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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|---|---|---|---|---|---|---|---|---|---|---|
single₀_obj_X_d_to (X : V) (j : ℕ) : ((single₀ V).obj X).d_to j = 0 | by { rw [d_to_eq ((single₀ V).obj X) rfl], simp, } | lemma | chain_complex.single₀_obj_X_d_to | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_obj_X_d_from (X : V) (i : ℕ) : ((single₀ V).obj X).d_from i = 0 | begin
cases i,
{ rw [d_from_eq_zero], simp, },
{ rw [d_from_eq ((single₀ V).obj X) rfl], simp, },
end | lemma | chain_complex.single₀_obj_X_d_from | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_map_f_0 {X Y : V} (f : X ⟶ Y) : ((single₀ V).map f).f 0 = f | rfl | lemma | chain_complex.single₀_map_f_0 | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_map_f_succ {X Y : V} (f : X ⟶ Y) (n : ℕ) :
((single₀ V).map f).f (n+1) = 0 | rfl | lemma | chain_complex.single₀_map_f_succ | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology_functor_0_single₀ : single₀ V ⋙ homology_functor V _ 0 ≅ (𝟭 V) | nat_iso.of_components (λ X, homology.congr _ _ (by simp) (by simp) ≪≫ homology_zero_zero)
(λ X Y f, by { ext, dsimp [homology_functor], simp, }) | def | chain_complex.homology_functor_0_single₀ | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [
"homology.congr",
"homology_functor",
"homology_zero_zero"
] | Sending objects to chain complexes supported at `0` then taking `0`-th homology
is the same as doing nothing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_functor_succ_single₀ (n : ℕ) : single₀ V ⋙ homology_functor V _ (n+1) ≅ 0 | nat_iso.of_components (λ X, homology.congr _ _ (by simp) (by simp) ≪≫
homology_zero_zero ≪≫ (functor.zero_obj _).iso_zero.symm)
(λ X Y f, by { exact (functor.zero_obj _).eq_of_tgt _ _ }) | def | chain_complex.homology_functor_succ_single₀ | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [
"homology.congr",
"homology_functor",
"homology_zero_zero"
] | Sending objects to chain complexes supported at `0` then taking `(n+1)`-st homology
is the same as the zero functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_single₀_equiv (C : chain_complex V ℕ) (X : V) :
(C ⟶ (single₀ V).obj X) ≃ { f : C.X 0 ⟶ X // C.d 1 0 ≫ f = 0 } | { to_fun := λ f, ⟨f.f 0, by { rw ←f.comm 1 0, simp, }⟩,
inv_fun := λ f,
{ f := λ i, match i with
| 0 := f.1
| (n+1) := 0
end,
comm' := λ i j h, begin
rcases i with _|_|i; cases j; unfold_aux; simp only [comp_zero, zero_comp, single₀_obj_X_d],
{ rw [C.shape, zero_comp], simp, },
{ e... | def | chain_complex.to_single₀_equiv | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [
"chain_complex",
"inv_fun"
] | Morphisms from a `ℕ`-indexed chain complex `C`
to a single object chain complex with `X` concentrated in degree 0
are the same as morphisms `f : C.X 0 ⟶ X` such that `C.d 1 0 ≫ f = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_single₀_ext {C : chain_complex V ℕ} {X : V}
(f g : (C ⟶ (single₀ V).obj X)) (h : f.f 0 = g.f 0) : f = g | (to_single₀_equiv C X).injective (by { ext, exact h, }) | lemma | chain_complex.to_single₀_ext | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_single₀_equiv (C : chain_complex V ℕ) (X : V) :
((single₀ V).obj X ⟶ C) ≃ (X ⟶ C.X 0) | { to_fun := λ f, f.f 0,
inv_fun := λ f,
{ f := λ i, match i with
| 0 := f
| (n+1) := 0
end,
comm' := λ i j h, begin
cases i; cases j; unfold_aux;
simp only [shape, complex_shape.down_rel, nat.one_ne_zero, not_false_iff,
comp_zero, zero_comp, nat.succ_ne_zero, single₀_obj_X_d],
... | def | chain_complex.from_single₀_equiv | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [
"chain_complex",
"inv_fun"
] | Morphisms from a single object chain complex with `X` concentrated in degree 0
to a `ℕ`-indexed chain complex `C` are the same as morphisms `f : X → C.X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single₀_iso_single : single₀ V ≅ single V _ 0 | nat_iso.of_components
(λ X,
{ hom := { f := λ i, by { cases i; simpa using 𝟙 _, } },
inv := { f := λ i, by { cases i; simpa using 𝟙 _, } },
hom_inv_id' := by { ext (_|i); { dsimp, simp, }, },
inv_hom_id' := begin
ext (_|i),
{ apply category.id_comp, },
{ apply has_zero_object.to_zero... | def | chain_complex.single₀_iso_single | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | `single₀` is the same as `single V _ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single₀ : V ⥤ cochain_complex V ℕ | { obj := λ X,
{ X := λ n, match n with
| 0 := X
| (n+1) := 0
end,
d := λ i j, 0, },
map := λ X Y f,
{ f := λ n, match n with
| 0 := f
| (n+1) := 0
end, },
map_id' := λ X, by { ext n, cases n, refl, dsimp, unfold_aux, simp, },
map_comp' := λ X Y Z f g, by { ext n, cases n, refl, dsi... | def | cochain_complex.single₀ | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [
"cochain_complex"
] | `cochain_complex.single₀ V` is the embedding of `V` into `cochain_complex V ℕ`
as cochain complexes supported in degree 0.
