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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mem_sup (x : M) : x ∈ N ⊔ N' ↔ ∃ (y ∈ N) (z ∈ N'), y + z = x | by { rw [← mem_coe_submodule, sup_coe_to_submodule, submodule.mem_sup], exact iff.rfl, } | lemma | lie_submodule.mem_sup | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_bot_iff : N = ⊥ ↔ ∀ (m : M), m ∈ N → m = 0 | by { rw eq_bot_iff, exact iff.rfl, } | lemma | lie_submodule.eq_bot_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_of_bot : subsingleton (lie_submodule R L ↥(⊥ : lie_submodule R L M)) | begin
apply subsingleton_of_bot_eq_top,
ext ⟨x, hx⟩, change x ∈ ⊥ at hx, rw lie_submodule.mem_bot at hx, subst hx,
simp only [true_iff, eq_self_iff_true, submodule.mk_eq_zero, lie_submodule.mem_bot],
end | instance | lie_submodule.subsingleton_of_bot | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"lie_submodule.mem_bot",
"submodule.mk_eq_zero",
"subsingleton_of_bot_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
well_founded_of_noetherian [is_noetherian R M] :
well_founded ((>) : lie_submodule R L M → lie_submodule R L M → Prop) | let f : ((>) : lie_submodule R L M → lie_submodule R L M → Prop) →r
((>) : submodule R M → submodule R M → 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_submodule.well_founded_of_noetherian | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"is_noetherian",
"lie_submodule",
"rel_hom_class.well_founded",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_iff : subsingleton (lie_submodule R L M) ↔ subsingleton M | have h : subsingleton (lie_submodule R L M) ↔ subsingleton (submodule R M),
{ rw [← subsingleton_iff_bot_eq_top, ← subsingleton_iff_bot_eq_top, ← coe_to_submodule_eq_iff,
top_coe_submodule, bot_coe_submodule], },
h.trans $ submodule.subsingleton_iff R | lemma | lie_submodule.subsingleton_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule",
"submodule.subsingleton_iff",
"subsingleton_iff",
"subsingleton_iff_bot_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial_iff : nontrivial (lie_submodule R L M) ↔ nontrivial M | not_iff_not.mp (
(not_nontrivial_iff_subsingleton.trans $ subsingleton_iff R L M).trans
not_nontrivial_iff_subsingleton.symm) | lemma | lie_submodule.nontrivial_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"nontrivial",
"nontrivial_iff",
"subsingleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial_iff_ne_bot {N : lie_submodule R L M} : nontrivial N ↔ N ≠ ⊥ | begin
split;
contrapose!,
{ rintros rfl ⟨⟨m₁, h₁ : m₁ ∈ (⊥ : lie_submodule R L M)⟩,
⟨m₂, h₂ : m₂ ∈ (⊥ : lie_submodule R L M)⟩, h₁₂⟩,
simpa [(lie_submodule.mem_bot _).mp h₁, (lie_submodule.mem_bot _).mp h₂] using h₁₂, },
{ rw [not_nontrivial_iff_subsingleton, lie_submodule.eq_bot_iff],
r... | lemma | lie_submodule.nontrivial_iff_ne_bot | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"lie_submodule.eq_bot_iff",
"lie_submodule.mem_bot",
"nontrivial",
"not_nontrivial_iff_subsingleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl : N →ₗ⁅R,L⁆ M | { map_lie' := λ x m, rfl,
..submodule.subtype (N : submodule R M) } | def | lie_submodule.incl | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule",
"submodule.subtype"
] | The inclusion of a Lie submodule into its ambient space is a morphism of Lie modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl_coe : (N.incl : N →ₗ[R] M) = (N : submodule R M).subtype | rfl | lemma | lie_submodule.incl_coe | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl_apply (m : N) : N.incl m = m | rfl | lemma | lie_submodule.incl_apply | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl_eq_val : (N.incl : N → M) = subtype.val | rfl | lemma | lie_submodule.incl_eq_val | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_le : N →ₗ⁅R,L⁆ N' | { map_lie' := λ x m, rfl,
..submodule.of_le (show N.to_submodule ≤ N'.to_submodule, from h) } | def | lie_submodule.hom_of_le | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule.of_le"
