₁ * m₂) (s₁ * s₂) | mk'_mul_mk'_of_map_mul f.to_linear_map f.map_mul m₁ m₂ s₁ s₂ | lemma | mk'_mul_mk' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra",
"is_localized_module",
"mk'",
"mk'_mul_mk'_of_map_mul",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_iff {m : M} {s : S} {m' : M'} :
mk' f m s = m' ↔ f m = s • m' | by rw [← smul_inj f s, submonoid.smul_def, ← mk'_smul, ← submonoid.smul_def, mk'_cancel] | lemma | mk'_eq_iff | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"mk'",
"mk'_cancel",
"mk'_smul",
"smul_inj",
"submonoid.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_zero {m : M} (s : S) :
mk' f m s = 0 ↔ f m = 0 | by rw [mk'_eq_iff, smul_zero] | lemma | mk'_eq_zero | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"mk'",
"mk'_eq_iff",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_zero' {m : M} (s : S) :
mk' f m s = 0 ↔ ∃ s' : S, s' • m = 0 | by simp_rw [← mk'_zero f (1 : S), mk'_eq_mk'_iff, smul_zero, one_smul, eq_comm] | lemma | mk'_eq_zero' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"mk'",
"mk'_eq_mk'_iff",
"mk'_zero",
"one_smul",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_mk' (s : S) (m : M) :
localized_module.mk m s = mk' (localized_module.mk_linear_map S M) m s | by rw [eq_comm, mk'_eq_iff, submonoid.smul_def, localized_module.smul'_mk,
← submonoid.smul_def, localized_module.mk_cancel, localized_module.mk_linear_map_apply] | lemma | mk_eq_mk' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module.mk",
"localized_module.mk_cancel",
"localized_module.mk_linear_map",
"localized_module.smul'_mk",
"mk'",
"mk'_eq_iff",
"submonoid.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff {m : M} :
f m = 0 ↔ ∃ s' : S, s' • m = 0 | (mk'_eq_zero (1 : S)).symm.trans (mk'_eq_zero' f _) | lemma | eq_zero_iff | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"mk'_eq_zero",
"mk'_eq_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_surjective : function.surjective (function.uncurry $ mk' f : M × S → M') | begin
intro x,
obtain ⟨⟨m, s⟩, e : s • x = f m⟩ := is_localized_module.surj S f x,
exact ⟨⟨m, s⟩, mk'_eq_iff.mpr e.symm⟩
end | lemma | mk'_surjective | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_algebra {R S S' : Type*} [comm_ring R] [comm_ring S] [comm_ring S']
[algebra R S] [algebra R S'] (M : submonoid R) (f : S →ₐ[R] S')
(h₁ : ∀ x ∈ M, is_unit (algebra_map R S' x))
(h₂ : ∀ y, ∃ (x : S × M), x.2 • y = f x.1)
(h₃ : ∀ x, f x = 0 → ∃ m : M, m • x = 0) :
is_localized_module M f.to_linear_map | begin
replace h₃ := λ x, iff.intro (h₃ x) (λ ⟨⟨m, hm⟩, e⟩, (h₁ m hm).mul_left_cancel $
by { rw ← algebra.smul_def, simpa [submonoid.smul_def] using f.congr_arg e }),
constructor,
{ intro x,
rw module.End_is_unit_iff,
split,
{ rintros a b (e : x • a = x • b), simp_rw [submonoid.smul_def, algebra.sm... | lemma | mk_of_algebra | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"comm_ring",
"is_localized_module",
"is_unit",
"is_unit.mul_coe_inv",
"module.End_is_unit_iff",
"mul_assoc",
"mul_left_cancel",
"one_mul",
"smul_sub",
"submonoid",
"submonoid.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_linear_equiv : M ≃ₗ[R] Mᵐᵒᵖ | { map_smul' := mul_opposite.op_smul, .. op_add_equiv } | def | mul_opposite.op_linear_equiv | algebra.module | src/algebra/module/opposites.lean | [
"algebra.module.equiv",
"group_theory.group_action.opposite"
] | [
"mul_opposite.op_smul"
] | The function `op` is a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_op_linear_equiv :
(op_linear_equiv R : M → Mᵐᵒᵖ) = op | rfl | lemma | mul_opposite.coe_op_linear_equiv | algebra.module | src/algebra/module/opposites.lean | [
"algebra.module.equiv",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_op_linear_equiv_symm :
((op_linear_equiv R).symm : Mᵐᵒᵖ → M) = unop | rfl | lemma | mul_opposite.coe_op_linear_equiv_symm | algebra.module | src/algebra/module/opposites.lean | [
"algebra.module.equiv",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_op_linear_equiv_to_linear_map :
