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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Inf_glb (S : set (lie_subalgebra R L)) : is_glb S (Inf S) | begin
have h : ∀ (K K' : lie_subalgebra R L), (K : set L) ≤ K' ↔ K ≤ K', { intros, exact iff.rfl, },
apply is_glb.of_image h,
simp only [Inf_coe],
exact is_glb_binfi
end | lemma | lie_subalgebra.Inf_glb | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"is_glb",
"is_glb.of_image",
"is_glb_binfi",
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_sup : K + K' = K ⊔ K' | rfl | lemma | lie_subalgebra.add_eq_sup | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"add_eq_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_coe_to_submodule :
(↑(K ⊓ K') : submodule R L) = (K : submodule R L) ⊓ (K' : submodule R L) | rfl | lemma | lie_subalgebra.inf_coe_to_submodule | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf (x : L) : x ∈ K ⊓ K' ↔ x ∈ K ∧ x ∈ K' | by rw [← mem_coe_submodule, ← mem_coe_submodule, ← mem_coe_submodule, inf_coe_to_submodule,
submodule.mem_inf] | lemma | lie_subalgebra.mem_inf | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule.mem_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_bot_iff : K = ⊥ ↔ ∀ (x : L), x ∈ K → x = 0 | by { rw eq_bot_iff, exact iff.rfl, } | lemma | lie_subalgebra.eq_bot_iff | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_of_bot : subsingleton (lie_subalgebra R ↥(⊥ : lie_subalgebra R L)) | begin
apply subsingleton_of_bot_eq_top,
ext ⟨x, hx⟩, change x ∈ ⊥ at hx, rw lie_subalgebra.mem_bot at hx, subst hx,
simp only [true_iff, eq_self_iff_true, submodule.mk_eq_zero, mem_bot],
end | instance | lie_subalgebra.subsingleton_of_bot | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"lie_subalgebra.mem_bot",
"submodule.mk_eq_zero",
"subsingleton_of_bot_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_bot : subsingleton ↥(⊥ : lie_subalgebra R L) | show subsingleton ((⊥ : lie_subalgebra R L) : set L), by simp | lemma | lie_subalgebra.subsingleton_bot | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
well_founded_of_noetherian [is_noetherian R L] :
well_founded ((>) : lie_subalgebra R L → lie_subalgebra R L → Prop) | let f : ((>) : lie_subalgebra R L → lie_subalgebra R L → Prop) →r
((>) : submodule R L → submodule R L → Prop) :=
{ to_fun := coe,
map_rel' := λ N N' h, h, }
in rel_hom_class.well_founded f (is_noetherian_iff_well_founded.mp infer_instance) | lemma | lie_subalgebra.well_founded_of_noetherian | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"is_noetherian",
"lie_subalgebra",
"rel_hom_class.well_founded",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_le : K →ₗ⁅R⁆ K' | { map_lie' := λ x y, rfl,
..submodule.of_le h } | def | lie_subalgebra.hom_of_le | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule.of_le"
] | Given two nested Lie subalgebras `K ⊆ K'`, the inclusion `K ↪ K'` is a morphism of Lie
algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_hom_of_le (x : K) : (hom_of_le h x : L) = x | rfl | lemma | lie_subalgebra.coe_hom_of_le | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_le_apply (x : K) : hom_of_le h x = ⟨x.1, h x.2⟩ | rfl | lemma | lie_subalgebra.hom_of_le_apply | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_le_injective : function.injective (hom_of_le h) | λ x y, by simp only [hom_of_le_apply, imp_self, subtype.mk_eq_mk, set_like.coe_eq_coe,
subtype.val_eq_coe] | lemma | lie_subalgebra.hom_of_le_injective | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"imp_self",
"set_like.coe_eq_coe",
"subtype.mk_eq_mk",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le : lie_subalgebra R K' | (hom_of_le h).range | def | lie_subalgebra.of_le | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | Given two nested Lie subalgebras `K ⊆ K'`, we can view `K` as a Lie subalgebra of `K'`,