] | Given two nested Lie submodules `N ⊆ N'`, the inclusion `N ↪ N'` is a morphism of Lie modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_hom_of_le (m : N) : (hom_of_le h m : M) = m | rfl | lemma | lie_submodule.coe_hom_of_le | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_le_apply (m : N) : hom_of_le h m = ⟨m.1, h m.2⟩ | rfl | lemma | lie_submodule.hom_of_le_apply | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_span : lie_submodule R L M | Inf {N | s ⊆ N} | def | lie_submodule.lie_span | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | The `lie_span` of a set `s ⊆ M` is the smallest Lie submodule of `M` that contains `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_lie_span {x : M} : x ∈ lie_span R L s ↔ ∀ N : lie_submodule R L M, s ⊆ N → x ∈ N | by { change x ∈ (lie_span R L s : set M) ↔ _, erw Inf_coe, exact mem_Inter₂, } | lemma | lie_submodule.mem_lie_span | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule"
] | 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 N hN, exact hN hm, } | lemma | lie_submodule.subset_lie_span | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_span_le {N} : lie_span R L s ≤ N ↔ s ⊆ N | begin
split,
{ exact subset.trans subset_lie_span, },
{ intros hs m hm, rw mem_lie_span at hm, exact hm _ hs, },
end | lemma | lie_submodule.lie_span_le | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_span_mono {t : set M} (h : s ⊆ t) : lie_span R L s ≤ lie_span R L t | by { rw lie_span_le, exact subset.trans h subset_lie_span, } | lemma | lie_submodule.lie_span_mono | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_span_eq : lie_span R L (N : set M) = N | le_antisymm (lie_span_le.mpr rfl.subset) subset_lie_span | lemma | lie_submodule.lie_span_eq | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lie_span_submodule_eq_iff {p : submodule R M} :
(lie_span R L (p : set M) : submodule R M) = p ↔ ∃ (N : lie_submodule R L M), ↑N = p | begin
rw p.exists_lie_submodule_coe_eq_iff L, 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_submodule.coe_lie_span_submodule_eq_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gi : galois_insertion (lie_span R L : set M → lie_submodule R L M) 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_submodule.gi | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"galois_insertion",
"lie_submodule"
] | `lie_span` forms a Galois insertion with the coercion from `lie_submodule` to `set`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_empty : lie_span R L (∅ : set M) = ⊥ | (lie_submodule.gi R L M).gc.l_bot | lemma | lie_submodule.span_empty | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_univ : lie_span R L (set.univ : set M) = ⊤ | eq_top_iff.2 $ set_like.le_def.2 $ subset_lie_span | lemma | lie_submodule.span_univ | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_span_eq_bot_iff : lie_span R L s = ⊥ ↔ ∀ (m ∈ s), m = (0 : M) | by rw [_root_.eq_bot_iff, lie_span_le, bot_coe, subset_singleton_iff] | lemma | lie_submodule.lie_span_eq_bot_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_union (s t : set M) : lie_span R L (s ∪ t) = lie_span R L s ⊔ lie_span R L t | (lie_submodule.gi R L M).gc.l_sup | lemma | lie_submodule.span_union | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_Union {ι} (s : ι → set M) : lie_span R L (⋃ i, s i) = ⨆ i, lie_span R L (s i) | (lie_submodule.gi R L M).gc.l_supr | lemma | lie_submodule.span_Union | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map : lie_submodule R L M' | { lie_mem := λ x m' h, by
{ rcases h with ⟨m, hm, hfm⟩, use ⁅x, m⁆, split,
{ apply N.lie_mem hm, },
{ norm_cast at hfm, simp [hfm], }, },
..(N : submodule R M).map (f : M →ₗ[R] M') } | def | lie_submodule.map | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | A morphism of Lie modules `f : M → M'` pushes forward Lie submodules of `M` to Lie submodules
of `M'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_submodule_map :
