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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
to_add_monoid_hom_commutes :
e.to_linear_map.to_add_monoid_hom = e.to_add_equiv.to_add_monoid_hom | rfl | lemma | linear_equiv.to_add_monoid_hom_commutes | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | The two paths coercion can take to an `add_monoid_hom` are equivalent | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (c : M₁) :
(e₁₂.trans e₂₃ : M₁ ≃ₛₗ[σ₁₃] M₃) c = e₂₃ (e₁₂ c) | rfl | theorem | linear_equiv.trans_apply | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_trans :
(e₁₂.trans e₂₃ : M₁ →ₛₗ[σ₁₃] M₃) = (e₂₃ : M₂ →ₛₗ[σ₂₃] M₃).comp (e₁₂ : M₁ →ₛₗ[σ₁₂] M₂) | rfl | theorem | linear_equiv.coe_trans | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply (c : M₂) : e (e.symm c) = c | e.right_inv c | theorem | linear_equiv.apply_symm_apply | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (b : M) : e.symm (e b) = b | e.left_inv b | theorem | linear_equiv.symm_apply_apply | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_symm : (e₁₂.trans e₂₃ : M₁ ≃ₛₗ[σ₁₃] M₃).symm = e₂₃.symm.trans e₁₂.symm | rfl | lemma | linear_equiv.trans_symm | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_apply
(c : M₃) : (e₁₂.trans e₂₃ : M₁ ≃ₛₗ[σ₁₃] M₃).symm c = e₁₂.symm (e₂₃.symm c) | rfl | lemma | linear_equiv.symm_trans_apply | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl : e.trans (refl S M₂) = e | to_equiv_injective e.to_equiv.trans_refl | lemma | linear_equiv.trans_refl | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_trans : (refl R M).trans e = e | to_equiv_injective e.to_equiv.refl_trans | lemma | linear_equiv.refl_trans | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_eq {x y} : e.symm x = y ↔ x = e y | e.to_equiv.symm_apply_eq | lemma | linear_equiv.symm_apply_eq | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_symm_apply {x y} : y = e.symm x ↔ e y = x | e.to_equiv.eq_symm_apply | lemma | linear_equiv.eq_symm_apply | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_comp_symm {α : Type*} (f : M₂ → α) (g : M₁ → α) :
f = g ∘ e₁₂.symm ↔ f ∘ e₁₂ = g | e₁₂.to_equiv.eq_comp_symm f g | lemma | linear_equiv.eq_comp_symm | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_symm_eq {α : Type*} (f : M₂ → α) (g : M₁ → α) :
g ∘ e₁₂.symm = f ↔ g = f ∘ e₁₂ | e₁₂.to_equiv.comp_symm_eq f g | lemma | linear_equiv.comp_symm_eq | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_symm_comp {α : Type*} (f : α → M₁) (g : α → M₂) :
f = e₁₂.symm ∘ g ↔ e₁₂ ∘ f = g | e₁₂.to_equiv.eq_symm_comp f g | lemma | linear_equiv.eq_symm_comp | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_comp_eq {α : Type*} (f : α → M₁) (g : α → M₂) :
e₁₂.symm ∘ g = f ↔ g = e₁₂ ∘ f | e₁₂.to_equiv.symm_comp_eq f g | lemma | linear_equiv.symm_comp_eq | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_comp_to_linear_map_symm (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₃] M₃) :
f = g.comp e₁₂.symm.to_linear_map ↔ f.comp e₁₂.to_linear_map = g | begin
split; intro H; ext,
{ simp [H, e₁₂.to_equiv.eq_comp_symm f g] },
{ simp [←H, ←e₁₂.to_equiv.eq_comp_symm f g] }
end | lemma | linear_equiv.eq_comp_to_linear_map_symm | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_to_linear_map_symm_eq (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₃] M₃) :
g.comp e₁₂.symm.to_linear_map = f ↔ g = f.comp e₁₂.to_linear_map | begin
split; intro H; ext,
{ simp [←H, ←e₁₂.to_equiv.comp_symm_eq f g] },
{ simp [H, e₁₂.to_equiv.comp_symm_eq f g] }
end | lemma | linear_equiv.comp_to_linear_map_symm_eq | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_linear_map_symm_comp (f : M₃ →ₛₗ[σ₃₁] M₁) (g : M₃ →ₛₗ[σ₃₂] M₂) :
f = e₁₂.symm.to_linear_map.comp g ↔ e₁₂.to_linear_map.comp f = g | begin
split; intro H; ext,
{ simp [H, e₁₂.to_equiv.eq_symm_comp f g] },
{ simp [←H, ←e₁₂.to_equiv.eq_symm_comp f g] }
end | lemma | linear_equiv.eq_to_linear_map_symm_comp | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_symm_comp_eq (f : M₃ →ₛₗ[σ₃₁] M₁) (g : M₃ →ₛₗ[σ₃₂] M₂) :
e₁₂.symm.to_linear_map.comp g = f ↔ g = e₁₂.to_linear_map.comp f | begin
split; intro H; ext,