This is naturally isomorphic to `single V _ 0`, but has better definitional properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single₀_obj_X_d_from (X : V) (j : ℕ) : ((single₀ V).obj X).d_from j = 0 | by { rw [d_from_eq ((single₀ V).obj X) rfl], simp, } | lemma | cochain_complex.single₀_obj_X_d_from | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_obj_X_d_to (X : V) (i : ℕ) : ((single₀ V).obj X).d_to i = 0 | begin
cases i,
{ rw [d_to_eq_zero], simp, },
{ rw [d_to_eq ((single₀ V).obj X) rfl], simp, },
end | lemma | cochain_complex.single₀_obj_X_d_to | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_single₀_equiv (C : cochain_complex V ℕ) (X : V) :
((single₀ V).obj X ⟶ C) ≃ { f : X ⟶ C.X 0 // f ≫ C.d 0 1 = 0 } | { to_fun := λ f, ⟨f.f 0, by { rw f.comm 0 1, simp, }⟩,
inv_fun := λ f,
{ f := λ i, match i with
| 0 := f.1
| (n+1) := 0
end,
comm' := λ i j h, begin
rcases j with _|_|j; cases i; unfold_aux; simp only [comp_zero, zero_comp, single₀_obj_X_d],
{ convert comp_zero, rw [C.shape], simp, },
... | def | cochain_complex.from_single₀_equiv | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [
"cochain_complex",
"inv_fun",
"nat.one_lt_succ_succ"
] | Morphisms from a single object cochain complex with `X` concentrated in degree 0
to a `ℕ`-indexed cochain complex `C`
are the same as morphisms `f : X ⟶ C.X 0` such that `f ≫ C.d 0 1 = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_short_exact_of_is_iso_of_is_iso (h : short_exact f g) (h' : short_exact f' g')
(i₁ : A ⟶ A') (i₂ : B ⟶ B') (i₃ : C ⟶ C')
(comm₁ : i₁ ≫ f' = f ≫ i₂) (comm₂ : i₂ ≫ g' = g ≫ i₃) [is_iso i₁] [is_iso i₃] :
is_iso i₂ | begin
obtain ⟨_⟩ := h,
obtain ⟨_⟩ := h',
resetI,
refine @abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso 𝒜 _ _ 0 _ _ _ 0 _ _ _
0 f g 0 f' g' 0 i₁ i₂ i₃ _ comm₁ comm₂ 0 0 0 0 0 _ _ _ _ _ _ _ _ _ _ _;
try { simp };
try { apply exact_zero_left_of_mono };
try { assumption };
rwa ← epi_iff_exact_... | lemma | category_theory.is_iso_of_short_exact_of_is_iso_of_is_iso | algebra.homology.short_exact | src/algebra/homology/short_exact/abelian.lean | [
"algebra.homology.short_exact.preadditive",
"category_theory.abelian.diagram_lemmas.four"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
splitting.mk' (h : short_exact f g) (i : B ⟶ A ⊞ C)
(h1 : f ≫ i = biprod.inl) (h2 : i ≫ biprod.snd = g) : splitting f g | { iso :=
begin
refine @as_iso _ _ _ _ i (id _),
refine is_iso_of_short_exact_of_is_iso_of_is_iso h _ _ _ _
(h1.trans (category.id_comp _).symm).symm (h2.trans (category.comp_id _).symm),
split,
apply exact_inl_snd
end,
comp_iso_eq_inl := by { rwa as_iso_hom, },
iso_comp_snd_eq := h2 } | def | category_theory.splitting.mk' | algebra.homology.short_exact | src/algebra/homology/short_exact/abelian.lean | [
"algebra.homology.short_exact.preadditive",
"category_theory.abelian.diagram_lemmas.four"
] | [
"iso"
] | To construct a splitting of `A -f⟶ B -g⟶ C` it suffices to supply
a *morphism* `i : B ⟶ A ⊞ C` such that `f ≫ i` is the canonical map `biprod.inl : A ⟶ A ⊞ C` and
`i ≫ q = g`, where `q` is the canonical map `biprod.snd : A ⊞ C ⟶ C`,
together with proofs that `f` is mono and `g` is epi.
The morphism `i` is then automat... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
splitting.mk'' (h : short_exact f g) (i : A ⊞ C ⟶ B)
(h1 : biprod.inl ≫ i = f) (h2 : i ≫ g = biprod.snd) : splitting f g | { iso :=
begin
refine (@as_iso _ _ _ _ i (id _)).symm,
refine is_iso_of_short_exact_of_is_iso_of_is_iso _ h _ _ _
(h1.trans (category.id_comp _).symm).symm (h2.trans (category.comp_id _).symm),
split,
apply exact_inl_snd
end,
comp_iso_eq_inl := by rw [iso.symm_hom, as_iso_inv, is_iso.comp_in... | def | category_theory.splitting.mk'' | algebra.homology.short_exact | src/algebra/homology/short_exact/abelian.lean | [
"algebra.homology.short_exact.preadditive",
"category_theory.abelian.diagram_lemmas.four"
] | [
"iso"
] | To construct a splitting of `A -f⟶ B -g⟶ C` it suffices to supply
a *morphism* `i : A ⊞ C ⟶ B` such that `p ≫ i = f` where `p` is the canonical map
`biprod.inl : A ⟶ A ⊞ C`, and `i ≫ g` is the canonical map `biprod.snd : A ⊞ C ⟶ C`,
together with proofs that `f` is mono and `g` is epi.