regarded as Lie algebra in its own right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_of_le (x : K') : x ∈ of_le h ↔ (x : L) ∈ K | begin
simp only [of_le, hom_of_le_apply, lie_hom.mem_range],
split,
{ rintros ⟨y, rfl⟩, exact y.property, },
{ intros h, use ⟨(x : L), h⟩, simp, },
end | lemma | lie_subalgebra.mem_of_le | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_hom.mem_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_eq_comap_incl : of_le h = K.comap K'.incl | by { ext, rw mem_of_le, refl, } | lemma | lie_subalgebra.of_le_eq_comap_incl | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_le : (of_le h : submodule R K') = (submodule.of_le h).range | rfl | lemma | lie_subalgebra.coe_of_le | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule",
"submodule.of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_le : K ≃ₗ⁅R⁆ of_le h | (hom_of_le h).equiv_range_of_injective (hom_of_le_injective h) | def | lie_subalgebra.equiv_of_le | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | Given nested Lie subalgebras `K ⊆ K'`, there is a natural equivalence from `K` to its image in
`K'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_le_apply (x : K) :
equiv_of_le h x = ⟨hom_of_le h x, (hom_of_le h).mem_range_self x⟩ | rfl | lemma | lie_subalgebra.equiv_of_le_apply | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_iff_le_comap {K : lie_subalgebra R L} {K' : lie_subalgebra R L₂} :
map f K ≤ K' ↔ K ≤ comap f K' | set.image_subset_iff | lemma | lie_subalgebra.map_le_iff_le_comap | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"set.image_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_map_comap : galois_connection (map f) (comap f) | λ K K', map_le_iff_le_comap | lemma | lie_subalgebra.gc_map_comap | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_span : lie_subalgebra R L | Inf {N | s ⊆ N} | def | lie_subalgebra.lie_span | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | The Lie subalgebra of a Lie algebra `L` generated by a subset `s ⊆ L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_lie_span {x : L} : x ∈ lie_span R L s ↔ ∀ K : lie_subalgebra R L, s ⊆ K → x ∈ K | by { change x ∈ (lie_span R L s : set L) ↔ _, erw Inf_coe, exact set.mem_Inter₂, } | lemma | lie_subalgebra.mem_lie_span | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"set.mem_Inter₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_lie_span : s ⊆ lie_span R L s | by { intros m hm, erw mem_lie_span, intros K hK, exact hK hm, } | lemma | lie_subalgebra.subset_lie_span | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule_span_le_lie_span : submodule.span R s ≤ lie_span R L s | by { rw submodule.span_le, apply subset_lie_span, } | lemma | lie_subalgebra.submodule_span_le_lie_span | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule.span",
"submodule.span_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_span_le {K} : lie_span R L s ≤ K ↔ s ⊆ K | begin
split,
{ exact set.subset.trans subset_lie_span, },
{ intros hs m hm, rw mem_lie_span at hm, exact hm _ hs, },
end | lemma | lie_subalgebra.lie_span_le | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_span_mono {t : set L} (h : s ⊆ t) : lie_span R L s ≤ lie_span R L t | by { rw lie_span_le, exact set.subset.trans h subset_lie_span, } | lemma | lie_subalgebra.lie_span_mono | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_span_eq : lie_span R L (K : set L) = K | le_antisymm (lie_span_le.mpr rfl.subset) subset_lie_span | lemma | lie_subalgebra.lie_span_eq | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lie_span_submodule_eq_iff {p : submodule R L} :
