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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|---|---|---|---|---|---|---|---|---|---|---|
skew_adjoint_matrices_lie_subalgebra_equiv_transpose_apply
{m : Type w} [decidable_eq m] [fintype m]
(e : matrix n n R ≃ₐ[R] matrix m m R) (h : ∀ A, (e A)ᵀ = e (Aᵀ))
(A : skew_adjoint_matrices_lie_subalgebra J) :
(skew_adjoint_matrices_lie_subalgebra_equiv_transpose J e h A : matrix m m R) = e A | rfl | lemma | skew_adjoint_matrices_lie_subalgebra_equiv_transpose_apply | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"fintype",
"matrix",
"skew_adjoint_matrices_lie_subalgebra",
"skew_adjoint_matrices_lie_subalgebra_equiv_transpose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_skew_adjoint_matrices_lie_subalgebra_unit_smul (u : Rˣ) (J A : matrix n n R) :
A ∈ skew_adjoint_matrices_lie_subalgebra (u • J) ↔
A ∈ skew_adjoint_matrices_lie_subalgebra J | begin
change A ∈ skew_adjoint_matrices_submodule (u • J) ↔ A ∈ skew_adjoint_matrices_submodule J,
simp only [mem_skew_adjoint_matrices_submodule, matrix.is_skew_adjoint, matrix.is_adjoint_pair],
split; intros h,
{ simpa using congr_arg (λ B, u⁻¹ • B) h, },
{ simp [h], },
end | lemma | mem_skew_adjoint_matrices_lie_subalgebra_unit_smul | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"matrix",
"matrix.is_adjoint_pair",
"matrix.is_skew_adjoint",
"mem_skew_adjoint_matrices_submodule",
"skew_adjoint_matrices_lie_subalgebra",
"skew_adjoint_matrices_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_ideal (k : ℕ) : lie_ideal R L → lie_ideal R L | (λ I, ⁅I, I⁆)^[k] | def | lie_algebra.derived_series_of_ideal | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"lie_ideal"
] | A generalisation of the derived series of a Lie algebra, whose zeroth term is a specified ideal.
It can be more convenient to work with this generalisation when considering the derived series of
an ideal since it provides a type-theoretic expression of the fact that the terms of the ideal's
derived series are also ide... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
derived_series_of_ideal_zero :
derived_series_of_ideal R L 0 I = I | rfl | lemma | lie_algebra.derived_series_of_ideal_zero | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_ideal_succ (k : ℕ) :
derived_series_of_ideal R L (k + 1) I =
⁅derived_series_of_ideal R L k I, derived_series_of_ideal R L k I⁆ | function.iterate_succ_apply' (λ I, ⁅I, I⁆) k I | lemma | lie_algebra.derived_series_of_ideal_succ | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"function.iterate_succ_apply'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series (k : ℕ) : lie_ideal R L | derived_series_of_ideal R L k ⊤ | abbreviation | lie_algebra.derived_series | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series",
"lie_ideal"
] | The derived series of Lie ideals of a Lie algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
derived_series_def (k : ℕ) :
derived_series R L k = derived_series_of_ideal R L k ⊤ | rfl | lemma | lie_algebra.derived_series_def | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_ideal_add (k l : ℕ) : D (k + l) I = D k (D l I) | begin
induction k with k ih,
{ rw [zero_add, derived_series_of_ideal_zero], },
{ rw [nat.succ_add k l, derived_series_of_ideal_succ, derived_series_of_ideal_succ, ih], },
end | lemma | lie_algebra.derived_series_of_ideal_add | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_ideal_le {I J : lie_ideal R L} {k l : ℕ} (h₁ : I ≤ J) (h₂ : l ≤ k) :
D k I ≤ D l J | begin
revert l, induction k with k ih; intros l h₂,
{ rw le_zero_iff at h₂, rw [h₂, derived_series_of_ideal_zero], exact h₁, },
{ have h : l = k.succ ∨ l ≤ k, by rwa [le_iff_eq_or_lt, nat.lt_succ_iff] at h₂,
cases h,
{ rw [h, derived_series_of_ideal_succ, derived_series_of_ideal_succ],
exact lie_sub... | lemma | lie_algebra.derived_series_of_ideal_le | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"ih",
"le_iff_eq_or_lt",
"le_zero_iff",
"lie_ideal",
"lie_submodule.lie_le_left",
"lie_submodule.mono_lie",