The morphism `i` is then automat... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_split.splitting {f : A ⟶ B} {g : B ⟶ C} (h : left_split f g) : splitting f g | splitting.mk' h.short_exact (biprod.lift h.left_split.some g)
(by { ext,
{ simpa only [biprod.inl_fst, biprod.lift_fst, category.assoc] using h.left_split.some_spec },
{ simp only [biprod.inl_snd, biprod.lift_snd, category.assoc, h.exact.w], } })
(by { simp only [biprod.lift_snd], }) | def | category_theory.left_split.splitting | algebra.homology.short_exact | src/algebra/homology/short_exact/abelian.lean | [
"algebra.homology.short_exact.preadditive",
"category_theory.abelian.diagram_lemmas.four"
] | [] | A short exact sequence that is left split admits a splitting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_split.splitting {f : A ⟶ B} {g : B ⟶ C} (h : right_split f g) : splitting f g | splitting.mk'' h.short_exact (biprod.desc f h.right_split.some)
(biprod.inl_desc _ _)
(by { ext,
{ rw [biprod.inl_snd, ← category.assoc, biprod.inl_desc, h.exact.w] },
{ rw [biprod.inr_snd, ← category.assoc, biprod.inr_desc, h.right_split.some_spec] } }) | def | category_theory.right_split.splitting | algebra.homology.short_exact | src/algebra/homology/short_exact/abelian.lean | [
"algebra.homology.short_exact.preadditive",
"category_theory.abelian.diagram_lemmas.four"
] | [] | A short exact sequence that is right split admits a splitting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
short_exact : Prop | [mono : mono f]
[epi : epi g]
(exact : exact f g) | structure | category_theory.short_exact | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | If `f : A ⟶ B` and `g : B ⟶ C` then `short_exact f g` is the proposition saying
the resulting diagram `0 ⟶ A ⟶ B ⟶ C ⟶ 0` is an exact sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_split : Prop | (left_split : ∃ φ : B ⟶ A, f ≫ φ = 𝟙 A)
[epi : epi g]
(exact : exact f g) | structure | category_theory.left_split | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | An exact sequence `A -f⟶ B -g⟶ C` is *left split*
if there exists a morphism `φ : B ⟶ A` such that `f ≫ φ = 𝟙 A` and `g` is epi.
Such a sequence is automatically short exact (i.e., `f` is mono). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_split.short_exact {f : A ⟶ B} {g : B ⟶ C} (h : left_split f g) : short_exact f g | { mono :=
begin
obtain ⟨φ, hφ⟩ := h.left_split,
haveI : mono (f ≫ φ) := by { rw hφ, apply_instance },
exact mono_of_mono f φ,
end,
epi := h.epi,
exact := h.exact } | lemma | category_theory.left_split.short_exact | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_split : Prop | (right_split : ∃ χ : C ⟶ B, χ ≫ g = 𝟙 C)
[mono : mono f]
(exact : exact f g) | structure | category_theory.right_split | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | An exact sequence `A -f⟶ B -g⟶ C` is *right split*
if there exists a morphism `φ : C ⟶ B` such that `f ≫ φ = 𝟙 A` and `f` is mono.
Such a sequence is automatically short exact (i.e., `g` is epi). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_split.short_exact {f : A ⟶ B} {g : B ⟶ C} (h : right_split f g) : short_exact f g | { epi :=
begin
obtain ⟨χ, hχ⟩ := h.right_split,
haveI : epi (χ ≫ g) := by { rw hχ, apply_instance },
exact epi_of_epi χ g,
end,
mono := h.mono,
exact := h.exact } | lemma | category_theory.right_split.short_exact | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split : Prop | (split : ∃ (φ : B ⟶ A) (χ : C ⟶ B),
f ≫ φ = 𝟙 A ∧ χ ≫ g = 𝟙 C ∧ f ≫ g = 0 ∧ χ ≫ φ = 0 ∧ φ ≫ f + g ≫ χ = 𝟙 B) | structure | category_theory.split | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | An exact sequence `A -f⟶ B -g⟶ C` is *split* if there exist
`φ : B ⟶ A` and `χ : C ⟶ B` such that:
* `f ≫ φ = 𝟙 A`
* `χ ≫ g = 𝟙 C`
* `f ≫ g = 0`
* `χ ≫ φ = 0`
* `φ ≫ f + g ≫ χ = 𝟙 B`
Such a sequence is automatically short exact (i.e., `f` is mono and `g` is epi). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_of_split {A B C : 𝒜} {f : A ⟶ B} {g : B ⟶ C} {χ : C ⟶ B} {φ : B ⟶ A}
(hfg : f ≫ g = 0) (H : φ ≫ f + g ≫ χ = 𝟙 B) : exact f g | { w := hfg,
epi :=
begin
let ψ : (kernel_subobject g : 𝒜) ⟶ image_subobject f :=
subobject.arrow _ ≫ φ ≫ factor_thru_image_subobject f,
suffices : ψ ≫ image_to_kernel f g hfg = 𝟙 _,
{ convert epi_of_epi ψ _, rw this, apply_instance },
rw ← cancel_mono (subobject.arrow _), swap, { apply_insta... | lemma | category_theory.exact_of_split | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [
"image_to_kernel",
"image_to_kernel_arrow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split.exact (h : split f g) : exact f g | by { obtain ⟨φ, χ, -, -, h1, -, h2⟩ := h, exact exact_of_split h1 h2 } | lemma | category_theory.split.exact | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split.left_split (h : split f g) : left_split f g | { left_split := by { obtain ⟨φ, χ, h1, -⟩ := h, exact ⟨φ, h1⟩, },
epi := begin
obtain ⟨φ, χ, -, h2, -⟩ := h,
have : epi (χ ≫ g), { rw h2, apply_instance },
exactI epi_of_epi χ g,
end,
exact := h.exact } | lemma | category_theory.split.left_split | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split.right_split (h : split f g) : right_split f g | { right_split := by { obtain ⟨φ, χ, -, h1, -⟩ := h, exact ⟨χ, h1⟩, },