(lie_span R L (p : set L) : submodule R L) = p ↔ ∃ (K : lie_subalgebra R L), ↑K = p | begin
rw p.exists_lie_subalgebra_coe_eq_iff, split; intros h,
{ intros x m hm, rw [← h, mem_coe_submodule], exact lie_mem _ (subset_lie_span hm), },
{ rw [← coe_to_submodule_mk p h, coe_to_submodule, coe_to_submodule_eq_iff, lie_span_eq], },
end | lemma | lie_subalgebra.coe_lie_span_submodule_eq_iff | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gi : galois_insertion (lie_span R L : set L → lie_subalgebra R L) coe | { choice := λ s _, lie_span R L s,
gc := λ s t, lie_span_le,
le_l_u := λ s, subset_lie_span,
choice_eq := λ s h, rfl } | def | lie_subalgebra.gi | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"galois_insertion",
"lie_subalgebra"
] | `lie_span` forms a Galois insertion with the coercion from `lie_subalgebra` to `set`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_empty : lie_span R L (∅ : set L) = ⊥ | (lie_subalgebra.gi R L).gc.l_bot | lemma | lie_subalgebra.span_empty | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_univ : lie_span R L (set.univ : set L) = ⊤ | eq_top_iff.2 $ set_like.le_def.2 $ subset_lie_span | lemma | lie_subalgebra.span_univ | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_union (s t : set L) : lie_span R L (s ∪ t) = lie_span R L s ⊔ lie_span R L t | (lie_subalgebra.gi R L).gc.l_sup | lemma | lie_subalgebra.span_union | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_Union {ι} (s : ι → set L) : lie_span R L (⋃ i, s i) = ⨆ i, lie_span R L (s i) | (lie_subalgebra.gi R L).gc.l_supr | lemma | lie_subalgebra.span_Union | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_injective (f : L₁ →ₗ⁅R⁆ L₂) (h : function.injective f) :
L₁ ≃ₗ⁅R⁆ f.range | { map_lie' := λ x y, by { apply set_coe.ext, simpa },
.. linear_equiv.of_injective (f : L₁ →ₗ[R] L₂) $ by rwa [lie_hom.coe_to_linear_map] } | def | lie_equiv.of_injective | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_hom.coe_to_linear_map",
"linear_equiv.of_injective",
"set_coe.ext"
] | An injective Lie algebra morphism is an equivalence onto its range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_injective_apply (f : L₁ →ₗ⁅R⁆ L₂) (h : function.injective f) (x : L₁) :
↑(of_injective f h x) = f x | rfl | lemma | lie_equiv.of_injective_apply | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_eq (h : (L₁' : set L₁) = L₁'') : L₁' ≃ₗ⁅R⁆ L₁'' | { map_lie' := λ x y, by { apply set_coe.ext, simp, },
..(linear_equiv.of_eq ↑L₁' ↑L₁''
(by {ext x, change x ∈ (L₁' : set L₁) ↔ x ∈ (L₁'' : set L₁), rw h, } )) } | def | lie_equiv.of_eq | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"linear_equiv.of_eq",
"of_eq",
"set_coe.ext"
] | Lie subalgebras that are equal as sets are equivalent as Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_eq_apply (L L' : lie_subalgebra R L₁) (h : (L : set L₁) = L') (x : L) :
(↑(of_eq L L' h x) : L₁) = x | rfl | lemma | lie_equiv.of_eq_apply | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_subalgebra_map : L₁'' ≃ₗ⁅R⁆ (L₁''.map e : lie_subalgebra R L₂) | { map_lie' := λ x y, by { apply set_coe.ext, exact lie_hom.map_lie (↑e : L₁ →ₗ⁅R⁆ L₂) ↑x ↑y, }
..(linear_equiv.submodule_map (e : L₁ ≃ₗ[R] L₂) ↑L₁'') } | def | lie_equiv.lie_subalgebra_map | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_hom.map_lie",
"lie_subalgebra",
"linear_equiv.submodule_map",
"set_coe.ext"
] | An equivalence of Lie algebras restricts to an equivalence from any Lie subalgebra onto its
image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_subalgebra_map_apply (x : L₁'') : ↑(e.lie_subalgebra_map _ x) = e x | rfl | lemma | lie_equiv.lie_subalgebra_map_apply | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subalgebras (h : L₁'.map ↑e = L₂') : L₁' ≃ₗ⁅R⁆ L₂' | { map_lie' := λ x y, by { apply set_coe.ext, exact lie_hom.map_lie (↑e : L₁ →ₗ⁅R⁆ L₂) ↑x ↑y, },