(N.map f : submodule R M') = (N : submodule R M).map (f : M →ₗ[R] M') | rfl | lemma | lie_submodule.coe_submodule_map | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap : lie_submodule R L M | { lie_mem := λ x m h, by { suffices : ⁅x, f m⁆ ∈ N', { simp [this], }, apply N'.lie_mem h, },
..(N' : submodule R M').comap (f : M →ₗ[R] M') } | def | lie_submodule.comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"submodule"
] | A morphism of Lie modules `f : M → M'` pulls back Lie submodules of `M'` to Lie submodules of
`M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_submodule_comap :
(N'.comap f : submodule R M) = (N' : submodule R M').comap (f : M →ₗ[R] M') | rfl | lemma | lie_submodule.coe_submodule_comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_iff_le_comap : map f N ≤ N' ↔ N ≤ comap f N' | set.image_subset_iff | lemma | lie_submodule.map_le_iff_le_comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"set.image_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_map_comap : galois_connection (map f) (comap f) | λ N N', map_le_iff_le_comap | lemma | lie_submodule.gc_map_comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup : (N ⊔ N₂).map f = N.map f ⊔ N₂.map f | (gc_map_comap f).l_sup | lemma | lie_submodule.map_sup | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map (m' : M') : m' ∈ N.map f ↔ ∃ m, m ∈ N ∧ f m = m' | submodule.mem_map | lemma | lie_submodule.mem_map | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"mem_map",
"submodule.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap {m : M} : m ∈ comap f N' ↔ f m ∈ N' | iff.rfl | lemma | lie_submodule.mem_comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_incl_eq_top : N₂.comap N.incl = ⊤ ↔ N ≤ N₂ | by simpa only [← lie_submodule.coe_to_submodule_eq_iff, lie_submodule.coe_submodule_comap,
lie_submodule.incl_coe, lie_submodule.top_coe_submodule, submodule.comap_subtype_eq_top] | lemma | lie_submodule.comap_incl_eq_top | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.coe_submodule_comap",
"lie_submodule.coe_to_submodule_eq_iff",
"lie_submodule.incl_coe",
"lie_submodule.top_coe_submodule",
"submodule.comap_subtype_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_incl_eq_bot : N₂.comap N.incl = ⊥ ↔ N ⊓ N₂ = ⊥ | by simpa only [_root_.eq_bot_iff, ← lie_submodule.coe_to_submodule_eq_iff,
lie_submodule.coe_submodule_comap, lie_submodule.incl_coe, lie_submodule.bot_coe_submodule,
← submodule.disjoint_iff_comap_eq_bot, disjoint_iff] | lemma | lie_submodule.comap_incl_eq_bot | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"disjoint_iff",
"lie_submodule.bot_coe_submodule",
"lie_submodule.coe_submodule_comap",
"lie_submodule.coe_to_submodule_eq_iff",
"lie_submodule.incl_coe",
"submodule.disjoint_iff_comap_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_coe_lie_subalgebra : ((⊤ : lie_ideal R L) : lie_subalgebra R L) = ⊤ | rfl | lemma | lie_ideal.top_coe_lie_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 | |
map : lie_ideal R L' | lie_submodule.lie_span R L' $ (I : submodule R L).map (f : L →ₗ[R] L') | def | lie_ideal.map | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_submodule.lie_span",
"submodule"
] | A morphism of Lie algebras `f : L → L'` pushes forward Lie ideals of `L` to Lie ideals of `L'`.
Note that unlike `lie_submodule.map`, we must take the `lie_span` of the image. Mathematically
this is because although `f` makes `L'` into a Lie module over `L`, in general the `L` submodules of
`L'` are not the same as th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap : lie_ideal R L | { lie_mem := λ x y h, by { suffices : ⁅f x, f y⁆ ∈ J, { simp [this], }, apply J.lie_mem h, },
..(J : submodule R L').comap (f : L →ₗ[R] L') } | def | lie_ideal.comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"submodule"
] | A morphism of Lie algebras `f : L → L'` pulls back Lie ideals of `L'` to Lie ideals of `L`.