((op_linear_equiv R).to_linear_map : M → Mᵐᵒᵖ) = op | rfl | lemma | mul_opposite.coe_op_linear_equiv_to_linear_map | algebra.module | src/algebra/module/opposites.lean | [
"algebra.module.equiv",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_op_linear_equiv_symm_to_linear_map :
((op_linear_equiv R).symm.to_linear_map : Mᵐᵒᵖ → M) = unop | rfl | lemma | mul_opposite.coe_op_linear_equiv_symm_to_linear_map | algebra.module | src/algebra/module/opposites.lean | [
"algebra.module.equiv",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_linear_equiv_to_add_equiv :
(op_linear_equiv R : M ≃ₗ[R] Mᵐᵒᵖ).to_add_equiv = op_add_equiv | rfl | lemma | mul_opposite.op_linear_equiv_to_add_equiv | algebra.module | src/algebra/module/opposites.lean | [
"algebra.module.equiv",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_linear_equiv_symm_to_add_equiv :
(op_linear_equiv R : M ≃ₗ[R] Mᵐᵒᵖ).symm.to_add_equiv = op_add_equiv.symm | rfl | lemma | mul_opposite.op_linear_equiv_symm_to_add_equiv | algebra.module | src/algebra/module/opposites.lean | [
"algebra.module.equiv",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_smul_regular.pi {α : Type*} [Π i, has_smul α $ f i] {k : α}
(hk : Π i, is_smul_regular (f i) k) : is_smul_regular (Π i, f i) k | λ _ _ h, funext $ λ i, hk i (congr_fun h i : _) | lemma | is_smul_regular.pi | algebra.module | src/algebra/module/pi.lean | [
"algebra.module.basic",
"algebra.regular.smul",
"algebra.ring.pi",
"group_theory.group_action.pi"
] | [
"has_smul",
"is_smul_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_with_zero (α) [has_zero α]
[Π i, has_zero (f i)] [Π i, smul_with_zero α (f i)] :
smul_with_zero α (Π i, f i) | { smul_zero := λ _, funext $ λ _, smul_zero _,
zero_smul := λ _, funext $ λ _, zero_smul _ _,
..pi.has_smul } | instance | pi.smul_with_zero | algebra.module | src/algebra/module/pi.lean | [
"algebra.module.basic",
"algebra.regular.smul",
"algebra.ring.pi",
"group_theory.group_action.pi"
] | [
"pi.has_smul",
"smul_with_zero",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_with_zero' {g : I → Type*} [Π i, has_zero (g i)]
[Π i, has_zero (f i)] [Π i, smul_with_zero (g i) (f i)] :
smul_with_zero (Π i, g i) (Π i, f i) | { smul_zero := λ _, funext $ λ _, smul_zero _,
zero_smul := λ _, funext $ λ _, zero_smul _ _,
..pi.has_smul' } | instance | pi.smul_with_zero' | algebra.module | src/algebra/module/pi.lean | [
"algebra.module.basic",
"algebra.regular.smul",
"algebra.ring.pi",
"group_theory.group_action.pi"
] | [
"pi.has_smul'",
"smul_with_zero",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_with_zero (α) [monoid_with_zero α]
[Π i, has_zero (f i)] [Π i, mul_action_with_zero α (f i)] :
mul_action_with_zero α (Π i, f i) | { ..pi.mul_action _,
..pi.smul_with_zero _ } | instance | pi.mul_action_with_zero | algebra.module | src/algebra/module/pi.lean | [
"algebra.module.basic",
"algebra.regular.smul",
"algebra.ring.pi",
"group_theory.group_action.pi"
] | [
"monoid_with_zero",
"mul_action_with_zero",
"pi.mul_action",
"pi.smul_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_with_zero' {g : I → Type*} [Π i, monoid_with_zero (g i)]
[Π i, has_zero (f i)] [Π i, mul_action_with_zero (g i) (f i)] :
mul_action_with_zero (Π i, g i) (Π i, f i) | { ..pi.mul_action',
..pi.smul_with_zero' } | instance | pi.mul_action_with_zero' | algebra.module | src/algebra/module/pi.lean | [
"algebra.module.basic",
"algebra.regular.smul",
"algebra.ring.pi",
"group_theory.group_action.pi"
] | [
"monoid_with_zero",
"mul_action_with_zero",
"pi.mul_action'",
"pi.smul_with_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module (α) {r : semiring α} {m : ∀ i, add_comm_monoid $ f i}
[∀ i, module α $ f i] :
@module α (Π i : I, f i) r (@pi.add_comm_monoid I f m) | { add_smul := λ c f g, funext $ λ i, add_smul _ _ _,
zero_smul := λ f, funext $ λ i, zero_smul α _,
..pi.distrib_mul_action _ } | instance | pi.module | algebra.module | src/algebra/module/pi.lean | [
"algebra.module.basic",
"algebra.regular.smul",
"algebra.ring.pi",