{ simp [←H, ←e₁₂.to_equiv.symm_comp_eq f g] },
{ simp [H, e₁₂.to_equiv.symm_comp_eq f g] }
end | lemma | linear_equiv.to_linear_map_symm_comp_eq | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_symm [module R M] : (refl R M).symm = linear_equiv.refl R M | rfl | lemma | linear_equiv.refl_symm | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"linear_equiv.refl",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_trans_symm (f : M₁ ≃ₛₗ[σ₁₂] M₂) :
f.trans f.symm = linear_equiv.refl R₁ M₁ | by { ext x, simp } | lemma | linear_equiv.self_trans_symm | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"linear_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_self (f : M₁ ≃ₛₗ[σ₁₂] M₂) :
f.symm.trans f = linear_equiv.refl R₂ M₂ | by { ext x, simp } | lemma | linear_equiv.symm_trans_self | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"linear_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_to_linear_map [module R M] :
(linear_equiv.refl R M : M →ₗ[R] M) = linear_map.id | rfl | lemma | linear_equiv.refl_to_linear_map | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"linear_equiv.refl",
"linear_map.id",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_coe [module R M] [module R M₂] [module R M₃] (f : M ≃ₗ[R] M₂)
(f' : M₂ ≃ₗ[R] M₃) : (f' : M₂ →ₗ[R] M₃).comp (f : M →ₗ[R] M₂) = (f.trans f' : M ≃ₗ[R] M₃) | rfl | lemma | linear_equiv.comp_coe | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (h₁ h₂ f h₃ h₄) :
(linear_equiv.mk e h₁ h₂ f h₃ h₄ : M ≃ₛₗ[σ] M₂) = e | ext $ λ _, rfl | lemma | linear_equiv.mk_coe | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (a b : M) : e (a + b) = e a + e b | map_add e a b | theorem | linear_equiv.map_add | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smulₛₗ (c : R) (x : M) : e (c • x) = (σ c) • e x | e.map_smul' c x | theorem | linear_equiv.map_smulₛₗ | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul (e : N₁ ≃ₗ[R₁] N₂) (c : R₁) (x : N₁) :
e (c • x) = c • e x | map_smulₛₗ e c x | theorem | linear_equiv.map_smul | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff {x : M} : e x = 0 ↔ x = 0 | e.to_add_equiv.map_eq_zero_iff | theorem | linear_equiv.map_eq_zero_iff | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne_zero_iff {x : M} : e x ≠ 0 ↔ x ≠ 0 | e.to_add_equiv.map_ne_zero_iff | theorem | linear_equiv.map_ne_zero_iff | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm (e : M ≃ₛₗ[σ] M₂): e.symm.symm = e | by { cases e, refl } | theorem | linear_equiv.symm_symm | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_bijective [module R M] [module S M₂] [ring_hom_inv_pair σ' σ]
[ring_hom_inv_pair σ σ'] : function.bijective (symm : (M ≃ₛₗ[σ] M₂) → (M₂ ≃ₛₗ[σ'] M)) | equiv.bijective ⟨(symm : (M ≃ₛₗ[σ] M₂) →
(M₂ ≃ₛₗ[σ'] M)), (symm : (M₂ ≃ₛₗ[σ'] M) → (M ≃ₛₗ[σ] M₂)), symm_symm, symm_symm⟩ | lemma | linear_equiv.symm_bijective | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"equiv.bijective",
"module",
"ring_hom_inv_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe' (f h₁ h₂ h₃ h₄) : (linear_equiv.mk f h₁ h₂ ⇑e h₃ h₄ :
M₂ ≃ₛₗ[σ'] M) = e.symm | symm_bijective.injective $ ext $ λ x, rfl | lemma | linear_equiv.mk_coe' | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_mk (f h₁ h₂ h₃ h₄) :
(⟨e, h₁, h₂, f, h₃, h₄⟩ : M ≃ₛₗ[σ] M₂).symm =
{ to_fun := f, inv_fun := e,
..(⟨e, h₁, h₂, f, h₃, h₄⟩ : M ≃ₛₗ[σ] M₂).symm } | rfl | theorem | linear_equiv.symm_mk | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_symm_mk [module R M] [module R M₂]
{to_fun inv_fun map_add map_smul left_inv right_inv} :
⇑((⟨to_fun, map_add, map_smul, inv_fun, left_inv, right_inv⟩ : M ≃ₗ[R] M₂).symm) = inv_fun | rfl | lemma | linear_equiv.coe_symm_mk | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"inv_fun",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijective : function.bijective e | e.to_equiv.bijective | lemma | linear_equiv.bijective | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective : function.injective e | e.to_equiv.injective | lemma | linear_equiv.injective | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective : function.surjective e | e.to_equiv.surjective | lemma | linear_equiv.surjective | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_eq_preimage (s : set M) : e '' s = e.symm ⁻¹' s | e.to_equiv.image_eq_preimage s | lemma | linear_equiv.image_eq_preimage | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_symm_eq_preimage (s : set M₂) : e.symm '' s = e ⁻¹' s | e.to_equiv.symm.image_eq_preimage s | lemma | linear_equiv.image_symm_eq_preimage | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ring_equiv.to_semilinear_equiv (f : R ≃+* S) :