mono := begin
obtain ⟨φ, χ, h1, -⟩ := h,
have : mono (f ≫ φ), { rw h1, apply_instance },
exactI mono_of_mono f φ,
end,
exact := h.exact } | lemma | category_theory.split.right_split | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split.short_exact (h : split f g) : short_exact f g | h.left_split.short_exact | lemma | category_theory.split.short_exact | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split.map {𝒜 ℬ : Type*} [category 𝒜] [preadditive 𝒜] [category ℬ] [preadditive ℬ]
(F : 𝒜 ⥤ ℬ) [functor.additive F] {A B C : 𝒜} {f : A ⟶ B} {g : B ⟶ C} (h : split f g) :
split (F.map f) (F.map g) | begin
obtain ⟨φ, χ, h1, h2, h3, h4, h5⟩ := h,
refine ⟨⟨F.map φ, F.map χ, _⟩⟩,
simp only [← F.map_comp, ← F.map_id, ← F.map_add, F.map_zero, *, eq_self_iff_true, and_true],
end | lemma | category_theory.split.map | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_inl_snd [has_binary_biproducts 𝒜] (A B : 𝒜) :
exact (biprod.inl : A ⟶ A ⊞ B) biprod.snd | exact_of_split biprod.inl_snd biprod.total | lemma | category_theory.exact_inl_snd | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | The sequence `A ⟶ A ⊞ B ⟶ B` is exact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_inr_fst [has_binary_biproducts 𝒜] (A B : 𝒜) :
exact (biprod.inr : B ⟶ A ⊞ B) biprod.fst | exact_of_split biprod.inr_fst ((add_comm _ _).trans biprod.total) | lemma | category_theory.exact_inr_fst | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | The sequence `B ⟶ A ⊞ B ⟶ A` is exact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
splitting [has_zero_morphisms 𝒜] [has_binary_biproducts 𝒜] | (iso : B ≅ A ⊞ C)
(comp_iso_eq_inl : f ≫ iso.hom = biprod.inl)
(iso_comp_snd_eq : iso.hom ≫ biprod.snd = g) | structure | category_theory.splitting | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [
"iso"
] | A *splitting* of a sequence `A -f⟶ B -g⟶ C` is an isomorphism
to the short exact sequence `0 ⟶ A ⟶ A ⊞ C ⟶ C ⟶ 0` such that
the vertical maps on the left and the right are the identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_comp_iso_eq : biprod.inl ≫ h.iso.inv = f | by rw [iso.comp_inv_eq, h.comp_iso_eq_inl] | lemma | category_theory.splitting.inl_comp_iso_eq | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_comp_eq_snd : h.iso.inv ≫ g = biprod.snd | by rw [iso.inv_comp_eq, h.iso_comp_snd_eq] | lemma | category_theory.splitting.iso_comp_eq_snd | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.category_theory.splitting.section : C ⟶ B | biprod.inr ≫ h.iso.inv | def | category_theory.splitting.section | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | If `h` is a splitting of `A -f⟶ B -g⟶ C`,
then `h.section : C ⟶ B` is the morphism satisfying `h.section ≫ g = 𝟙 C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
retraction : B ⟶ A | h.iso.hom ≫ biprod.fst | def | category_theory.splitting.retraction | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | If `h` is a splitting of `A -f⟶ B -g⟶ C`,
then `h.retraction : B ⟶ A` is the morphism satisfying `f ≫ h.retraction = 𝟙 A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
section_π : h.section ≫ g = 𝟙 C | by { delta splitting.section, simp } | lemma | category_theory.splitting.section_π | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_retraction : f ≫ h.retraction = 𝟙 A | by { delta retraction, simp } | lemma | category_theory.splitting.ι_retraction | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
section_retraction : h.section ≫ h.retraction = 0 | by { delta splitting.section retraction, simp } | lemma | category_theory.splitting.section_retraction | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_mono : split_mono f | ⟨h.retraction, by simp⟩ | def | category_theory.splitting.split_mono | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | The retraction in a splitting is a split mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_epi : split_epi g | ⟨h.section, by simp⟩ | def | category_theory.splitting.split_epi | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | The section in a splitting is a split epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr_iso_inv : biprod.inr ≫ h.iso.inv = h.section | rfl | lemma | category_theory.splitting.inr_iso_inv | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_hom_fst : h.iso.hom ≫ biprod.fst = h.retraction | rfl | lemma | category_theory.splitting.iso_hom_fst | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
splitting_of_is_iso_zero {X Y Z : 𝒜} (f : X ⟶ Y) [is_iso f] (hZ : is_zero Z) :
splitting f (0 : Y ⟶ Z) | ⟨(as_iso f).symm ≪≫ iso_biprod_zero hZ, by simp [hZ.eq_of_tgt _ 0], by simp⟩ | def | category_theory.splitting.splitting_of_is_iso_zero | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | A short exact sequence of the form `X -f⟶ Y -0⟶ Z` where `f` is an iso and `Z` is zero
has a splitting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono : mono f | begin
apply mono_of_mono _ h.retraction,
rw h.ι_retraction,
apply_instance
end | lemma | category_theory.splitting.mono | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi : epi g | begin