"group_theory.group_action.pi"
] | [
"add_comm_monoid",
"add_smul",
"module",
"pi.distrib_mul_action",
"semiring",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.module (α β : Type*) [semiring α] [add_comm_monoid β] [module α β] :
module α (I → β) | pi.module _ _ _ | instance | function.module | algebra.module | src/algebra/module/pi.lean | [
"algebra.module.basic",
"algebra.regular.smul",
"algebra.ring.pi",
"group_theory.group_action.pi"
] | [
"add_comm_monoid",
"module",
"pi.module",
"semiring"
] | A special case of `pi.module` for non-dependent types. Lean struggles to elaborate
definitions elsewhere in the library without this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module' {g : I → Type*} {r : Π i, semiring (f i)} {m : Π i, add_comm_monoid (g i)}
[Π i, module (f i) (g i)] :
module (Π i, f i) (Π i, g i) | { add_smul := by { intros, ext1, apply add_smul },
zero_smul := by { intros, ext1, apply zero_smul } } | instance | pi.module' | algebra.module | src/algebra/module/pi.lean | [
"algebra.module.basic",
"algebra.regular.smul",
"algebra.ring.pi",
"group_theory.group_action.pi"
] | [
"add_comm_monoid",
"add_smul",
"module",
"semiring",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.no_zero_smul_divisors {ι α β : Type*} {r : semiring α}
{m : add_comm_monoid β} [module α β] [no_zero_smul_divisors α β] :
no_zero_smul_divisors α (ι → β) | pi.no_zero_smul_divisors _ | instance | function.no_zero_smul_divisors | algebra.module | src/algebra/module/pi.lean | [
"algebra.module.basic",
"algebra.regular.smul",
"algebra.ring.pi",
"group_theory.group_action.pi"
] | [
"add_comm_monoid",
"module",
"no_zero_smul_divisors",
"semiring"
] | A special case of `pi.no_zero_smul_divisors` for non-dependent types. Lean struggles to
synthesize this instance by itself elsewhere in the library. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.is_internal_prime_power_torsion_of_pid
[module.finite R M] (hM : module.is_torsion R M) :
direct_sum.is_internal (λ p : (factors (⊤ : submodule R M).annihilator).to_finset,
torsion_by R M
(is_principal.generator (p : ideal R)
^ (factors (⊤ : submodule R M).annihilator).count p)) | begin
convert is_internal_prime_power_torsion hM,
ext p : 1,
rw [← torsion_by_span_singleton_eq, ideal.submodule_span_eq, ← ideal.span_singleton_pow,
ideal.span_singleton_generator],
end | theorem | submodule.is_internal_prime_power_torsion_of_pid | algebra.module | src/algebra/module/pid.lean | [
"algebra.module.dedekind_domain",
"linear_algebra.free_module.pid",
"algebra.module.projective",
"algebra.category.Module.biproducts"
] | [
"direct_sum.is_internal",
"ideal",
"ideal.span_singleton_generator",
"ideal.span_singleton_pow",
"ideal.submodule_span_eq",
"module.finite",
"module.is_torsion",
"submodule"
] | A finitely generated torsion module over a PID is an internal direct sum of its
`p i ^ e i`-torsion submodules for some primes `p i` and numbers `e i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.exists_is_internal_prime_power_torsion_of_pid
[module.finite R M] (hM : module.is_torsion R M) :
∃ (ι : Type u) [fintype ι] [decidable_eq ι] (p : ι → R) (h : ∀ i, irreducible $ p i) (e : ι → ℕ),
by exactI direct_sum.is_internal (λ i, torsion_by R M $ p i ^ e i) | begin
refine ⟨_, _, _, _, _, _, submodule.is_internal_prime_power_torsion_of_pid hM⟩,
exact finset.fintype_coe_sort _,
{ rintro ⟨p, hp⟩,
have hP := prime_of_factor p (multiset.mem_to_finset.mp hp),
haveI := ideal.is_prime_of_prime hP,
exact (is_principal.prime_generator_of_is_prime p hP.ne_zero).irred... | theorem | submodule.exists_is_internal_prime_power_torsion_of_pid | algebra.module | src/algebra/module/pid.lean | [
"algebra.module.dedekind_domain",
"linear_algebra.free_module.pid",
"algebra.module.projective",
"algebra.category.Module.biproducts"
] | [
"direct_sum.is_internal",
"finset.fintype_coe_sort",
"fintype",
"ideal.is_prime_of_prime",
"irreducible",
"module.finite",
"module.is_torsion",
"submodule.is_internal_prime_power_torsion_of_pid"