..(linear_equiv.of_submodules (e : L₁ ≃ₗ[R] L₂) ↑L₁' ↑L₂' (by { rw ←h, refl, })) } | def | lie_equiv.of_subalgebras | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_hom.map_lie",
"linear_equiv.of_submodules",
"set_coe.ext"
] | An equivalence of Lie algebras restricts to an equivalence from any Lie subalgebra onto its
image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_subalgebras_apply (h : L₁'.map ↑e = L₂') (x : L₁') :
↑(e.of_subalgebras _ _ h x) = e x | rfl | lemma | lie_equiv.of_subalgebras_apply | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subalgebras_symm_apply (h : L₁'.map ↑e = L₂') (x : L₂') :
↑((e.of_subalgebras _ _ h).symm x) = e.symm x | rfl | lemma | lie_equiv.of_subalgebras_symm_apply | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_submodule extends submodule R M | (lie_mem : ∀ {x : L} {m : M}, m ∈ carrier → ⁅x, m⁆ ∈ carrier) | structure | lie_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | A Lie submodule of a Lie module is a submodule that is closed under the Lie bracket.
This is a sufficient condition for the subset itself to form a Lie module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_submodule : has_coe (lie_submodule R L M) (submodule R M) | ⟨to_submodule⟩ | instance | lie_submodule.coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submodule_eq_coe : N.to_submodule = N | rfl | lemma | lie_submodule.to_submodule_eq_coe | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submodule : ((N : submodule R M) : set M) = N | rfl | lemma | lie_submodule.coe_to_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_carrier {x : M} : x ∈ N.carrier ↔ x ∈ (N : set M) | iff.rfl | lemma | lie_submodule.mem_carrier | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk_iff (S : set M) (h₁ h₂ h₃ h₄) {x : M} :
x ∈ (⟨S, h₁, h₂, h₃, h₄⟩ : lie_submodule R L M) ↔ x ∈ S | iff.rfl | lemma | lie_submodule.mem_mk_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe_submodule {x : M} : x ∈ (N : submodule R M) ↔ x ∈ N | iff.rfl | lemma | lie_submodule.mem_coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe {x : M} : x ∈ (N : set M) ↔ x ∈ N | iff.rfl | lemma | lie_submodule.mem_coe | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem : (0 : M) ∈ N | zero_mem N | lemma | lie_submodule.zero_mem | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_zero {x} (h : x ∈ N) : (⟨x, h⟩ : N) = 0 ↔ x = 0 | subtype.ext_iff_val | lemma | lie_submodule.mk_eq_zero | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"subtype.ext_iff_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_set_mk (S : set M) (h₁ h₂ h₃ h₄) :
((⟨S, h₁, h₂, h₃, h₄⟩ : lie_submodule R L M) : set M) = S | rfl | lemma | lie_submodule.coe_to_set_mk | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submodule_mk (p : submodule R M) (h) :
(({lie_mem := h, ..p} : lie_submodule R L M) : submodule R M) = p | by { cases p, refl, } | lemma | lie_submodule.coe_to_submodule_mk | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_injective :
function.injective (to_submodule : lie_submodule R L M → submodule R M) | λ x y h, by { cases x, cases y, congr, injection h } | lemma | lie_submodule.coe_submodule_injective | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (h : ∀ m, m ∈ N ↔ m ∈ N') : N = N' | set_like.ext h | lemma | lie_submodule.ext | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submodule_eq_iff : (N : submodule R M) = (N' : submodule R M) ↔ N = N' | coe_submodule_injective.eq_iff | lemma | lie_submodule.coe_to_submodule_eq_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (s : set M) (hs : s = ↑N) : lie_submodule R L M | { carrier := s,
zero_mem' := hs.symm ▸ N.zero_mem',
add_mem' := λ _ _, hs.symm ▸ N.add_mem',