Note that `f` makes `L'` into a Lie module over `L` (turning `f` into a morphism of Lie modules)
and so this is a special case of `lie_submodule.comap` but we do not exploit this fact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_coe_submodule (h : ↑(map f I) = f '' I) :
(map f I : submodule R L') = (I : submodule R L).map (f : L →ₗ[R] L') | by { rw [set_like.ext'_iff, lie_submodule.coe_to_submodule, h, submodule.map_coe], refl, } | lemma | lie_ideal.map_coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.coe_to_submodule",
"set_like.ext'_iff",
"submodule",
"submodule.map_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_coe_submodule :
(comap f J : submodule R L) = (J : submodule R L').comap (f : L →ₗ[R] L') | rfl | lemma | lie_ideal.comap_coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le : map f I ≤ J ↔ f '' I ⊆ J | lie_submodule.lie_span_le | lemma | lie_ideal.map_le | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.lie_span_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {x : L} (hx : x ∈ I) : f x ∈ map f I | by { apply lie_submodule.subset_lie_span, use x, exact ⟨hx, rfl⟩, } | lemma | lie_ideal.mem_map | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.subset_lie_span",
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap {x : L} : x ∈ comap f J ↔ f x ∈ J | iff.rfl | lemma | lie_ideal.mem_comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_iff_le_comap : map f I ≤ J ↔ I ≤ comap f J | by { rw map_le, exact set.image_subset_iff, } | lemma | lie_ideal.map_le_iff_le_comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"set.image_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_map_comap : galois_connection (map f) (comap f) | λ I I', map_le_iff_le_comap | lemma | lie_ideal.gc_map_comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup : (I ⊔ I₂).map f = I.map f ⊔ I₂.map f | (gc_map_comap f).l_sup | lemma | lie_ideal.map_sup | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_le : map f (comap f J) ≤ J | by { rw map_le_iff_le_comap, exact le_rfl, } | lemma | lie_ideal.map_comap_le | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_map_le : I ≤ comap f (map f I) | by { rw ← map_le_iff_le_comap, exact le_rfl, } | lemma | lie_ideal.comap_map_le | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"le_rfl"
] | See also `lie_ideal.map_comap_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mono : monotone (map f) | λ I₁ I₂ h,
by { rw set_like.le_def at h, apply lie_submodule.lie_span_mono (set.image_subset ⇑f h), } | lemma | lie_ideal.map_mono | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.lie_span_mono",
"monotone",
"set.image_subset",
"set_like.le_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_mono : monotone (comap f) | λ J₁ J₂ h, by { rw ← set_like.coe_subset_coe at h ⊢, exact set.preimage_mono h, } | lemma | lie_ideal.comap_mono | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"monotone",
"set.preimage_mono",
"set_like.coe_subset_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_of_image (h : f '' I = J) : I.map f = J | begin
apply le_antisymm,
{ erw [lie_submodule.lie_span_le, submodule.map_coe, h], },
{ rw [← set_like.coe_subset_coe, ← h], exact lie_submodule.subset_lie_span, },
end | lemma | lie_ideal.map_of_image | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.lie_span_le",
"lie_submodule.subset_lie_span",
"set_like.coe_subset_coe",
"submodule.map_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_of_bot : subsingleton (lie_ideal R (⊥ : lie_ideal R L)) | begin
apply subsingleton_of_bot_eq_top,
ext ⟨x, hx⟩, change x ∈ ⊥ at hx, rw lie_submodule.mem_bot at hx, subst hx,
simp only [true_iff, eq_self_iff_true, submodule.mk_eq_zero, lie_submodule.mem_bot],
end | instance | lie_ideal.subsingleton_of_bot | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_submodule.mem_bot",
"submodule.mk_eq_zero",
"subsingleton_of_bot_eq_top"
] | Note that this is not a special case of `lie_submodule.subsingleton_of_bot`. Indeed, given
`I : lie_ideal R L`, in general the two lattices `lie_ideal R I` and `lie_submodule R L I` are
different (though the latter does naturally inject into the former).