"nat.lt_succ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_ideal_succ_le (k : ℕ) : D (k + 1) I ≤ D k I | derived_series_of_ideal_le (le_refl I) k.le_succ | lemma | lie_algebra.derived_series_of_ideal_succ_le | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_ideal_le_self (k : ℕ) : D k I ≤ I | derived_series_of_ideal_le (le_refl I) (zero_le k) | lemma | lie_algebra.derived_series_of_ideal_le_self | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_ideal_mono {I J : lie_ideal R L} (h : I ≤ J) (k : ℕ) : D k I ≤ D k J | derived_series_of_ideal_le h (le_refl k) | lemma | lie_algebra.derived_series_of_ideal_mono | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_ideal_antitone {k l : ℕ} (h : l ≤ k) : D k I ≤ D l I | derived_series_of_ideal_le (le_refl I) h | lemma | lie_algebra.derived_series_of_ideal_antitone | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_ideal_add_le_add (J : lie_ideal R L) (k l : ℕ) :
D (k + l) (I + J) ≤ (D k I) + (D l J) | begin
let D₁ : lie_ideal R L →o lie_ideal R L :=
{ to_fun := λ I, ⁅I, I⁆,
monotone' := λ I J h, lie_submodule.mono_lie I J I J h h, },
have h₁ : ∀ (I J : lie_ideal R L), D₁ (I ⊔ J) ≤ (D₁ I) ⊔ J,
{ simp [lie_submodule.lie_le_right, lie_submodule.lie_le_left, le_sup_of_le_right], },
rw ← D₁.iterate_sup_l... | lemma | lie_algebra.derived_series_of_ideal_add_le_add | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"le_sup_of_le_right",
"lie_ideal",
"lie_submodule.lie_le_left",
"lie_submodule.lie_le_right",
"lie_submodule.mono_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_of_bot_eq_bot (k : ℕ) : derived_series_of_ideal R L k ⊥ = ⊥ | by { rw eq_bot_iff, exact derived_series_of_ideal_le_self ⊥ k, } | lemma | lie_algebra.derived_series_of_bot_eq_bot | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abelian_iff_derived_one_eq_bot : is_lie_abelian I ↔ derived_series_of_ideal R L 1 I = ⊥ | by rw [derived_series_of_ideal_succ, derived_series_of_ideal_zero,
lie_submodule.lie_abelian_iff_lie_self_eq_bot] | lemma | lie_algebra.abelian_iff_derived_one_eq_bot | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_lie_abelian",
"lie_submodule.lie_abelian_iff_lie_self_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abelian_iff_derived_succ_eq_bot (I : lie_ideal R L) (k : ℕ) :
is_lie_abelian (derived_series_of_ideal R L k I) ↔ derived_series_of_ideal R L (k + 1) I = ⊥ | by rw [add_comm, derived_series_of_ideal_add I 1 k, abelian_iff_derived_one_eq_bot] | lemma | lie_algebra.abelian_iff_derived_succ_eq_bot | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_lie_abelian",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_eq_derived_series_of_ideal_comap (k : ℕ) :
derived_series R I k = (derived_series_of_ideal R L k I).comap I.incl | begin
induction k with k ih,
{ simp only [derived_series_def, comap_incl_self, derived_series_of_ideal_zero], },
{ simp only [derived_series_def, derived_series_of_ideal_succ] at ⊢ ih, rw ih,
exact comap_bracket_incl_of_le I
(derived_series_of_ideal_le_self I k) (derived_series_of_ideal_le_self I k), },... | lemma | lie_ideal.derived_series_eq_derived_series_of_ideal_comap | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_eq_derived_series_of_ideal_map (k : ℕ) :
(derived_series R I k).map I.incl = derived_series_of_ideal R L k I | by { rw [derived_series_eq_derived_series_of_ideal_comap, map_comap_incl, inf_eq_right],
apply derived_series_of_ideal_le_self, } | lemma | lie_ideal.derived_series_eq_derived_series_of_ideal_map | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series",
"inf_eq_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_eq_bot_iff (k : ℕ) :
derived_series R I k = ⊥ ↔ derived_series_of_ideal R L k I = ⊥ | by rw [← derived_series_eq_derived_series_of_ideal_map, map_eq_bot_iff, ker_incl, eq_bot_iff] | lemma | lie_ideal.derived_series_eq_bot_iff | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series",
"eq_bot_iff",
"map_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_add_eq_bot {k l : ℕ} {I J : lie_ideal R L}
(hI : derived_series R I k = ⊥) (hJ : derived_series R J l = ⊥) :