by haveI | ring_hom_inv_pair.of_ring_equiv f;
haveI := ring_hom_inv_pair.symm (↑f : R →+* S) (f.symm : S →+* R);
exact (R ≃ₛₗ[(↑f : R →+* S)] S) :=
by exact
{ to_fun := f,
map_smul' := f.map_mul,
.. f} | def | ring_equiv.to_semilinear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"ring_hom_inv_pair.of_ring_equiv",
"ring_hom_inv_pair.symm"
] | Interpret a `ring_equiv` `f` as an `f`-semilinear equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_involutive {σ σ' : R →+* R} [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ]
{module_M : module R M} (f : M →ₛₗ[σ] M) (hf : involutive f) :
M ≃ₛₗ[σ] M | { .. f, .. hf.to_perm f } | def | linear_equiv.of_involutive | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"involutive",
"module",
"ring_hom_inv_pair"
] | An involutive linear map is a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_involutive {σ σ' : R →+* R} [ring_hom_inv_pair σ σ']
[ring_hom_inv_pair σ' σ] {module_M : module R M} (f : M →ₛₗ[σ] M) (hf : involutive f) :
⇑(of_involutive f hf) = f | rfl | lemma | linear_equiv.coe_of_involutive | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"involutive",
"module",
"ring_hom_inv_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars (f : M ≃ₗ[S] M₂) : M ≃ₗ[R] M₂ | { to_fun := f,
inv_fun := f.symm,
left_inv := f.left_inv,
right_inv := f.right_inv,
.. f.to_linear_map.restrict_scalars R } | def | linear_equiv.restrict_scalars | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"inv_fun",
"restrict_scalars"
] | If `M` and `M₂` are both `R`-semimodules and `S`-semimodules and `R`-semimodule structures
are defined by an action of `R` on `S` (formally, we have two scalar towers), then any `S`-linear
equivalence from `M` to `M₂` is also an `R`-linear equivalence.
See also `linear_map.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 f g h, ext (linear_equiv.congr_fun h : _) | lemma | linear_equiv.restrict_scalars_injective | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"linear_equiv.congr_fun",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_inj (f g : M ≃ₗ[S] M₂) :
f.restrict_scalars R = g.restrict_scalars R ↔ f = g | (restrict_scalars_injective R).eq_iff | lemma | linear_equiv.restrict_scalars_inj | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
automorphism_group : group (M ≃ₗ[R] M) | { mul := λ f g, g.trans f,
one := linear_equiv.refl R M,
inv := λ f, f.symm,
mul_assoc := λ f g h, rfl,
mul_one := λ f, ext $ λ x, rfl,
one_mul := λ f, ext $ λ x, rfl,
mul_left_inv := λ f, ext $ f.left_inv } | instance | linear_equiv.automorphism_group | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"group",
"linear_equiv.refl",
"mul_assoc",
"mul_left_inv",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
automorphism_group.to_linear_map_monoid_hom : (M ≃ₗ[R] M) →* (M →ₗ[R] M) | { to_fun := coe,
map_one' := rfl,
map_mul' := λ _ _, rfl } | def | linear_equiv.automorphism_group.to_linear_map_monoid_hom | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | Restriction from `R`-linear automorphisms of `M` to `R`-linear endomorphisms of `M`,
promoted to a monoid hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_distrib_mul_action : distrib_mul_action (M ≃ₗ[R] M) M | { smul := ($),
smul_zero := linear_equiv.map_zero,
smul_add := linear_equiv.map_add,
one_smul := λ _, rfl,
mul_smul := λ _ _ _, rfl } | instance | linear_equiv.apply_distrib_mul_action | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"distrib_mul_action",
"linear_equiv.map_add",
"linear_equiv.map_zero",
"one_smul",
"smul_add",
"smul_zero"
] | The tautological action by `M ≃ₗ[R] M` on `M`.