] | A finitely generated torsion module over a PID is an internal direct sum of its
`p i ^ e i`-torsion submodules for some primes `p i` and numbers `e i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.ideal.torsion_of_eq_span_pow_p_order (x : M) :
torsion_of R M x = span {p ^ p_order hM x} | begin
dunfold p_order,
rw [← (torsion_of R M x).span_singleton_generator, ideal.span_singleton_eq_span_singleton,
← associates.mk_eq_mk_iff_associated, associates.mk_pow],
have prop : (λ n : ℕ, p ^ n • x = 0) =
λ n : ℕ, (associates.mk $ generator $ torsion_of R M x) ∣ associates.mk p ^ n,
{ ext n, rw [←... | lemma | ideal.torsion_of_eq_span_pow_p_order | algebra.module | src/algebra/module/pid.lean | [
"algebra.module.dedekind_domain",
"linear_algebra.free_module.pid",
"algebra.module.projective",
"algebra.category.Module.biproducts"
] | [
"associates.eq_pow_find_of_dvd_irreducible_pow",
"associates.irreducible_mk",
"associates.mk",
"associates.mk_dvd_mk",
"associates.mk_eq_mk_iff_associated",
"associates.mk_pow",
"ideal.span_singleton_eq_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
p_pow_smul_lift {x y : M} {k : ℕ} (hM' : module.is_torsion_by R M (p ^ p_order hM y))
(h : p ^ k • x ∈ R ∙ y) : ∃ a : R, p ^ k • x = p ^ k • a • y | begin
by_cases hk : k ≤ p_order hM y,
{ let f := ((R ∙ p ^ (p_order hM y - k) * p ^ k).quot_equiv_of_eq _ _).trans
(quot_torsion_of_equiv_span_singleton R M y),
have : f.symm ⟨p ^ k • x, h⟩ ∈
R ∙ ideal.quotient.mk (R ∙ p ^ (p_order hM y - k) * p ^ k) (p ^ k),
{ rw [← quotient.torsion_by_eq_span_... | lemma | module.p_pow_smul_lift | algebra.module | src/algebra/module/pid.lean | [
"algebra.module.dedekind_domain",
"linear_algebra.free_module.pid",
"algebra.module.projective",
"algebra.category.Module.biproducts"
] | [
"ideal.quotient.mk",
"ideal.quotient.mk_eq_mk",
"ideal.torsion_of_eq_span_pow_p_order",
"linear_equiv.trans_apply",
"mem_non_zero_divisors_of_ne_zero",
"module.is_torsion_by",
"mul_comm",
"pow_add",
"pow_ne_zero",
"smul_eq_mul",
"smul_smul",
"smul_zero",
"submodule.mem_span_singleton",
"su... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_smul_eq_zero_and_mk_eq {z : M} (hz : module.is_torsion_by R M (p ^ p_order hM z))
{k : ℕ} (f : (R ⧸ R ∙ p ^ k) →ₗ[R] M ⧸ R ∙ z) :
∃ x : M, p ^ k • x = 0 ∧ submodule.quotient.mk x = f 1 | begin
have f1 := mk_surjective (R ∙ z) (f 1),
have : p ^ k • f1.some ∈ R ∙ z,
{ rw [← quotient.mk_eq_zero, mk_smul, f1.some_spec, ← f.map_smul],
convert f.map_zero, change _ • submodule.quotient.mk _ = _,
rw [← mk_smul, quotient.mk_eq_zero, algebra.id.smul_eq_mul, mul_one],
exact submodule.mem_span_si... | lemma | module.exists_smul_eq_zero_and_mk_eq | algebra.module | src/algebra/module/pid.lean | [
"algebra.module.dedekind_domain",
"linear_algebra.free_module.pid",
"algebra.module.projective",
"algebra.category.Module.biproducts"
] | [
"algebra.id.smul_eq_mul",
"module.is_torsion_by",
"mul_one",
"smul_sub",
"smul_zero",
"submodule.mem_span_singleton_self",
"submodule.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_prime_power_decomposition (hN : module.is_torsion' N (submonoid.powers p))
[h' : module.finite R N] :
∃ (d : ℕ) (k : fin d → ℕ), nonempty $ N ≃ₗ[R] ⨁ (i : fin d), R ⧸ R ∙ (p ^ (k i : ℕ)) | begin
obtain ⟨d, s, hs⟩ := @module.finite.exists_fin _ _ _ _ _ h', use d, clear h',
unfreezingI { induction d with d IH generalizing N },
{ use λ i, fin_zero_elim i,
rw [set.range_eq_empty, submodule.span_empty] at hs,
haveI : unique N := ⟨⟨0⟩, λ x, by { rw [← mem_bot _, hs], trivial }⟩,
exact ⟨0⟩ },
... | theorem | module.torsion_by_prime_power_decomposition | algebra.module | src/algebra/module/pid.lean | [
"algebra.module.dedekind_domain",
"linear_algebra.free_module.pid",
"algebra.module.projective",
"algebra.category.Module.biproducts"
] | [
"direct_sum.lequiv_congr_left",
"direct_sum.lequiv_prod_direct_sum",
"direct_sum.lof",
"direct_sum.to_module",
"direct_sum.to_module_lof",
"fin.range_succ_above",
"fin_succ_equiv",
"fin_zero_elim",
"function.comp_apply",
"ideal.quotient.mk_eq_mk",