derived_series R ↥(I + J) (k + l) = ⊥ | begin
rw lie_ideal.derived_series_eq_bot_iff at hI hJ ⊢,
rw ← le_bot_iff,
let D := derived_series_of_ideal R L, change D k I = ⊥ at hI, change D l J = ⊥ at hJ,
calc D (k + l) (I + J) ≤ (D k I) + (D l J) : derived_series_of_ideal_add_le_add I J k l
... ≤ ⊥ : by { rw [hI, hJ], simp, },
end | lemma | lie_ideal.derived_series_add_eq_bot | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series",
"le_bot_iff",
"lie_ideal",
"lie_ideal.derived_series_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_map_le (k : ℕ) :
(derived_series R L' k).map f ≤ derived_series R L k | begin
induction k with k ih,
{ simp only [derived_series_def, derived_series_of_ideal_zero, le_top], },
{ simp only [derived_series_def, derived_series_of_ideal_succ] at ih ⊢,
exact le_trans (map_bracket_le f) (lie_submodule.mono_lie _ _ _ _ ih ih), },
end | lemma | lie_ideal.derived_series_map_le | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series",
"ih",
"le_top",
"lie_submodule.mono_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_map_eq (k : ℕ) (h : function.surjective f) :
(derived_series R L' k).map f = derived_series R L k | begin
induction k with k ih,
{ change (⊤ : lie_ideal R L').map f = ⊤,
rw ←f.ideal_range_eq_map,
exact f.ideal_range_eq_top_of_surjective h, },
{ simp only [derived_series_def, map_bracket_eq f h, ih, derived_series_of_ideal_succ], },
end | lemma | lie_ideal.derived_series_map_eq | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series",
"ih",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_solvable : Prop | (solvable : ∃ k, derived_series R L k = ⊥) | class | lie_algebra.is_solvable | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series",
"is_solvable"
] | A Lie algebra is solvable if its derived series reaches 0 (in a finite number of steps). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_solvable_bot : is_solvable R ↥(⊥ : lie_ideal R L) | ⟨⟨0, subsingleton.elim _ ⊥⟩⟩ | instance | lie_algebra.is_solvable_bot | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_solvable",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_solvable_add {I J : lie_ideal R L} [hI : is_solvable R I] [hJ : is_solvable R J] :
is_solvable R ↥(I + J) | begin
obtain ⟨k, hk⟩ := id hI, obtain ⟨l, hl⟩ := id hJ,
exact ⟨⟨k+l, lie_ideal.derived_series_add_eq_bot hk hl⟩⟩,
end | instance | lie_algebra.is_solvable_add | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_solvable",
"lie_ideal",
"lie_ideal.derived_series_add_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective.lie_algebra_is_solvable [h₁ : is_solvable R L] (h₂ : injective f) :
is_solvable R L' | begin
obtain ⟨k, hk⟩ := id h₁,
use k,
apply lie_ideal.bot_of_map_eq_bot h₂, rw [eq_bot_iff, ← hk],
apply lie_ideal.derived_series_map_le,
end | lemma | function.injective.lie_algebra_is_solvable | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"eq_bot_iff",
"is_solvable",
"lie_ideal.bot_of_map_eq_bot",
"lie_ideal.derived_series_map_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective.lie_algebra_is_solvable [h₁ : is_solvable R L'] (h₂ : surjective f) :
is_solvable R L | begin
obtain ⟨k, hk⟩ := id h₁,
use k,
rw [← lie_ideal.derived_series_map_eq k h₂, hk],
simp only [lie_ideal.map_eq_bot_iff, bot_le],
end | lemma | function.surjective.lie_algebra_is_solvable | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"bot_le",
"is_solvable",
"lie_ideal.derived_series_map_eq",
"lie_ideal.map_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_hom.is_solvable_range (f : L' →ₗ⁅R⁆ L) [h : lie_algebra.is_solvable R L'] :
lie_algebra.is_solvable R f.range | f.surjective_range_restrict.lie_algebra_is_solvable | lemma | lie_hom.is_solvable_range | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"lie_algebra.is_solvable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
solvable_iff_equiv_solvable (e : L' ≃ₗ⁅R⁆ L) : is_solvable R L' ↔ is_solvable R L | begin
split; introsI h,
{ exact e.symm.injective.lie_algebra_is_solvable, },
{ exact e.injective.lie_algebra_is_solvable, },