In other words, in general, ideals of `I`, regarded as a Lie alg... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker : lie_ideal R L | lie_ideal.comap f ⊥ | def | lie_hom.ker | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_ideal.comap"
] | The kernel of a morphism of Lie algebras, as an ideal in the domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_range : lie_ideal R L' | lie_submodule.lie_span R L' f.range | def | lie_hom.ideal_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_submodule.lie_span"
] | The range of a morphism of Lie algebras as an ideal in the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_range_eq_lie_span_range :
f.ideal_range = lie_submodule.lie_span R L' f.range | rfl | lemma | lie_hom.ideal_range_eq_lie_span_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.lie_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_range_eq_map :
f.ideal_range = lie_ideal.map f ⊤ | by { ext, simp only [ideal_range, range_eq_map], refl } | lemma | lie_hom.ideal_range_eq_map | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ideal_morphism : Prop | (f.ideal_range : lie_subalgebra R L') = f.range | def | lie_hom.is_ideal_morphism | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | The condition that the image of a morphism of Lie algebras is an ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_ideal_morphism_def :
f.is_ideal_morphism ↔ (f.ideal_range : lie_subalgebra R L') = f.range | iff.rfl | lemma | lie_hom.is_ideal_morphism_def | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ideal_morphism_iff :
f.is_ideal_morphism ↔ ∀ (x : L') (y : L), ∃ (z : L), ⁅x, f y⁆ = f z | begin
simp only [is_ideal_morphism_def, ideal_range_eq_lie_span_range,
← lie_subalgebra.coe_to_submodule_eq_iff, ← f.range.coe_to_submodule,
lie_ideal.coe_to_lie_subalgebra_to_submodule, lie_submodule.coe_lie_span_submodule_eq_iff,
lie_subalgebra.mem_coe_submodule, mem_range, exists_imp_distrib,
submo... | lemma | lie_hom.is_ideal_morphism_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"exists_imp_distrib",
"lie_ideal.coe_to_lie_subalgebra_to_submodule",
"lie_subalgebra.coe_to_submodule_eq_iff",
"lie_subalgebra.mem_coe_submodule",
"lie_submodule.coe_lie_span_submodule_eq_iff",
"submodule.exists_lie_submodule_coe_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_subset_ideal_range : (f.range : set L') ⊆ f.ideal_range | lie_submodule.subset_lie_span | lemma | lie_hom.range_subset_ideal_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.subset_lie_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_ideal_range : I.map f ≤ f.ideal_range | begin
rw f.ideal_range_eq_map,
exact lie_ideal.map_mono le_top,
end | lemma | lie_hom.map_le_ideal_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"le_top",
"lie_ideal.map_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_le_comap : f.ker ≤ J.comap f | lie_ideal.comap_mono bot_le | lemma | lie_hom.ker_le_comap | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"bot_le",
"lie_ideal.comap_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_coe_submodule : (ker f : submodule R L) = (f : L →ₗ[R] L').ker | rfl | lemma | lie_hom.ker_coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ker {x : L} : x ∈ ker f ↔ f x = 0 | show x ∈ (f.ker : submodule R L) ↔ _,
by simp only [ker_coe_submodule, linear_map.mem_ker, coe_to_linear_map] | lemma | lie_hom.mem_ker | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"linear_map.mem_ker",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ideal_range {x : L} : f x ∈ ideal_range f | begin
rw ideal_range_eq_map,
exact lie_ideal.mem_map (lie_submodule.mem_top x)
end | lemma | lie_hom.mem_ideal_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal.mem_map",