apply_with (epi_of_epi h.section) { instances := ff },
rw h.section_π,
apply_instance
end | lemma | category_theory.splitting.epi | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_add : h.retraction ≫ f + g ≫ h.section = 𝟙 _ | begin
delta splitting.section retraction,
rw [← cancel_mono h.iso.hom, ← cancel_epi h.iso.inv],
simp only [category.comp_id, category.id_comp, category.assoc,
iso.inv_hom_id_assoc, iso.inv_hom_id, limits.biprod.total,
preadditive.comp_add, preadditive.add_comp,
splitting.comp_iso_eq_inl, splitting.iso... | lemma | category_theory.splitting.split_add | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
retraction_ι_eq_id_sub :
h.retraction ≫ f = 𝟙 _ - g ≫ h.section | eq_sub_iff_add_eq.mpr h.split_add | lemma | category_theory.splitting.retraction_ι_eq_id_sub | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π_section_eq_id_sub :
g ≫ h.section = 𝟙 _ - h.retraction ≫ f | eq_sub_iff_add_eq.mpr ((add_comm _ _).trans h.split_add) | lemma | category_theory.splitting.π_section_eq_id_sub | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
splittings_comm (h h' : splitting f g) :
h'.section ≫ h.retraction = - h.section ≫ h'.retraction | begin
haveI := h.mono,
rw ← cancel_mono f,
simp [retraction_ι_eq_id_sub],
end | lemma | category_theory.splitting.splittings_comm | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split : split f g | begin
let φ := h.iso.hom ≫ biprod.fst,
let χ := biprod.inr ≫ h.iso.inv,
refine ⟨⟨h.retraction, h.section, h.ι_retraction, h.section_π, _,
h.section_retraction, h.split_add⟩⟩,
rw [← h.inl_comp_iso_eq, category.assoc, h.iso_comp_eq_snd, biprod.inl_snd],
end | lemma | category_theory.splitting.split | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_zero : f ≫ g = 0 | h.split.1.some_spec.some_spec.2.2.1 | lemma | category_theory.splitting.comp_eq_zero | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact : exact f g | begin
rw exact_iff_exact_of_iso f g (biprod.inl : A ⟶ A ⊞ C) (biprod.snd : A ⊞ C ⟶ C) _ _ _,
{ exact exact_inl_snd _ _ },
{ refine arrow.iso_mk (iso.refl _) h.iso _,
simp only [iso.refl_hom, arrow.mk_hom, category.id_comp, comp_iso_eq_inl], },
{ refine arrow.iso_mk h.iso (iso.refl _) _,
dsimp, simp, },
... | lemma | category_theory.splitting.exact | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_exact : short_exact f g | { mono := h.mono, epi := h.epi, exact := h.exact } | lemma | category_theory.splitting.short_exact | algebra.homology.short_exact | src/algebra/homology/short_exact/preadditive.lean | [
"algebra.homology.exact",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jordan [has_mul A] | (lmul_comm_rmul : ∀ a b : A, (a * b) * a = a * (b * a))
(lmul_lmul_comm_lmul : ∀ a b : A, (a * a) * (a * b) = a * ((a * a) * b))
(lmul_lmul_comm_rmul : ∀ a b : A, (a * a) * (b * a) = ((a * a) * b) * a)
(lmul_comm_rmul_rmul : ∀ a b : A, (a * b) * (a * a) = a * (b * (a * a)))
(rmul_comm_rmul_rmul : ∀ a b : A, (b * a) * (... | class | is_jordan | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [] | A (non-commutative) Jordan multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_comm_jordan [has_mul A] | (mul_comm : ∀ a b : A, a * b = b * a)
(lmul_comm_rmul_rmul : ∀ a b : A, (a * b) * (a * a) = a * (b * (a * a))) | class | is_comm_jordan | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"mul_comm"
] | A commutative Jordan multipication | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_comm_jordan.to_is_jordan [has_mul A] [is_comm_jordan A] : is_jordan A | { lmul_comm_rmul := λ a b, by rw [is_comm_jordan.mul_comm, is_comm_jordan.mul_comm a b],
lmul_lmul_comm_lmul := λ a b, by rw [is_comm_jordan.mul_comm (a * a) (a * b),
is_comm_jordan.lmul_comm_rmul_rmul, is_comm_jordan.mul_comm b (a * a)],
lmul_comm_rmul_rmul := is_comm_jordan.lmul_comm_rmul_rmul,
lmul_lmul_co... | instance | is_comm_jordan.to_is_jordan | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"is_comm_jordan",
"is_jordan"
] | A (commutative) Jordan multiplication is also a Jordan multipication | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semigroup.is_jordan [semigroup A] : is_jordan A | { lmul_comm_rmul := λ a b, by rw mul_assoc,
lmul_lmul_comm_lmul := λ a b, by rw [mul_assoc, mul_assoc],
lmul_comm_rmul_rmul := λ a b, by rw [mul_assoc],
lmul_lmul_comm_rmul := λ a b, by rw [←mul_assoc],
rmul_comm_rmul_rmul := λ a b, by rw [← mul_assoc, ← mul_assoc], } | instance | semigroup.is_jordan | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"is_jordan",
"mul_assoc",
"semigroup"
] | Semigroup multiplication satisfies the (non-commutative) Jordan axioms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_semigroup.is_comm_jordan [comm_semigroup A] : is_comm_jordan A | { mul_comm := mul_comm,
lmul_comm_rmul_rmul := λ a b, mul_assoc _ _ _, } | instance | comm_semigroup.is_comm_jordan | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"comm_semigroup",
"is_comm_jordan",