This generalizes `function.End.apply_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def (f : M ≃ₗ[R] M) (a : M) :
f • a = f a | rfl | lemma | linear_equiv.smul_def | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_has_faithful_smul : has_faithful_smul (M ≃ₗ[R] M) M | ⟨λ _ _, linear_equiv.ext⟩ | instance | linear_equiv.apply_has_faithful_smul | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"has_faithful_smul"
] | `linear_equiv.apply_distrib_mul_action` is faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_smul_comm_class : smul_comm_class R (M ≃ₗ[R] M) M | { smul_comm := λ r e m, (e.map_smul r m).symm } | instance | linear_equiv.apply_smul_comm_class | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_smul_comm_class' : smul_comm_class (M ≃ₗ[R] M) R M | { smul_comm := linear_equiv.map_smul } | instance | linear_equiv.apply_smul_comm_class' | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"linear_equiv.map_smul",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subsingleton : M ≃ₗ[R] M₂ | { to_fun := λ _, 0,
inv_fun := λ _, 0,
left_inv := λ x, subsingleton.elim _ _,
right_inv := λ x, subsingleton.elim _ _,
.. (0 : M →ₗ[R] M₂)} | def | linear_equiv.of_subsingleton | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"inv_fun"
] | Any two modules that are subsingletons are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_subsingleton_self : of_subsingleton M M = refl R M | by { ext, simp } | lemma | linear_equiv.of_subsingleton_self | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_hom.to_linear_equiv {R S : Type*} [semiring R] [semiring S] (g : R ≃+* S) :
(by haveI := comp_hom S (↑g : R →+* S); exact (R ≃ₗ[R] S)) | by exact
{ to_fun := (g : R → S),
inv_fun := (g.symm : S → R),
map_smul' := g.map_mul,
..g } | def | module.comp_hom.to_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"inv_fun",
"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_equiv (s : S) : M ≃ₗ[R] M | { ..to_add_equiv M s,
..to_linear_map R M s } | def | distrib_mul_action.to_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | Each element of the group defines a linear equivalence.
This is a stronger version of `distrib_mul_action.to_add_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_module_aut : S →* M ≃ₗ[R] M | { to_fun := to_linear_equiv R M,
map_one' := linear_equiv.ext $ one_smul _,
map_mul' := λ a b, linear_equiv.ext $ mul_smul _ _ } | def | distrib_mul_action.to_module_aut | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"linear_equiv.ext",
"one_smul"
] | Each element of the group defines a module automorphism.
This is a stronger version of `distrib_mul_action.to_add_aut`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_equiv (h : ∀ (c : R) x, e (c • x) = c • e x) : M ≃ₗ[R] M₂ | { map_smul' := h, .. e, } | def | add_equiv.to_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | An additive equivalence whose underlying function preserves `smul` is a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_linear_equiv (h : ∀ (c : R) x, e (c • x) = c • e x) :
⇑(e.to_linear_equiv h) = e | rfl | lemma | add_equiv.coe_to_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_linear_equiv_symm (h : ∀ (c : R) x, e (c • x) = c • e x) :
⇑(e.to_linear_equiv h).symm = e.symm | rfl | lemma | add_equiv.coe_to_linear_equiv_symm | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_linear_equiv : M ≃ₗ[ℕ] M₂ | e.to_linear_equiv $ λ c a, by { erw e.to_add_monoid_hom.map_nsmul, refl } | def | add_equiv.to_nat_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | An additive equivalence between commutative additive monoids is a linear equivalence between
ℕ-modules | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_nat_linear_equiv :
⇑(e.to_nat_linear_equiv) = e | rfl | lemma | add_equiv.coe_to_nat_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_linear_equiv_to_add_equiv :