"lie_submodule.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ideal_range_iff (h : is_ideal_morphism f) {y : L'} :
y ∈ ideal_range f ↔ ∃ (x : L), f x = y | begin
rw f.is_ideal_morphism_def at h,
rw [← lie_submodule.mem_coe, ← lie_ideal.coe_to_subalgebra, h, f.range_coe, set.mem_range],
end | lemma | lie_hom.mem_ideal_range_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal.coe_to_subalgebra",
"lie_submodule.mem_coe",
"set.mem_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_ker_iff : I ≤ f.ker ↔ ∀ x, x ∈ I → f x = 0 | begin
split; intros h x hx,
{ specialize h hx, rw mem_ker at h, exact h, },
{ rw mem_ker, apply h x hx, },
end | lemma | lie_hom.le_ker_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_eq_bot : f.ker = ⊥ ↔ function.injective f | by rw [← lie_submodule.coe_to_submodule_eq_iff, ker_coe_submodule, lie_submodule.bot_coe_submodule,
linear_map.ker_eq_bot, coe_to_linear_map] | lemma | lie_hom.ker_eq_bot | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.bot_coe_submodule",
"lie_submodule.coe_to_submodule_eq_iff",
"linear_map.ker_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_coe_submodule : (f.range : submodule R L') = (f : L →ₗ[R] L').range | rfl | lemma | lie_hom.range_coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_top : f.range = ⊤ ↔ function.surjective f | begin
rw [← lie_subalgebra.coe_to_submodule_eq_iff, range_coe_submodule,
lie_subalgebra.top_coe_submodule],
exact linear_map.range_eq_top,
end | lemma | lie_hom.range_eq_top | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_subalgebra.coe_to_submodule_eq_iff",
"lie_subalgebra.top_coe_submodule",
"linear_map.range_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_range_eq_top_of_surjective (h : function.surjective f) : f.ideal_range = ⊤ | begin
rw ← f.range_eq_top at h,
rw [ideal_range_eq_lie_span_range, h, ← lie_subalgebra.coe_to_submodule,
← lie_submodule.coe_to_submodule_eq_iff, lie_submodule.top_coe_submodule,
lie_subalgebra.top_coe_submodule, lie_submodule.coe_lie_span_submodule_eq_iff],
use ⊤,
exact lie_submodule.top_coe_submodul... | lemma | lie_hom.ideal_range_eq_top_of_surjective | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_subalgebra.coe_to_submodule",
"lie_subalgebra.top_coe_submodule",
"lie_submodule.coe_lie_span_submodule_eq_iff",
"lie_submodule.coe_to_submodule_eq_iff",
"lie_submodule.top_coe_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ideal_morphism_of_surjective (h : function.surjective f) : f.is_ideal_morphism | by rw [is_ideal_morphism_def, f.ideal_range_eq_top_of_surjective h, f.range_eq_top.mpr h,
lie_ideal.top_coe_lie_subalgebra] | lemma | lie_hom.is_ideal_morphism_of_surjective | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal.top_coe_lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_bot_iff : I.map f = ⊥ ↔ I ≤ f.ker | by { rw ← le_bot_iff, exact lie_ideal.map_le_iff_le_comap } | lemma | lie_ideal.map_eq_bot_iff | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"le_bot_iff",
"lie_ideal.map_le_iff_le_comap",
"map_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_map_of_surjective (h : function.surjective f) :
(I.map f : submodule R L') = (I : submodule R L).map (f : L →ₗ[R] L') | begin
let J : lie_ideal R L' :=
{ lie_mem := λ x y hy,
begin
have hy' : ∃ (x : L), x ∈ I ∧ f x = y, { simpa [hy], },
obtain ⟨z₂, hz₂, rfl⟩ := hy',
obtain ⟨z₁, rfl⟩ := h x,
simp only [lie_hom.coe_to_linear_map, set_like.mem_coe, set.mem_image,
lie_submodule.mem_coe_submodule, subm... | lemma | lie_ideal.coe_map_of_surjective | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_hom.coe_to_linear_map",
"lie_ideal",
"lie_submodule.coe_lie_span_submodule_eq_iff",
"lie_submodule.coe_to_submodule_mk",