"ideal.torsion_of_eq_span_pow_p_order",
"lequi... | A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
`R ⧸ R ∙ (p ^ e i)` for some `e i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_direct_sum_of_is_torsion [h' : module.finite R N] (hN : module.is_torsion R N) :
∃ (ι : Type u) [fintype ι] (p : ι → R) (h : ∀ i, irreducible $ p i) (e : ι → ℕ),
nonempty $ N ≃ₗ[R] ⨁ (i : ι), R ⧸ R ∙ (p i ^ e i) | begin
obtain ⟨I, fI, _, p, hp, e, h⟩ := submodule.exists_is_internal_prime_power_torsion_of_pid hN,
haveI := fI,
have : ∀ i, ∃ (d : ℕ) (k : fin d → ℕ),
nonempty $ torsion_by R N (p i ^ e i) ≃ₗ[R] ⨁ j, R ⧸ R ∙ (p i ^ k j),
{ haveI := is_noetherian_of_fg_of_noetherian' (module.finite_def.mp h'),
haveI := ... | theorem | module.equiv_direct_sum_of_is_torsion | algebra.module | src/algebra/module/pid.lean | [
"algebra.module.dedekind_domain",
"linear_algebra.free_module.pid",
"algebra.module.projective",
"algebra.category.Module.biproducts"
] | [
"dfinsupp.map_range.linear_equiv",
"direct_sum.coe_linear_map",
"direct_sum.sigma_lcurry_equiv",
"fintype",
"irreducible",
"is_noetherian_of_fg_of_noetherian'",
"is_noetherian_submodule'",
"linear_equiv.of_bijective",
"module.finite",
"module.is_torsion",
"submodule.exists_is_internal_prime_powe... | A finitely generated torsion module over a PID is isomorphic to a direct sum of some
`R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_free_prod_direct_sum [h' : module.finite R N] :
∃ (n : ℕ) (ι : Type u) [fintype ι] (p : ι → R) (h : ∀ i, irreducible $ p i) (e : ι → ℕ),
nonempty $ N ≃ₗ[R] (fin n →₀ R) × ⨁ (i : ι), R ⧸ R ∙ (p i ^ e i) | begin
haveI := is_noetherian_of_fg_of_noetherian' (module.finite_def.mp h'),
haveI := is_noetherian_submodule' (torsion R N),
haveI := module.finite.of_surjective _ (torsion R N).mkq_surjective,
obtain ⟨I, fI, p, hp, e, ⟨h⟩⟩ := equiv_direct_sum_of_is_torsion (@torsion_is_torsion R N _ _ _),
obtain ⟨n, ⟨g⟩⟩ :=... | theorem | module.equiv_free_prod_direct_sum | algebra.module | src/algebra/module/pid.lean | [
"algebra.module.dedekind_domain",
"linear_algebra.free_module.pid",
"algebra.module.projective",
"algebra.category.Module.biproducts"
] | [
"fintype",
"irreducible",
"is_noetherian_of_fg_of_noetherian'",
"is_noetherian_submodule'",
"lequiv_prod_of_right_split_exact",
"linear_equiv.prod_comm",
"linear_map.id",
"module.basis_of_finite_type_torsion_free'",
"module.finite",
"module.finite.of_surjective",
"module.projective",
"module.p... | **Structure theorem of finitely generated modules over a PID** : A finitely generated
module over a PID is isomorphic to the product of a free module and a direct sum of some
`R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_pi_subset [∀ i, has_smul K (R i)] (r : K) (s : set ι) (t : Π i, set (R i)) :
r • pi s t ⊆ pi s (r • t) | begin
rintros x ⟨y, h, rfl⟩ i hi,
exact smul_mem_smul_set (h i hi),
end | lemma | smul_pi_subset | algebra.module | src/algebra/module/pointwise_pi.lean | [
"data.set.pointwise.smul",
"group_theory.group_action.pi"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_univ_pi [∀ i, has_smul K (R i)] (r : K) (t : Π i, set (R i)) :
r • pi (univ : set ι) t = pi (univ : set ι) (r • t) | subset.antisymm (smul_pi_subset _ _ _) $ λ x h, begin
refine ⟨λ i, classical.some (h i $ set.mem_univ _), λ i hi, _, funext $ λ i, _⟩,
{ exact (classical.some_spec (h i _)).left, },
{ exact (classical.some_spec (h i _)).right, },
end | lemma | smul_univ_pi | algebra.module | src/algebra/module/pointwise_pi.lean | [
"data.set.pointwise.smul",
"group_theory.group_action.pi"
] | [
"has_smul",
"set.mem_univ",
"smul_pi_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_pi [group K] [∀ i, mul_action K (R i)] (r : K) (S : set ι) (t : Π i, set (R i)) :
r • S.pi t = S.pi (r • t) | subset.antisymm (smul_pi_subset _ _ _) $ λ x h,
⟨r⁻¹ • x, λ i hiS, mem_smul_set_iff_inv_smul_mem.mp (h i hiS), smul_inv_smul _ _⟩ | lemma | smul_pi | algebra.module | src/algebra/module/pointwise_pi.lean | [
"data.set.pointwise.smul",