e.to_nat_linear_equiv.to_add_equiv = e | by { ext, refl } | lemma | add_equiv.to_nat_linear_equiv_to_add_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.linear_equiv.to_add_equiv_to_nat_linear_equiv
(e : M ≃ₗ[ℕ] M₂) : e.to_add_equiv.to_nat_linear_equiv = e | fun_like.coe_injective rfl | lemma | linear_equiv.to_add_equiv_to_nat_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_linear_equiv_symm :
(e.to_nat_linear_equiv).symm = e.symm.to_nat_linear_equiv | rfl | lemma | add_equiv.to_nat_linear_equiv_symm | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_linear_equiv_refl :
((add_equiv.refl M).to_nat_linear_equiv) = linear_equiv.refl ℕ M | rfl | lemma | add_equiv.to_nat_linear_equiv_refl | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"linear_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_linear_equiv_trans (e₂ : M₂ ≃+ M₃) :
(e.to_nat_linear_equiv).trans (e₂.to_nat_linear_equiv) = (e.trans e₂).to_nat_linear_equiv | rfl | lemma | add_equiv.to_nat_linear_equiv_trans | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_int_linear_equiv : M ≃ₗ[ℤ] M₂ | e.to_linear_equiv $ λ c a, e.to_add_monoid_hom.map_zsmul a c | def | add_equiv.to_int_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | An additive equivalence between commutative additive groups is a linear
equivalence between ℤ-modules | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_int_linear_equiv :
⇑(e.to_int_linear_equiv) = e | rfl | lemma | add_equiv.coe_to_int_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_int_linear_equiv_to_add_equiv :
e.to_int_linear_equiv.to_add_equiv = e | by { ext, refl } | lemma | add_equiv.to_int_linear_equiv_to_add_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.linear_equiv.to_add_equiv_to_int_linear_equiv
(e : M ≃ₗ[ℤ] M₂) : e.to_add_equiv.to_int_linear_equiv = e | fun_like.coe_injective rfl | lemma | linear_equiv.to_add_equiv_to_int_linear_equiv | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_int_linear_equiv_symm :
(e.to_int_linear_equiv).symm = e.symm.to_int_linear_equiv | rfl | lemma | add_equiv.to_int_linear_equiv_symm | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_int_linear_equiv_refl :
((add_equiv.refl M).to_int_linear_equiv) = linear_equiv.refl ℤ M | rfl | lemma | add_equiv.to_int_linear_equiv_refl | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [
"linear_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_int_linear_equiv_trans (e₂ : M₂ ≃+ M₃) :
(e.to_int_linear_equiv).trans (e₂.to_int_linear_equiv) = (e.trans e₂).to_int_linear_equiv | rfl | lemma | add_equiv.to_int_linear_equiv_trans | algebra.module | src/algebra/module/equiv.lean | [
"algebra.module.linear_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gdistrib_mul_action [add_monoid ι] [gmonoid A] [Π i, add_monoid (M i)]
extends gmul_action A M | (smul_add {i j} (a : A i) (b c : M j) : smul a (b + c) = smul a b + smul a c)
(smul_zero {i j} (a : A i) : smul a (0 : M j) = 0) | class | direct_sum.gdistrib_mul_action | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"add_monoid",
"smul_add",
"smul_zero"
] | A graded version of `distrib_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gmodule [add_monoid ι] [Π i, add_monoid (A i)] [Π i, add_monoid (M i)]
[gmonoid A] extends gdistrib_mul_action A M | (add_smul {i j} (a a' : A i) (b : M j) : smul (a + a') b = smul a b + smul a' b)
(zero_smul {i j} (b : M j) : smul (0 : A i) b = 0) | class | direct_sum.gmodule | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"add_monoid",
"add_smul",
"zero_smul"
] | A graded version of `module`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gsemiring.to_gmodule [decidable_eq ι] [add_monoid ι]
[Π (i : ι), add_comm_monoid (A i)] [gsemiring A] :
gmodule A A | { smul_add := λ _ _, gsemiring.mul_add,
smul_zero := λ i j, gsemiring.mul_zero,
add_smul := λ i j, gsemiring.add_mul,
zero_smul := λ i j, gsemiring.zero_mul,
..gmonoid.to_gmul_action A } | instance | direct_sum.gsemiring.to_gmodule | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"add_comm_monoid",