"mul_assoc",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_lmul_rmul (a : A) : commute (L a) (R a) | add_monoid_hom.ext $ λ b, (is_jordan.lmul_comm_rmul _ _).symm | lemma | commute_lmul_rmul | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_lmul_lmul_sq (a : A) : commute (L a) (L (a * a)) | add_monoid_hom.ext $ λ b, (is_jordan.lmul_lmul_comm_lmul _ _).symm | lemma | commute_lmul_lmul_sq | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_lmul_rmul_sq (a : A) : commute (L a) (R (a * a)) | add_monoid_hom.ext $ λ b, (is_jordan.lmul_comm_rmul_rmul _ _).symm | lemma | commute_lmul_rmul_sq | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_lmul_sq_rmul (a : A) : commute (L (a * a)) (R a) | add_monoid_hom.ext $ λ b, (is_jordan.lmul_lmul_comm_rmul _ _) | lemma | commute_lmul_sq_rmul | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_rmul_rmul_sq (a : A) : commute (R a) (R (a * a)) | add_monoid_hom.ext $ λ b, (is_jordan.rmul_comm_rmul_rmul _ _).symm | lemma | commute_rmul_rmul_sq | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add (a b : A) :
2•(⁅L a, L (a * b)⁆ + ⁅L b, L (b * a)⁆) = ⁅L (a * a), L b⁆ + ⁅L (b * b), L a⁆ | begin
suffices : 2 • ⁅L a, L (a * b)⁆ + 2 • ⁅L b, L (b * a)⁆ + ⁅L b, L (a * a)⁆ + ⁅L a, L (b * b)⁆ = 0,
{ rwa [← sub_eq_zero, ← sub_sub, sub_eq_add_neg, sub_eq_add_neg, lie_skew, lie_skew, nsmul_add] },
convert (commute_lmul_lmul_sq (a + b)).lie_eq,
simp only [add_mul, mul_add, map_add, lie_add, add_lie, is_com... | lemma | two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"add_lie",
"commute_lmul_lmul_sq",
"lie_add",
"lie_skew"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_lie_lmul_lmul_add_add_eq_zero (a b c : A) :
2•(⁅L a, L (b * c)⁆ + ⁅L b, L (c * a)⁆ + ⁅L c, L (a * b)⁆) = 0 | begin
symmetry,
calc 0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ :
by rw (commute_lmul_lmul_sq (a + b + c)).lie_eq
... = ⁅L a + L b + L c,
L (a * a) + L (a * b) + L (a * c) + (L (b * a) + L (b * b) + L (b * c))
+ (L (c * a) + L (c * b) + L (c * c))⁆ :
by rw [add_mul, add_mul, mul_add, mul... | lemma | two_nsmul_lie_lmul_lmul_add_add_eq_zero | algebra.jordan | src/algebra/jordan/basic.lean | [
"algebra.lie.of_associative"
] | [
"add_lie",
"commute_lmul_lmul_sq",
"lie_add",
"lie_nsmul",
"lie_skew",
"two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_module.is_trivial (L : Type v) (M : Type w) [has_bracket L M] [has_zero M] : Prop | (trivial : ∀ (x : L) (m : M), ⁅x, m⁆ = 0) | class | lie_module.is_trivial | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"has_bracket"
] | A Lie (ring) module is trivial iff all brackets vanish. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivial_lie_zero (L : Type v) (M : Type w)
[has_bracket L M] [has_zero M] [lie_module.is_trivial L M] (x : L) (m : M) : ⁅x, m⁆ = 0 | lie_module.is_trivial.trivial x m | lemma | trivial_lie_zero | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"has_bracket",
"lie_module.is_trivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lie_abelian (L : Type v) [has_bracket L L] [has_zero L] : Prop | lie_module.is_trivial L L | abbreviation | is_lie_abelian | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"has_bracket",
"lie_module.is_trivial"
] | A Lie algebra is Abelian iff it is trivial as a Lie module over itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ideal.is_lie_abelian_of_trivial (R : Type u) (L : Type v)
[comm_ring R] [lie_ring L] [lie_algebra R L] (I : lie_ideal R L) [h : lie_module.is_trivial L I] :
is_lie_abelian I | { trivial := λ x y, by apply h.trivial } | instance | lie_ideal.is_lie_abelian_of_trivial | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"comm_ring",
"is_lie_abelian",
"lie_algebra",
"lie_ideal",
"lie_module.is_trivial",
"lie_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.is_lie_abelian {R : Type u} {L₁ : Type v} {L₂ : Type w}
[comm_ring R] [lie_ring L₁] [lie_ring L₂] [lie_algebra R L₁] [lie_algebra R L₂]
{f : L₁ →ₗ⁅R⁆ L₂} (h₁ : function.injective f) (h₂ : is_lie_abelian L₂) :
is_lie_abelian L₁ | { trivial := λ x y, h₁ $
calc f ⁅x,y⁆ = ⁅f x, f y⁆ : lie_hom.map_lie f x y
... = 0 : trivial_lie_zero _ _ _ _
... = f 0 : f.map_zero.symm } | lemma | function.injective.is_lie_abelian | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"comm_ring",
"is_lie_abelian",
"lie_algebra",
"lie_hom.map_lie",
"lie_ring",
"trivial_lie_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.surjective.is_lie_abelian {R : Type u} {L₁ : Type v} {L₂ : Type w}
[comm_ring R] [lie_ring L₁] [lie_ring L₂] [lie_algebra R L₁] [lie_algebra R L₂]
{f : L₁ →ₗ⁅R⁆ L₂} (h₁ : function.surjective f) (h₂ : is_lie_abelian L₁) :
is_lie_abelian L₂ | { trivial := λ x y,
begin
obtain ⟨u, rfl⟩ := h₁ x,
obtain ⟨v, rfl⟩ := h₁ y,
rw [← lie_hom.map_lie, trivial_lie_zero, lie_hom.map_zero],
end } | lemma | function.surjective.is_lie_abelian | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"comm_ring",
"is_lie_abelian",
"lie_algebra",
"lie_hom.map_lie",
"lie_hom.map_zero",
"lie_ring",
"trivial_lie_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_abelian_iff_equiv_lie_abelian {R : Type u} {L₁ : Type v} {L₂ : Type w}
[comm_ring R] [lie_ring L₁] [lie_ring L₂] [lie_algebra R L₁] [lie_algebra R L₂]
(e : L₁ ≃ₗ⁅R⁆ L₂) : is_lie_abelian L₁ ↔ is_lie_abelian L₂ | ⟨e.symm.injective.is_lie_abelian, e.injective.is_lie_abelian⟩ | lemma | lie_abelian_iff_equiv_lie_abelian | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"comm_ring",