"lie_submodule.mem_coe_submodule",
"set.mem_image",
"set_like.mem_coe",
"submodule",
"submodule.map_coe",
"submodule.mem_carrier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map_of_surjective {y : L'} (h₁ : function.surjective f) (h₂ : y ∈ I.map f) :
∃ (x : I), f x = y | begin
rw [← lie_submodule.mem_coe_submodule, coe_map_of_surjective h₁, submodule.mem_map] at h₂,
obtain ⟨x, hx, rfl⟩ := h₂,
use ⟨x, hx⟩,
refl,
end | lemma | lie_ideal.mem_map_of_surjective | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.mem_coe_submodule",
"submodule.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_of_map_eq_bot {I : lie_ideal R L} (h₁ : function.injective f) (h₂ : I.map f = ⊥) :
I = ⊥ | begin
rw ← f.ker_eq_bot at h₁, change comap f ⊥ = ⊥ at h₁,
rw [eq_bot_iff, map_le_iff_le_comap, h₁] at h₂,
rw eq_bot_iff, exact h₂,
end | lemma | lie_ideal.bot_of_map_eq_bot | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"eq_bot_iff",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_le {I₁ I₂ : lie_ideal R L} (h : I₁ ≤ I₂) : I₁ →ₗ⁅R⁆ I₂ | { map_lie' := λ x y, rfl,
..submodule.of_le (show I₁.to_submodule ≤ I₂.to_submodule, from h), } | def | lie_ideal.hom_of_le | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"submodule.of_le"
] | Given two nested Lie ideals `I₁ ⊆ I₂`, the inclusion `I₁ ↪ I₂` is a morphism of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_hom_of_le {I₁ I₂ : lie_ideal R L} (h : I₁ ≤ I₂) (x : I₁) :
(hom_of_le h x : L) = x | rfl | lemma | lie_ideal.coe_hom_of_le | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_le_apply {I₁ I₂ : lie_ideal R L} (h : I₁ ≤ I₂) (x : I₁) :
hom_of_le h x = ⟨x.1, h x.2⟩ | rfl | lemma | lie_ideal.hom_of_le_apply | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_le_injective {I₁ I₂ : lie_ideal R L} (h : I₁ ≤ I₂) :
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_ideal.hom_of_le_injective | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"imp_self",
"lie_ideal",
"set_like.coe_eq_coe",
"subtype.mk_eq_mk",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup_ker_eq_map : lie_ideal.map f (I ⊔ f.ker) = lie_ideal.map f I | begin
suffices : lie_ideal.map f (I ⊔ f.ker) ≤ lie_ideal.map f I,
{ exact le_antisymm this (lie_ideal.map_mono le_sup_left), },
apply lie_submodule.lie_span_mono,
rintros x ⟨y, hy₁, hy₂⟩, rw ← hy₂,
erw lie_submodule.mem_sup at hy₁, obtain ⟨z₁, hz₁, z₂, hz₂, hy⟩ := hy₁, rw ← hy,
rw [f.coe_to_linear_map, f.ma... | lemma | lie_ideal.map_sup_ker_eq_map | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"le_sup_left",
"lie_ideal.map",
"lie_ideal.map_mono",
"lie_submodule.lie_span_mono",
"lie_submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_eq (h : f.is_ideal_morphism) : map f (comap f J) = f.ideal_range ⊓ J | begin
apply le_antisymm,
{ rw le_inf_iff, exact ⟨f.map_le_ideal_range _, map_comap_le⟩, },
{ rw f.is_ideal_morphism_def at h,
rw [← set_like.coe_subset_coe, lie_submodule.inf_coe, ← coe_to_subalgebra, h],
rintros y ⟨⟨x, h₁⟩, h₂⟩, rw ← h₁ at h₂ ⊢, exact mem_map h₂, },
end | lemma | lie_ideal.map_comap_eq | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"le_inf_iff",
"lie_submodule.inf_coe",
"mem_map",
"set_like.coe_subset_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_map_eq (h : ↑(map f I) = f '' I) : comap f (map f I) = I ⊔ f.ker | by rw [← lie_submodule.coe_to_submodule_eq_iff, comap_coe_submodule, I.map_coe_submodule f h,
lie_submodule.sup_coe_to_submodule, f.ker_coe_submodule, submodule.comap_map_eq] | lemma | lie_ideal.comap_map_eq | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.coe_to_submodule_eq_iff",
"lie_submodule.sup_coe_to_submodule",