"add_monoid",
"add_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | A graded version of `semiring.to_module`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gsmul_hom [gmonoid A] [gmodule A M] {i j} :
A i →+ M j →+ M (i + j) | { to_fun := λ a,
{ to_fun := λ b, ghas_smul.smul a b,
map_zero' := gdistrib_mul_action.smul_zero _,
map_add' := gdistrib_mul_action.smul_add _ },
map_zero' := add_monoid_hom.ext $ λ a, gmodule.zero_smul a,
map_add' := λ a₁ a₂, add_monoid_hom.ext $ λ b, gmodule.add_smul _ _ _} | def | direct_sum.gsmul_hom | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [] | The piecewise multiplication from the `has_mul` instance, as a bundled homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_add_monoid_hom
[decidable_eq ι] [gmonoid A] [gmodule A M] :
(⨁ i, A i) →+ (⨁ i, M i) →+ (⨁ i, M i) | to_add_monoid $ λ i, add_monoid_hom.flip $
to_add_monoid $ λ j, add_monoid_hom.flip $
(of M _).comp_hom.comp $ gsmul_hom A M | def | direct_sum.gmodule.smul_add_monoid_hom | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [] | For graded monoid `A` and a graded module `M` over `A`. `gmodule.smul_add_monoid_hom` is the
`⨁ᵢ Aᵢ`-scalar multiplication on `⨁ᵢ Mᵢ` induced by `gsmul_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def [decidable_eq ι] [gmonoid A] [gmodule A M]
(x : ⨁ i, A i) (y : ⨁ i, M i) : x • y = smul_add_monoid_hom _ _ x y | rfl | lemma | direct_sum.gmodule.smul_def | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_add_monoid_hom_apply_of_of [decidable_eq ι] [gmonoid A] [gmodule A M]
{i j} (x : A i) (y : M j) :
smul_add_monoid_hom A M (direct_sum.of A i x) (of M j y) =
of M (i + j) (ghas_smul.smul x y) | by simp [smul_add_monoid_hom] | lemma | direct_sum.gmodule.smul_add_monoid_hom_apply_of_of | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"direct_sum.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_smul_of [decidable_eq ι] [gmonoid A] [gmodule A M]
{i j} (x : A i) (y : M j) :
direct_sum.of A i x • of M j y = of M (i + j) (ghas_smul.smul x y) | smul_add_monoid_hom_apply_of_of _ _ _ _ | lemma | direct_sum.gmodule.of_smul_of | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"direct_sum.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_smul [decidable_eq ι] [gmonoid A] [gmodule A M] (x : ⨁ i, M i) :
(1 : ⨁ i, A i) • x = x | suffices smul_add_monoid_hom A M 1 = add_monoid_hom.id (⨁ i, M i),
from add_monoid_hom.congr_fun this x,
begin
apply direct_sum.add_hom_ext, intros i xi,
unfold has_one.one,
rw smul_add_monoid_hom_apply_of_of,
exact direct_sum.of_eq_of_graded_monoid_eq (one_smul (graded_monoid A) $ graded_monoid.mk i xi),
end | lemma | direct_sum.gmodule.one_smul | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"direct_sum.add_hom_ext",
"direct_sum.of_eq_of_graded_monoid_eq",
"graded_monoid",
"graded_monoid.mk",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_smul [decidable_eq ι] [gsemiring A] [gmodule A M]
(a b : ⨁ i, A i) (c : ⨁ i, M i) : (a * b) • c = a • (b • c) | suffices (smul_add_monoid_hom A M).comp_hom.comp (direct_sum.mul_hom A)
-- `λ a b c, (a * b) • c` as a bundled hom
= (add_monoid_hom.comp_hom add_monoid_hom.flip_hom $
(smul_add_monoid_hom A M).flip.comp_hom.comp $ smul_add_monoid_hom A M).flip,
-- `λ a b c, a • (b • c)` as a bundled hom
f... | lemma | direct_sum.gmodule.mul_smul | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"direct_sum.mul_hom",
"direct_sum.mul_hom_of_of",
"direct_sum.of_eq_of_graded_monoid_eq",
"graded_monoid.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module [decidable_eq ι] [gsemiring A] [gmodule A M] :
module (⨁ i, A i) (⨁ i, M i) | { smul := (•),
one_smul := one_smul _ _,
mul_smul := mul_smul _ _,
smul_add := λ r, (smul_add_monoid_hom A M r).map_add,
smul_zero := λ r, (smul_add_monoid_hom A M r).map_zero,
add_smul := λ r s x, by simp only [smul_def, map_add, add_monoid_hom.add_apply],