smul_mem' := hs.symm ▸ N.smul_mem',
lie_mem := λ _ _, hs.symm ▸ N.lie_mem, } | def | lie_submodule.copy | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | Copy of a lie_submodule with a new `carrier` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (S : lie_submodule R L M) (s : set M) (hs : s = ↑S) :
(S.copy s hs : set M) = s | rfl | lemma | lie_submodule.coe_copy | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (S : lie_submodule R L M) (s : set M) (hs : s = ↑S) : S.copy s hs = S | set_like.coe_injective hs | lemma | lie_submodule.copy_eq | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module' {S : Type*} [semiring S] [has_smul S R] [module S M] [is_scalar_tower S R M] :
module S N | N.to_submodule.module' | instance | lie_submodule.module' | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"has_smul",
"is_scalar_tower",
"module",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : N) : M) = (0 : M) | rfl | lemma | lie_submodule.coe_zero | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (m m' : N) : (↑(m + m') : M) = (m : M) + (m' : M) | rfl | lemma | lie_submodule.coe_add | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (m : N) : (↑(-m) : M) = -(m : M) | rfl | lemma | lie_submodule.coe_neg | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (m m' : N) : (↑(m - m') : M) = (m : M) - (m' : M) | rfl | lemma | lie_submodule.coe_sub | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (t : R) (m : N) : (↑(t • m) : M) = t • (m : M) | rfl | lemma | lie_submodule.coe_smul | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bracket (x : L) (m : N) : (↑⁅x, m⁆ : M) = ⁅x, ↑m⁆ | rfl | lemma | lie_submodule.coe_bracket | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal | lie_submodule R L L | abbreviation | lie_ideal | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | An ideal of a Lie algebra is a Lie submodule of the Lie algebra as a Lie module over itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_mem_right (I : lie_ideal R L) (x y : L) (h : y ∈ I) : ⁅x, y⁆ ∈ I | I.lie_mem h | lemma | lie_mem_right | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_mem_left (I : lie_ideal R L) (x y : L) (h : x ∈ I) : ⁅x, y⁆ ∈ I | by { rw [←lie_skew, ←neg_lie], apply lie_mem_right, assumption, } | lemma | lie_mem_left | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_mem_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal_subalgebra (I : lie_ideal R L) : lie_subalgebra R L | { lie_mem' := by { intros x y hx hy, apply lie_mem_right, exact hy, },
..I.to_submodule, } | def | lie_ideal_subalgebra | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_mem_right",
"lie_subalgebra"
] | An ideal of a Lie algebra is a Lie subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ideal.coe_to_subalgebra (I : lie_ideal R L) :
((I : lie_subalgebra R L) : set L) = I | rfl | lemma | lie_ideal.coe_to_subalgebra | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal.coe_to_lie_subalgebra_to_submodule (I : lie_ideal R L) :
((I : lie_subalgebra R L) : submodule R L) = I | rfl | lemma | lie_ideal.coe_to_lie_subalgebra_to_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_subalgebra",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal.lie_ring (I : lie_ideal R L) : lie_ring I | lie_subalgebra.lie_ring R L ↑I | instance | lie_ideal.lie_ring | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_ring"
] | An ideal of `L` is a Lie subalgebra of `L`, so it is a Lie ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ideal.lie_algebra (I : lie_ideal R L) : lie_algebra R I | lie_subalgebra.lie_algebra R L ↑I | instance | lie_ideal.lie_algebra | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_algebra",
"lie_ideal"