"submodule.comap_map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl : I →ₗ⁅R⁆ L | (I : lie_subalgebra R L).incl | def | lie_ideal.incl | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | Regarding an ideal `I` as a subalgebra, the inclusion map into its ambient space is a morphism
of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl_range : I.incl.range = I | (I : lie_subalgebra R L).incl_range | lemma | lie_ideal.incl_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl_apply (x : I) : I.incl x = x | rfl | lemma | lie_ideal.incl_apply | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl_coe : (I.incl : I →ₗ[R] L) = (I : submodule R L).subtype | rfl | lemma | lie_ideal.incl_coe | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_incl_self : comap I.incl I = ⊤ | by rw [← lie_submodule.coe_to_submodule_eq_iff, lie_submodule.top_coe_submodule,
lie_ideal.comap_coe_submodule, lie_ideal.incl_coe, submodule.comap_subtype_self] | lemma | lie_ideal.comap_incl_self | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal.comap_coe_submodule",
"lie_ideal.incl_coe",
"lie_submodule.coe_to_submodule_eq_iff",
"lie_submodule.top_coe_submodule",
"submodule.comap_subtype_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_incl : I.incl.ker = ⊥ | by rw [← lie_submodule.coe_to_submodule_eq_iff, I.incl.ker_coe_submodule,
lie_submodule.bot_coe_submodule, incl_coe, submodule.ker_subtype] | lemma | lie_ideal.ker_incl | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.bot_coe_submodule",
"lie_submodule.coe_to_submodule_eq_iff",
"submodule.ker_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl_ideal_range : I.incl.ideal_range = I | begin
rw [lie_hom.ideal_range_eq_lie_span_range, ← lie_subalgebra.coe_to_submodule,
← lie_submodule.coe_to_submodule_eq_iff, incl_range, coe_to_lie_subalgebra_to_submodule,
lie_submodule.coe_lie_span_submodule_eq_iff],
use I,
end | lemma | lie_ideal.incl_ideal_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_hom.ideal_range_eq_lie_span_range",
"lie_subalgebra.coe_to_submodule",
"lie_submodule.coe_lie_span_submodule_eq_iff",
"lie_submodule.coe_to_submodule_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl_is_ideal_morphism : I.incl.is_ideal_morphism | begin
rw [I.incl.is_ideal_morphism_def, incl_ideal_range],
exact (I : lie_subalgebra R L).incl_range.symm,
end | lemma | lie_ideal.incl_is_ideal_morphism | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker : lie_submodule R L M | lie_submodule.comap f ⊥ | def | lie_module_hom.ker | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"lie_submodule.comap"
] | The kernel of a morphism of Lie algebras, as an ideal in the domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_coe_submodule : (f.ker : submodule R M) = (f : M →ₗ[R] N).ker | rfl | lemma | lie_module_hom.ker_coe_submodule | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ker (m : M) : m ∈ f.ker ↔ f m = 0 | iff.rfl | lemma | lie_module_hom.mem_ker | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_id : (lie_module_hom.id : M →ₗ⁅R,L⁆ M).ker = ⊥ | rfl | lemma | lie_module_hom.ker_id | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_module_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_ker_incl : f.comp f.ker.incl = 0 | by { ext ⟨m, hm⟩, exact (mem_ker m).mp hm, } | lemma | lie_module_hom.comp_ker_incl | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_ker_iff_map (M' : lie_submodule R L M) :
M' ≤ f.ker ↔ lie_submodule.map f M' = ⊥ | by rw [ker, eq_bot_iff, lie_submodule.map_le_iff_le_comap] | lemma | lie_module_hom.le_ker_iff_map | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"eq_bot_iff",
"lie_submodule",
"lie_submodule.map",
"lie_submodule.map_le_iff_le_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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