zero_smul := λ x, by simp only [smul_def, map_zero... | instance | direct_sum.gmodule.module | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"add_smul",
"module",
"one_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | The `module` derived from `gmodule A M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gmul_action [add_monoid M] [distrib_mul_action A M]
[set_like σ M] [set_like.graded_monoid 𝓐] [set_like.has_graded_smul 𝓐 𝓜] :
graded_monoid.gmul_action (λ i, 𝓐 i) (λ i, 𝓜 i) | { one_smul := λ ⟨i, m⟩, sigma.subtype_ext (zero_add _) (one_smul _ _),
mul_smul := λ ⟨i, a⟩ ⟨j, a'⟩ ⟨k, b⟩, sigma.subtype_ext (add_assoc _ _ _) (mul_smul _ _ _),
..set_like.ghas_smul 𝓐 𝓜 } | instance | set_like.gmul_action | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"add_monoid",
"distrib_mul_action",
"graded_monoid.gmul_action",
"one_smul",
"set_like",
"set_like.ghas_smul",
"set_like.graded_monoid",
"set_like.has_graded_smul",
"sigma.subtype_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gdistrib_mul_action [add_monoid M] [distrib_mul_action A M]
[set_like σ M] [add_submonoid_class σ M] [set_like.graded_monoid 𝓐]
[set_like.has_graded_smul 𝓐 𝓜] :
direct_sum.gdistrib_mul_action (λ i, 𝓐 i) (λ i, 𝓜 i) | { smul_add := λ i j a b c, subtype.ext $ smul_add _ _ _,
smul_zero := λ i j a, subtype.ext $ smul_zero _,
..set_like.gmul_action 𝓐 𝓜 } | instance | set_like.gdistrib_mul_action | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"add_monoid",
"add_submonoid_class",
"direct_sum.gdistrib_mul_action",
"distrib_mul_action",
"set_like",
"set_like.gmul_action",
"set_like.graded_monoid",
"set_like.has_graded_smul",
"smul_add",
"smul_zero",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gmodule : direct_sum.gmodule (λ i, 𝓐 i) (λ i, 𝓜 i) | { smul := λ i j x y, ⟨(x : A) • (y : M), set_like.has_graded_smul.smul_mem x.2 y.2⟩,
add_smul := λ i j a a' b, subtype.ext $ add_smul _ _ _,
zero_smul := λ i j b, subtype.ext $ zero_smul _ _,
..set_like.gdistrib_mul_action 𝓐 𝓜} | instance | set_like.gmodule | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"add_smul",
"direct_sum.gmodule",
"set_like.gdistrib_mul_action",
"subtype.ext",
"zero_smul"
] | `[set_like.graded_monoid 𝓐] [set_like.has_graded_smul 𝓐 𝓜]` is the internal version of graded
module, the internal version can be translated into the external version `gmodule`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_module [decidable_eq ι] [graded_ring 𝓐] :
module A (⨁ i, 𝓜 i) | { smul := λ a b, direct_sum.decompose 𝓐 a • b,
.. module.comp_hom _ (direct_sum.decompose_ring_equiv 𝓐 : A ≃+* ⨁ i, 𝓐 i).to_ring_hom } | def | graded_module.is_module | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"direct_sum.decompose",
"direct_sum.decompose_ring_equiv",
"graded_ring",
"module",
"module.comp_hom"
] | The smul multiplication of `A` on `⨁ i, 𝓜 i` from `(⨁ i, 𝓐 i) →+ (⨁ i, 𝓜 i) →+ ⨁ i, 𝓜 i`
turns `⨁ i, 𝓜 i` into an `A`-module | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv [decidable_eq ι] [graded_ring 𝓐]
[direct_sum.decomposition 𝓜] :
M ≃ₗ[A] ⨁ i, 𝓜 i | { to_fun := direct_sum.decompose_add_equiv 𝓜,
map_smul' := λ x y, begin
classical,
rw [← direct_sum.sum_support_decompose 𝓐 x, map_sum, finset.sum_smul, map_sum,
finset.sum_smul, finset.sum_congr rfl (λ i hi, _)],
rw [ring_hom.id_apply, ← direct_sum.sum_support_decompose 𝓜 y, map_sum, finset.smul... | def | graded_module.linear_equiv | algebra.module | src/algebra/module/graded_module.lean | [
"ring_theory.graded_algebra.basic",
"algebra.graded_mul_action",
"algebra.direct_sum.decomposition",
"algebra.module.big_operators"
] | [
"direct_sum.decompose_add_equiv",
"direct_sum.decompose_coe",
"direct_sum.decomposition",
"direct_sum.gmodule.smul_add_monoid_hom_apply_of_of",
"direct_sum.sum_support_decompose",
"finset.smul_sum",
"finset.sum_smul",
"graded_ring",
"linear_equiv",
"ring_hom.id_apply"
] | `⨁ i, 𝓜 i` and `M` are isomorphic as `A`-modules.