"group_theory.group_action.pi"
] | [
"group",
"mul_action",
"smul_inv_smul",
"smul_pi_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_pi₀ [group_with_zero K] [∀ i, mul_action K (R i)] {r : K} (S : set ι)
(t : Π i, set (R i)) (hr : r ≠ 0) : r • S.pi t = S.pi (r • t) | smul_pi (units.mk0 r hr) S t | lemma | smul_pi₀ | algebra.module | src/algebra/module/pointwise_pi.lean | [
"data.set.pointwise.smul",
"group_theory.group_action.pi"
] | [
"group_with_zero",
"mul_action",
"smul_pi",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_with_zero [has_zero R] [has_zero M] [has_zero N]
[smul_with_zero R M] [smul_with_zero R N] : smul_with_zero R (M × N) | { smul_zero := λ r, prod.ext (smul_zero _) (smul_zero _),
zero_smul := λ ⟨m, n⟩, prod.ext (zero_smul _ _) (zero_smul _ _),
..prod.has_smul } | instance | prod.smul_with_zero | algebra.module | src/algebra/module/prod.lean | [
"algebra.module.basic",
"group_theory.group_action.prod"
] | [
"prod.ext",
"smul_with_zero",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_with_zero [monoid_with_zero R] [has_zero M] [has_zero N]
[mul_action_with_zero R M] [mul_action_with_zero R N] : mul_action_with_zero R (M × N) | { smul_zero := λ r, prod.ext (smul_zero _) (smul_zero _),
zero_smul := λ ⟨m, n⟩, prod.ext (zero_smul _ _) (zero_smul _ _),
..prod.mul_action } | instance | prod.mul_action_with_zero | algebra.module | src/algebra/module/prod.lean | [
"algebra.module.basic",
"group_theory.group_action.prod"
] | [
"monoid_with_zero",
"mul_action_with_zero",
"prod.ext",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.projective (R : Type*) [semiring R] (P : Type*) [add_comm_monoid P]
[module R P] : Prop | (out : ∃ s : P →ₗ[R] (P →₀ R), function.left_inverse (finsupp.total P P R id) s) | class | module.projective | algebra.module | src/algebra/module/projective.lean | [
"algebra.module.basic",
"linear_algebra.finsupp",
"linear_algebra.free_module.basic"
] | [
"add_comm_monoid",
"finsupp.total",
"module",
"semiring"
] | An R-module is projective if it is a direct summand of a free module, or equivalently
if maps from the module lift along surjections. There are several other equivalent
definitions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
projective_def : projective R P ↔
(∃ s : P →ₗ[R] (P →₀ R), function.left_inverse (finsupp.total P P R id) s) | ⟨λ h, h.1, λ h, ⟨h⟩⟩ | lemma | module.projective_def | algebra.module | src/algebra/module/projective.lean | [
"algebra.module.basic",
"linear_algebra.finsupp",
"linear_algebra.free_module.basic"
] | [
"finsupp.total"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
projective_def' : projective R P ↔
(∃ s : P →ₗ[R] (P →₀ R), (finsupp.total P P R id) ∘ₗ s = id) | by simp_rw [projective_def, fun_like.ext_iff, function.left_inverse, coe_comp, id_coe, id.def] | theorem | module.projective_def' | algebra.module | src/algebra/module/projective.lean | [
"algebra.module.basic",
"linear_algebra.finsupp",
"linear_algebra.free_module.basic"
] | [
"finsupp.total",
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
projective_lifting_property [h : projective R P] (f : M →ₗ[R] N) (g : P →ₗ[R] N)
(hf : function.surjective f) : ∃ (h : P →ₗ[R] M), f.comp h = g | begin
/-
Here's the first step of the proof.
Recall that `X →₀ R` is Lean's way of talking about the free `R`-module
on a type `X`. The universal property `finsupp.total` says that to a map
`X → N` from a type to an `R`-module, we get an associated R-module map
`(X →₀ R) →ₗ N`. Apply this to a (noncomputabl... | theorem | module.projective_lifting_property | algebra.module | src/algebra/module/projective.lean | [
"algebra.module.basic",
"linear_algebra.finsupp",
"linear_algebra.free_module.basic"
] | [
"finsupp.total",
"finsupp.total_apply"
] | A projective R-module has the property that maps from it lift along surjections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
projective_of_basis {ι : Type*} (b : basis ι R P) : projective R P | begin
-- need P →ₗ (P →₀ R) for definition of projective.