] | Transfer the `lie_algebra` instance from the coercion `lie_ideal → lie_subalgebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ideal.lie_ring_module {R L : Type*} [comm_ring R] [lie_ring L] [lie_algebra R L]
(I : lie_ideal R L) [lie_ring_module L M] : lie_ring_module I M | lie_subalgebra.lie_ring_module (I : lie_subalgebra R L) | instance | lie_ideal.lie_ring_module | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"comm_ring",
"lie_algebra",
"lie_ideal",
"lie_ring",
"lie_ring_module",
"lie_subalgebra"
] | Transfer the `lie_module` instance from the coercion `lie_ideal → lie_subalgebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ideal.coe_bracket_of_module {R L : Type*} [comm_ring R] [lie_ring L] [lie_algebra R L]
(I : lie_ideal R L) [lie_ring_module L M] (x : I) (m : M) :
⁅x,m⁆ = ⁅(↑x : L),m⁆ | lie_subalgebra.coe_bracket_of_module (I : lie_subalgebra R L) x m | theorem | lie_ideal.coe_bracket_of_module | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"comm_ring",
"lie_algebra",
"lie_ideal",
"lie_ring",
"lie_ring_module",
"lie_subalgebra",
"lie_subalgebra.coe_bracket_of_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal.lie_module (I : lie_ideal R L) : lie_module R I M | lie_subalgebra.lie_module (I : lie_subalgebra R L) | instance | lie_ideal.lie_module | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_module",
"lie_subalgebra"
] | Transfer the `lie_module` instance from the coercion `lie_ideal → lie_subalgebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.exists_lie_submodule_coe_eq_iff (p : submodule R M) :
(∃ (N : lie_submodule R L M), ↑N = p) ↔ ∀ (x : L) (m : M), m ∈ p → ⁅x, m⁆ ∈ p | begin
split,
{ rintros ⟨N, rfl⟩ _ _, exact N.lie_mem, },
{ intros h, use { lie_mem := h, ..p }, exact lie_submodule.coe_to_submodule_mk p _, },
end | lemma | submodule.exists_lie_submodule_coe_eq_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"lie_submodule.coe_to_submodule_mk",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_submodule : lie_submodule R K L | { lie_mem := λ x y hy, K.lie_mem x.property hy,
.. (K : submodule R L) } | def | lie_subalgebra.to_lie_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | Given a Lie subalgebra `K ⊆ L`, if we view `L` as a `K`-module by restriction, it contains
a distinguished Lie submodule for the action of `K`, namely `K` itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_lie_submodule :
(K.to_lie_submodule : submodule R L) = K | by { rcases K with ⟨⟨⟩⟩, refl, } | lemma | lie_subalgebra.coe_to_lie_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_lie_submodule (x : L) :
x ∈ K.to_lie_submodule ↔ x ∈ K | iff.rfl | lemma | lie_subalgebra.mem_to_lie_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lie_ideal_coe_eq_iff :
(∃ (I : lie_ideal R L), ↑I = K) ↔ ∀ (x y : L), y ∈ K → ⁅x, y⁆ ∈ K | begin
simp only [← coe_to_submodule_eq_iff, lie_ideal.coe_to_lie_subalgebra_to_submodule,
submodule.exists_lie_submodule_coe_eq_iff L],
exact iff.rfl,
end | lemma | lie_subalgebra.exists_lie_ideal_coe_eq_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_ideal.coe_to_lie_subalgebra_to_submodule",
"submodule.exists_lie_submodule_coe_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nested_lie_ideal_coe_eq_iff {K' : lie_subalgebra R L} (h : K ≤ K') :
(∃ (I : lie_ideal R K'), ↑I = of_le h) ↔ ∀ (x y : L), x ∈ K' → y ∈ K → ⁅x, y⁆ ∈ K | begin
simp only [exists_lie_ideal_coe_eq_iff, coe_bracket, mem_of_le],
split,
{ intros h' x y hx hy, exact h' ⟨x, hx⟩ ⟨y, h hy⟩ hy, },
{ rintros h' ⟨x, hx⟩ ⟨y, hy⟩ hy', exact h' x y hx hy', },