"The internal version" and "the external version" are isomorphism as `A`-modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_smul (r : R) (f : A →+ B) : ⇑(r • f) = r • f | rfl | lemma | add_monoid_hom.coe_smul | algebra.module | src/algebra/module/hom.lean | [
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (r : R) (f : A →+ B) (x : A) : (r • f) x = r • f x | rfl | lemma | add_monoid_hom.smul_apply | algebra.module | src/algebra/module/hom.lean | [
"algebra.module.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.injective : Prop | (out : ∀ (X Y : Type (max u v)) [add_comm_group X] [add_comm_group Y] [module R X] [module R Y]
(f : X →ₗ[R] Y) (hf : function.injective f) (g : X →ₗ[R] Q),
∃ (h : Y →ₗ[R] Q), ∀ x, h (f x) = g x) | class | module.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"
] | [
"add_comm_group",
"module"
] | An `R`-module `Q` is injective if and only if every injective `R`-linear map descends to a linear
map to `Q`, i.e. in the following diagram, if `f` is injective then there is an `R`-linear map
`h : Y ⟶ Q` such that `g = h ∘ f`
```
X --- f ---> Y
|
| g
v
Q
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module.injective_object_of_injective_module [module.injective.{u v} R Q] :
category_theory.injective.{max u v} (⟨Q⟩ : Module.{max u v} R) | { factors := λ X Y g f mn, begin
rcases module.injective.out X Y f ((Module.mono_iff_injective f).mp mn) g with ⟨h, eq1⟩,
exact ⟨h, linear_map.ext eq1⟩,
end } | lemma | module.injective_object_of_injective_module | 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.mono_iff_injective",
"linear_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.injective_module_of_injective_object
[category_theory.injective.{max u v} (⟨Q⟩ : Module.{max u v} R)] :
module.injective.{u v} R Q | { out := λ X Y ins1 ins2 ins3 ins4 f hf g, begin
resetI,
rcases @category_theory.injective.factors (Module R) _ ⟨Q⟩ _ ⟨X⟩ ⟨Y⟩ g f
((Module.mono_iff_injective _).mpr hf) with ⟨h, rfl⟩,
exact ⟨h, λ x, rfl⟩
end } | lemma | module.injective_module_of_injective_object | 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",
"Module.mono_iff_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.injective_iff_injective_object :
module.injective.{u v} R Q ↔ category_theory.injective.{max u v} (⟨Q⟩ : Module.{max u v} R) | ⟨λ h, @@module.injective_object_of_injective_module R _ Q _ _ h,
λ h, @@module.injective_module_of_injective_object R _ Q _ _ h⟩ | lemma | module.injective_iff_injective_object | 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.injective_module_of_injective_object",
"module.injective_object_of_injective_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.Baer : Prop | ∀ (I : ideal R) (g : I →ₗ[R] Q), ∃ (g' : R →ₗ[R] Q),
∀ (x : R) (mem : x ∈ I), g' x = g ⟨x, mem⟩ | def | module.Baer | 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"
] | An `R`-module `Q` satisfies Baer's criterion if any `R`-linear map from an `ideal R` extends to
an `R`-linear map `R ⟶ Q` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of extends linear_pmap R N Q | (le : i.range ≤ domain)
(is_extension : ∀ (m : M), f m = to_linear_pmap ⟨i m, le ⟨m, rfl⟩⟩) | structure | module.Baer.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"
] | [
"linear_pmap"
] | If we view `M` as a submodule of `N` via the injective linear map `i : M ↪ N`, then a submodule
between `M` and `N` is a submodule `N'` of `N`. To prove Baer's criterion, we need to consider
pairs of `(N', f')` such that `M ≤ N' ≤ N` and `f'` extends `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_of.ext {a b : extension_of i f}
(domain_eq : a.domain = b.domain)
(to_fun_eq : ∀ ⦃x : a.domain⦄ ⦃y : b.domain⦄,
(x : N) = y → a.to_linear_pmap x = b.to_linear_pmap y) : a = b | begin
rcases a with ⟨a, a_le, e1⟩,
rcases b with ⟨b, b_le, e2⟩,
congr,
exact linear_pmap.ext domain_eq to_fun_eq,
end | lemma | module.Baer.extension_of.ext | 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_pmap.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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