-- get it from `ι → (P →₀ R)` coming from `b`.
use b.constr ℕ (λ i, finsupp.single (b i) (1 : R)),
intro m,
simp only [b.constr_apply, mul_one, id.def, finsupp.smul_single', finsupp.total_single,
linear_map.map_finsupp_sum],
exact b.total_repr m,... | theorem | module.projective_of_basis | algebra.module | src/algebra/module/projective.lean | [
"algebra.module.basic",
"linear_algebra.finsupp",
"linear_algebra.free_module.basic"
] | [
"basis",
"finsupp.single",
"finsupp.smul_single'",
"finsupp.total_single",
"linear_map.map_finsupp_sum",
"mul_one"
] | Free modules are projective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
projective_of_free [module.free R P] : module.projective R P | projective_of_basis $ module.free.choose_basis R P | instance | module.projective_of_free | algebra.module | src/algebra/module/projective.lean | [
"algebra.module.basic",
"linear_algebra.finsupp",
"linear_algebra.free_module.basic"
] | [
"module.free",
"module.free.choose_basis",
"module.projective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
projective_of_lifting_property'
{R : Type u} [semiring R] {P : Type (max u v)} [add_comm_monoid P] [module R P]
-- If for all surjections of `R`-modules `M →ₗ N`, all maps `P →ₗ N` lift to `P →ₗ M`,
(huniv : ∀ {M : Type (max v u)} {N : Type (max u v)} [add_comm_monoid M] [add_comm_monoid N],
by exactI
∀ [... | begin
-- let `s` be the universal map `(P →₀ R) →ₗ P` coming from the identity map `P →ₗ P`.
obtain ⟨s, hs⟩ : ∃ (s : P →ₗ[R] P →₀ R),
(finsupp.total P P R id).comp s = linear_map.id :=
huniv (finsupp.total P P R (id : P → P)) (linear_map.id : P →ₗ[R] P) _,
-- This `s` works.
{ use s,
rwa linear_map.... | theorem | module.projective_of_lifting_property' | algebra.module | src/algebra/module/projective.lean | [
"algebra.module.basic",
"linear_algebra.finsupp",
"linear_algebra.free_module.basic"
] | [
"add_comm_monoid",
"finsupp.single",
"finsupp.total",
"linear_map.ext_iff",
"linear_map.id",
"module",
"semiring"
] | A module which satisfies the universal property is projective. Note that the universe variables
in `huniv` are somewhat restricted. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
projective_of_lifting_property
{R : Type u} [ring R] {P : Type (max u v)} [add_comm_group P] [module R P]
-- If for all surjections of `R`-modules `M →ₗ N`, all maps `P →ₗ N` lift to `P →ₗ M`,
(huniv : ∀ {M : Type (max v u)} {N : Type (max u v)} [add_comm_group M] [add_comm_group N],
by exactI
∀ [module R... | -- We could try and prove this *using* `of_lifting_property`,
-- but this quickly leads to typeclass hell,
-- so we just prove it over again.
begin
-- let `s` be the universal map `(P →₀ R) →ₗ P` coming from the identity map `P →ₗ P`.
obtain ⟨s, hs⟩ : ∃ (s : P →ₗ[R] P →₀ R),
(finsupp.total P P R id).comp s = li... | theorem | module.projective_of_lifting_property | algebra.module | src/algebra/module/projective.lean | [
"algebra.module.basic",
"linear_algebra.finsupp",
"linear_algebra.free_module.basic"
] | [
"add_comm_group",
"finsupp.single",
"finsupp.total",
"linear_map.ext_iff",
"linear_map.id",
"module",
"ring"
] | A variant of `of_lifting_property'` when we're working over a `[ring R]`,
which only requires quantifying over modules with an `add_comm_group` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion_of (x : M) : ideal R | (linear_map.to_span_singleton R M x).ker | def | ideal.torsion_of | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"ideal",
"linear_map.to_span_singleton"
] | The torsion ideal of `x`, containing all `a` such that `a • x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion_of_zero : torsion_of R M (0 : M) = ⊤ | by simp [torsion_of] | lemma | ideal.torsion_of_zero | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_torsion_of_iff (x : M) (a : R) : a ∈ torsion_of R M x ↔ a • x = 0 | iff.rfl | lemma | ideal.mem_torsion_of_iff | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_of_eq_top_iff (m : M) : torsion_of R M m = ⊤ ↔ m = 0 | begin
refine ⟨λ h, _, λ h, by simp [h]⟩,
rw [← one_smul R m, ← mem_torsion_of_iff m (1 : R), h],
exact submodule.mem_top,
end | lemma | ideal.torsion_of_eq_top_iff | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"one_smul",
"submodule.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_of_eq_bot_iff_of_no_zero_smul_divisors
[nontrivial R] [no_zero_smul_divisors R M] (m : M) :
torsion_of R M m = ⊥ ↔ m ≠ 0 | begin
refine ⟨λ h contra, _, λ h, (submodule.eq_bot_iff _).mpr $ λ r hr, _⟩,
{ rw [contra, torsion_of_zero] at h,
exact bot_ne_top.symm h, },
{ rw [mem_torsion_of_iff, smul_eq_zero] at hr,
tauto, },
end | lemma | ideal.torsion_of_eq_bot_iff_of_no_zero_smul_divisors | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"no_zero_smul_divisors",
"nontrivial",
"smul_eq_zero",
"submodule.eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_lattice.independent.linear_independent' {ι R M : Type*} {v : ι → M}