"is_lie_abelian",
"lie_algebra",
"lie_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutative_ring_iff_abelian_lie_ring {A : Type v} [ring A] :
is_commutative A (*) ↔ is_lie_abelian A | have h₁ : is_commutative A (*) ↔ ∀ (a b : A), a * b = b * a := ⟨λ h, h.1, λ h, ⟨h⟩⟩,
have h₂ : is_lie_abelian A ↔ ∀ (a b : A), ⁅a, b⁆ = 0 := ⟨λ h, h.1, λ h, ⟨h⟩⟩,
by simp only [h₁, h₂, lie_ring.of_associative_ring_bracket, sub_eq_zero] | lemma | commutative_ring_iff_abelian_lie_ring | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"is_lie_abelian",
"lie_ring.of_associative_ring_bracket",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.is_lie_abelian_bot (R : Type u) (L : Type v)
[comm_ring R] [lie_ring L] [lie_algebra R L] : is_lie_abelian (⊥ : lie_ideal R L) | ⟨λ ⟨x, hx⟩ _, by convert zero_lie _⟩ | lemma | lie_algebra.is_lie_abelian_bot | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"comm_ring",
"is_lie_abelian",
"lie_algebra",
"lie_ideal",
"lie_ring",
"zero_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker : lie_ideal R L | (to_endomorphism R L M).ker | def | lie_module.ker | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_ideal"
] | The kernel of the action of a Lie algebra `L` on a Lie module `M` as a Lie ideal in `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ker (x : L) : x ∈ lie_module.ker R L M ↔ ∀ (m : M), ⁅x, m⁆ = 0 | by simp only [lie_module.ker, lie_hom.mem_ker, linear_map.ext_iff, linear_map.zero_apply,
to_endomorphism_apply_apply] | lemma | lie_module.mem_ker | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_hom.mem_ker",
"lie_module.ker",
"linear_map.ext_iff",
"linear_map.zero_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_triv_submodule : lie_submodule R L M | { carrier := { m | ∀ (x : L), ⁅x, m⁆ = 0 },
zero_mem' := λ x, lie_zero x,
add_mem' := λ x y hx hy z, by rw [lie_add, hx, hy, add_zero],
smul_mem' := λ c x hx y, by rw [lie_smul, hx, smul_zero],
lie_mem := λ x m hm y, by rw [hm, lie_zero], } | def | lie_module.max_triv_submodule | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_add",
"lie_smul",
"lie_submodule",
"lie_zero",
"smul_zero"
] | The largest submodule of a Lie module `M` on which the Lie algebra `L` acts trivially. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_max_triv_submodule (m : M) :
m ∈ max_triv_submodule R L M ↔ ∀ (x : L), ⁅x, m⁆ = 0 | iff.rfl | lemma | lie_module.mem_max_triv_submodule | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_oper_max_triv_submodule_eq_bot (I : lie_ideal R L) :
⁅I, max_triv_submodule R L M⁆ = ⊥ | begin
rw [← lie_submodule.coe_to_submodule_eq_iff, lie_submodule.lie_ideal_oper_eq_linear_span,
lie_submodule.bot_coe_submodule, submodule.span_eq_bot],
rintros m ⟨⟨x, hx⟩, ⟨⟨m, hm⟩, rfl⟩⟩,
exact hm x,
end | lemma | lie_module.ideal_oper_max_triv_submodule_eq_bot | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_ideal",
"lie_submodule.bot_coe_submodule",
"lie_submodule.coe_to_submodule_eq_iff",
"lie_submodule.lie_ideal_oper_eq_linear_span",
"submodule.span_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_max_triv_iff_bracket_eq_bot {N : lie_submodule R L M} :
N ≤ max_triv_submodule R L M ↔ ⁅(⊤ : lie_ideal R L), N⁆ = ⊥ | begin
refine ⟨λ h, _, λ h m hm, _⟩,
{ rw [← le_bot_iff, ← ideal_oper_max_triv_submodule_eq_bot R L M ⊤],
exact lie_submodule.mono_lie_right _ _ ⊤ h, },
{ rw mem_max_triv_submodule,
rw lie_submodule.lie_eq_bot_iff at h,
exact λ x, h x (lie_submodule.mem_top x) m hm, },
end | lemma | lie_module.le_max_triv_iff_bracket_eq_bot | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"le_bot_iff",
"lie_ideal",
"lie_submodule",
"lie_submodule.lie_eq_bot_iff",
"lie_submodule.mem_top",
"lie_submodule.mono_lie_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivial_iff_le_maximal_trivial (N : lie_submodule R L M) :
is_trivial L N ↔ N ≤ max_triv_submodule R L M | ⟨ λ h m hm x, is_trivial.dcases_on h (λ h, subtype.ext_iff.mp (h x ⟨m, hm⟩)),
λ h, { trivial := λ x m, subtype.ext (h m.2 x) }⟩ | lemma | lie_module.trivial_iff_le_maximal_trivial | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_submodule",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_trivial_iff_max_triv_eq_top :
is_trivial L M ↔ max_triv_submodule R L M = ⊤ | begin
split,
{ rintros ⟨h⟩, ext,
simp only [mem_max_triv_submodule, h, forall_const, true_iff, eq_self_iff_true], },
{ intros h, constructor, intros x m, revert x,
rw [← mem_max_triv_submodule R L M, h], exact lie_submodule.mem_top m, },
end | lemma | lie_module.is_trivial_iff_max_triv_eq_top | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"forall_const",
"lie_submodule.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_triv_hom (f : M →ₗ⁅R,L⁆ N) :
max_triv_submodule R L M →ₗ⁅R,L⁆ max_triv_submodule R L N | { to_fun := λ m, ⟨f m, λ x, (lie_module_hom.map_lie _ _ _).symm.trans $
(congr_arg f (m.property x)).trans (lie_module_hom.map_zero _)⟩,
map_add' := λ m n, by simpa,
map_smul' := λ t m, by simpa,
map_lie' := λ x m, by simp, } | def | lie_module.max_triv_hom | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_module_hom.map_lie",
"lie_module_hom.map_zero"