end | lemma | lie_subalgebra.exists_nested_lie_ideal_coe_eq_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : function.injective (coe : lie_submodule R L M → set M) | set_like.coe_injective | lemma | lie_submodule.coe_injective | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_le_coe_submodule : (N : submodule R M) ≤ N' ↔ N ≤ N' | iff.rfl | lemma | lie_submodule.coe_submodule_le_coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_coe : ((⊥ : lie_submodule R L M) : set M) = {0} | rfl | lemma | lie_submodule.bot_coe | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_coe_submodule : ((⊥ : lie_submodule R L M) : submodule R M) = ⊥ | rfl | lemma | lie_submodule.bot_coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot (x : M) : x ∈ (⊥ : lie_submodule R L M) ↔ x = 0 | mem_singleton_iff | lemma | lie_submodule.mem_bot | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_coe : ((⊤ : lie_submodule R L M) : set M) = univ | rfl | lemma | lie_submodule.top_coe | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_coe_submodule : ((⊤ : lie_submodule R L M) : submodule R M) = ⊤ | rfl | lemma | lie_submodule.top_coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top (x : M) : x ∈ (⊤ : lie_submodule R L M) | mem_univ x | lemma | lie_submodule.mem_top | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_coe : (↑(N ⊓ N') : set M) = N ∩ N' | rfl | theorem | lie_submodule.inf_coe | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_coe_to_submodule (S : set (lie_submodule R L M)) :
(↑(Inf S) : submodule R M) = Inf {(s : submodule R M) | s ∈ S} | rfl | lemma | lie_submodule.Inf_coe_to_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_coe (S : set (lie_submodule R L M)) : (↑(Inf S) : set M) = ⋂ s ∈ S, (s : set M) | begin
rw [← lie_submodule.coe_to_submodule, Inf_coe_to_submodule, submodule.Inf_coe],
ext m,
simpa only [mem_Inter, mem_set_of_eq, forall_apply_eq_imp_iff₂, exists_imp_distrib],
end | lemma | lie_submodule.Inf_coe | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"exists_imp_distrib",
"forall_apply_eq_imp_iff₂",
"lie_submodule",
"lie_submodule.coe_to_submodule",
"submodule.Inf_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_glb (S : set (lie_submodule R L M)) : is_glb S (Inf S) | begin
have h : ∀ (N N' : lie_submodule R L M), (N : set M) ≤ N' ↔ N ≤ N', { intros, refl },
apply is_glb.of_image h,
simp only [Inf_coe],
exact is_glb_binfi
end | lemma | lie_submodule.Inf_glb | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"is_glb",
"is_glb.of_image",
"is_glb_binfi",
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_sup : N + N' = N ⊔ N' | rfl | lemma | lie_submodule.add_eq_sup | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"add_eq_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_coe_to_submodule :
(↑(N ⊔ N') : submodule R M) = (N : submodule R M) ⊔ (N' : submodule R M) | begin
have aux : ∀ (x : L) m, m ∈ (N ⊔ N' : submodule R M) → ⁅x,m⁆ ∈ (N ⊔ N' : submodule R M),
{ simp only [submodule.mem_sup],
rintro x m ⟨y, hy, z, hz, rfl⟩,
refine ⟨⁅x, y⁆, N.lie_mem hy, ⁅x, z⁆, N'.lie_mem hz, (lie_add _ _ _).symm⟩ },
refine le_antisymm (Inf_le ⟨{ lie_mem := aux, ..(N ⊔ N' : submodule ... | lemma | lie_submodule.sup_coe_to_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"Inf_le",
"aux",
"exists_prop",
"le_sup_left",
"le_sup_right",
"lie_add",
"submodule",
"submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_coe_to_submodule :
(↑(N ⊓ N') : submodule R M) = (N : submodule R M) ⊓ (N' : submodule R M) | rfl | lemma | lie_submodule.inf_coe_to_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf (x : M) : x ∈ N ⊓ N' ↔ x ∈ N ∧ x ∈ N' | by rw [← mem_coe_submodule, ← mem_coe_submodule, ← mem_coe_submodule, inf_coe_to_submodule,
submodule.mem_inf] | lemma | lie_submodule.mem_inf | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule.mem_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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