[ring R] [add_comm_group M] [module R M]
(hv : complete_lattice.independent $ λ i, (R ∙ v i))
(h_ne_zero : ∀ i, ideal.torsion_of R M (v i) = ⊥) :
linear_independent R v | begin
refine linear_independent_iff_not_smul_mem_span.mpr (λ i r hi, _),
replace hv := complete_lattice.independent_def.mp hv i,
simp only [supr_subtype', ← submodule.span_range_eq_supr, disjoint_iff] at hv,
have : r • v i ∈ ⊥,
{ rw [← hv, submodule.mem_inf],
refine ⟨submodule.mem_span_singleton.mpr ⟨r, r... | lemma | ideal.complete_lattice.independent.linear_independent' | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"add_comm_group",
"complete_lattice.independent",
"disjoint_iff",
"ideal.torsion_of",
"linear_independent",
"module",
"ring",
"submodule.mem_bot",
"submodule.mem_inf",
"submodule.span_range_eq_supr",
"supr_subtype'"
] | See also `complete_lattice.independent.linear_independent` which provides the same conclusion
but requires the stronger hypothesis `no_zero_smul_divisors R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_torsion_of_equiv_span_singleton (x : M) :
(R ⧸ torsion_of R M x) ≃ₗ[R] (R ∙ x) | (linear_map.to_span_singleton R M x).quot_ker_equiv_range.trans $
linear_equiv.of_eq _ _ (linear_map.span_singleton_eq_range R M x).symm | def | ideal.quot_torsion_of_equiv_span_singleton | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"linear_equiv.of_eq",
"linear_map.span_singleton_eq_range",
"linear_map.to_span_singleton"
] | The span of `x` in `M` is isomorphic to `R` quotiented by the torsion ideal of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_torsion_of_equiv_span_singleton_apply_mk (x : M) (a : R) :
quot_torsion_of_equiv_span_singleton R M x (submodule.quotient.mk a) =
a • ⟨x, submodule.mem_span_singleton_self x⟩ | rfl | lemma | ideal.quot_torsion_of_equiv_span_singleton_apply_mk | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"submodule.mem_span_singleton_self",
"submodule.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by (a : R) : submodule R M | (distrib_mul_action.to_linear_map R M a).ker | def | submodule.torsion_by | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"distrib_mul_action.to_linear_map",
"submodule"
] | The `a`-torsion submodule for `a` in `R`, containing all elements `x` of `M` such that
`a • x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion_by_set (s : set R) : submodule R M | Inf (torsion_by R M '' s) | def | submodule.torsion_by_set | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"submodule"
] | The submodule containing all elements `x` of `M` such that `a • x = 0` for all `a` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion' (S : Type*)
[comm_monoid S] [distrib_mul_action S M] [smul_comm_class S R M] :
submodule R M | { carrier := { x | ∃ a : S, a • x = 0 },
zero_mem' := ⟨1, smul_zero _⟩,
add_mem' := λ x y ⟨a, hx⟩ ⟨b, hy⟩,
⟨b * a,
by rw [smul_add, mul_smul, mul_comm, mul_smul, hx, hy, smul_zero, smul_zero, add_zero]⟩,
smul_mem' := λ a x ⟨b, h⟩, ⟨b, by rw [smul_comm, h, smul_zero]⟩ } | def | submodule.torsion' | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"comm_monoid",
"distrib_mul_action",
"mul_comm",
"smul_add",
"smul_comm_class",
"smul_zero",
"submodule"
] | The `S`-torsion submodule, containing all elements `x` of `M` such that `a • x = 0` for some
`a` in `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion | torsion' R M R⁰ | def | submodule.torsion | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | The torsion submodule, containing all elements `x` of `M` such that `a • x = 0` for some
non-zero-divisor `a` in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_torsion_by (a : R) | ∀ ⦃x : M⦄, a • x = 0 | def | module.is_torsion_by | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | A `a`-torsion module is a module where every element is `a`-torsion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_torsion_by_set (s : set R) | ∀ ⦃x : M⦄ ⦃a : s⦄, (a : R) • x = 0 | def | module.is_torsion_by_set | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | A module where every element is `a`-torsion for all `a` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_torsion' (S : Type*) [has_smul S M] | ∀ ⦃x : M⦄, ∃ a : S, a • x = 0 | def | module.is_torsion' | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"has_smul"
] | A `S`-torsion module is a module where every element is `a`-torsion for some `a` in `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_torsion | ∀ ⦃x : M⦄, ∃ a : R⁰, a • x = 0 | def | module.is_torsion | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | A torsion module is a module where every element is `a`-torsion for some non-zero-divisor `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_torsion_by (x : torsion_by R M a) : a • x = 0 | subtype.ext x.prop | lemma | submodule.smul_torsion_by | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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