] | `max_triv_submodule` is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_max_triv_hom_apply
(f : M →ₗ⁅R,L⁆ N) (m : max_triv_submodule R L M) :
(max_triv_hom f m : N) = f m | rfl | lemma | lie_module.coe_max_triv_hom_apply | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_triv_equiv (e : M ≃ₗ⁅R,L⁆ N) :
max_triv_submodule R L M ≃ₗ⁅R,L⁆ max_triv_submodule R L N | { to_fun := max_triv_hom (e : M →ₗ⁅R,L⁆ N),
inv_fun := max_triv_hom (e.symm : N →ₗ⁅R,L⁆ M),
left_inv := λ m, by { ext, simp, },
right_inv := λ n, by { ext, simp, },
.. max_triv_hom (e : M →ₗ⁅R,L⁆ N), } | def | lie_module.max_triv_equiv | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"inv_fun"
] | The maximal trivial submodules of Lie-equivalent Lie modules are Lie-equivalent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_max_triv_equiv_apply
(e : M ≃ₗ⁅R,L⁆ N) (m : max_triv_submodule R L M) :
(max_triv_equiv e m : N) = e ↑m | rfl | lemma | lie_module.coe_max_triv_equiv_apply | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_triv_equiv_of_refl_eq_refl :
max_triv_equiv (lie_module_equiv.refl : M ≃ₗ⁅R,L⁆ M) = lie_module_equiv.refl | by { ext, simp only [coe_max_triv_equiv_apply, lie_module_equiv.refl_apply], } | lemma | lie_module.max_triv_equiv_of_refl_eq_refl | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_module_equiv.refl",
"lie_module_equiv.refl_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_triv_equiv_of_equiv_symm_eq_symm (e : M ≃ₗ⁅R,L⁆ N) :
(max_triv_equiv e).symm = max_triv_equiv e.symm | rfl | lemma | lie_module.max_triv_equiv_of_equiv_symm_eq_symm | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_triv_linear_map_equiv_lie_module_hom :
(max_triv_submodule R L (M →ₗ[R] N)) ≃ₗ[R] (M →ₗ⁅R,L⁆ N) | { to_fun := λ f,
{ to_linear_map := f.val,
map_lie' := λ x m, by
{ have hf : ⁅x, f.val⁆ m = 0, { rw [f.property x, linear_map.zero_apply], },
rw [lie_hom.lie_apply, sub_eq_zero, ← linear_map.to_fun_eq_coe] at hf, exact hf.symm, }, },
map_add' := λ f g, by { ext, simp, },
map_smul' := λ F... | def | lie_module.max_triv_linear_map_equiv_lie_module_hom | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"inv_fun",
"lie_hom.lie_apply",
"linear_map.to_fun_eq_coe",
"linear_map.zero_apply"
] | A linear map between two Lie modules is a morphism of Lie modules iff the Lie algebra action
on it is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_max_triv_linear_map_equiv_lie_module_hom
(f : max_triv_submodule R L (M →ₗ[R] N)) :
((max_triv_linear_map_equiv_lie_module_hom f) : M → N) = f | by { ext, refl, } | lemma | lie_module.coe_max_triv_linear_map_equiv_lie_module_hom | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_max_triv_linear_map_equiv_lie_module_hom_symm
(f : M →ₗ⁅R,L⁆ N) :
((max_triv_linear_map_equiv_lie_module_hom.symm f) : M → N) = f | rfl | lemma | lie_module.coe_max_triv_linear_map_equiv_lie_module_hom_symm | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_map_max_triv_linear_map_equiv_lie_module_hom
(f : max_triv_submodule R L (M →ₗ[R] N)) :
((max_triv_linear_map_equiv_lie_module_hom f) : M →ₗ[R] N) = (f : M →ₗ[R] N) | by { ext, refl, } | lemma | lie_module.coe_linear_map_max_triv_linear_map_equiv_lie_module_hom | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_map_max_triv_linear_map_equiv_lie_module_hom_symm
(f : M →ₗ⁅R,L⁆ N) :
((max_triv_linear_map_equiv_lie_module_hom.symm f) : M →ₗ[R] N) = (f : M →ₗ[R] N) | rfl | lemma | lie_module.coe_linear_map_max_triv_linear_map_equiv_lie_module_hom_symm | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center : lie_ideal R L | lie_module.max_triv_submodule R L L | abbreviation | lie_algebra.center | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_ideal",
"lie_module.max_triv_submodule"
] | The center of a Lie algebra is the set of elements that commute with everything. It can
be viewed as the maximal trivial submodule of the Lie algebra as a Lie module over itself via the
adjoint representation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ad_ker_eq_self_module_ker : (ad R L).ker = lie_module.ker R L L | rfl | lemma | lie_algebra.ad_ker_eq_self_module_ker | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_module.ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_module_ker_eq_center : lie_module.ker R L L = center R L | begin
ext y,
simp only [lie_module.mem_max_triv_submodule, lie_module.mem_ker, ← lie_skew _ y, neg_eq_zero],
end | lemma | lie_algebra.self_module_ker_eq_center | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_module.ker",
"lie_module.mem_ker",
"lie_module.mem_max_triv_submodule",
"lie_skew"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abelian_of_le_center (I : lie_ideal R L) (h : I ≤ center R L) : is_lie_abelian I | begin
haveI : lie_module.is_trivial L I := (lie_module.trivial_iff_le_maximal_trivial R L L I).mpr h,
exact lie_ideal.is_lie_abelian_of_trivial R L I,
end | lemma | lie_algebra.abelian_of_le_center | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"is_lie_abelian",
"lie_ideal",
"lie_ideal.is_lie_abelian_of_trivial",
"lie_module.is_trivial",
"lie_module.trivial_iff_le_maximal_trivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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