end | lemma | lie_algebra.solvable_iff_equiv_solvable | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_solvable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_solvable_ideal_solvable {I J : lie_ideal R L} (h₁ : I ≤ J) (h₂ : is_solvable R J) :
is_solvable R I | (lie_ideal.hom_of_le_injective h₁).lie_algebra_is_solvable | lemma | lie_algebra.le_solvable_ideal_solvable | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_solvable",
"lie_ideal",
"lie_ideal.hom_of_le_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_abelian_is_solvable [is_lie_abelian L] : is_solvable R L | begin
use 1,
rw [← abelian_iff_derived_one_eq_bot, lie_abelian_iff_equiv_lie_abelian lie_ideal.top_equiv],
apply_instance,
end | instance | lie_algebra.of_abelian_is_solvable | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_lie_abelian",
"is_solvable",
"lie_abelian_iff_equiv_lie_abelian",
"lie_ideal.top_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical | Sup { I : lie_ideal R L | is_solvable R I } | def | lie_algebra.radical | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_solvable",
"lie_ideal"
] | The (solvable) radical of Lie algebra is the `Sup` of all solvable ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
radical_is_solvable [is_noetherian R L] : is_solvable R (radical R L) | begin
have hwf := lie_submodule.well_founded_of_noetherian R L L,
rw ← complete_lattice.is_sup_closed_compact_iff_well_founded at hwf,
refine hwf { I : lie_ideal R L | is_solvable R I } ⟨⊥, _⟩ (λ I hI J hJ, _),
{ exact lie_algebra.is_solvable_bot R L, },
{ apply lie_algebra.is_solvable_add R L, exacts [hI, hJ... | instance | lie_algebra.radical_is_solvable | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"complete_lattice.is_sup_closed_compact_iff_well_founded",
"is_noetherian",
"is_solvable",
"lie_algebra.is_solvable_add",
"lie_algebra.is_solvable_bot",
"lie_ideal",
"lie_submodule.well_founded_of_noetherian"
] | The radical of a Noetherian Lie algebra is solvable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ideal.solvable_iff_le_radical [is_noetherian R L] (I : lie_ideal R L) :
is_solvable R I ↔ I ≤ radical R L | ⟨λ h, le_Sup h, λ h, le_solvable_ideal_solvable h infer_instance⟩ | lemma | lie_algebra.lie_ideal.solvable_iff_le_radical | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_noetherian",
"is_solvable",
"le_Sup",
"lie_ideal"
] | The `→` direction of this lemma is actually true without the `is_noetherian` assumption. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
center_le_radical : center R L ≤ radical R L | have h : is_solvable R (center R L), { apply_instance, }, le_Sup h | lemma | lie_algebra.center_le_radical | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_solvable",
"le_Sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_length_of_ideal (I : lie_ideal R L) : ℕ | Inf {k | derived_series_of_ideal R L k I = ⊥} | def | lie_algebra.derived_length_of_ideal | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"lie_ideal"
] | Given a solvable Lie ideal `I` with derived series `I = D₀ ≥ D₁ ≥ ⋯ ≥ Dₖ = ⊥`, this is the
natural number `k` (the number of inclusions).
For a non-solvable ideal, the value is 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
derived_length : ℕ | derived_length_of_ideal R L ⊤ | abbreviation | lie_algebra.derived_length | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [] | The derived length of a Lie algebra is the derived length of its 'top' Lie ideal.
See also `lie_algebra.derived_length_eq_derived_length_of_ideal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
derived_series_of_derived_length_succ (I : lie_ideal R L) (k : ℕ) :
derived_length_of_ideal R L I = k + 1 ↔
is_lie_abelian (derived_series_of_ideal R L k I) ∧ derived_series_of_ideal R L k I ≠ ⊥ | begin
rw abelian_iff_derived_succ_eq_bot,
let s := {k | derived_series_of_ideal R L k I = ⊥}, change Inf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s,
have hs : ∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s,
{ intros k₁ k₂ h₁₂ h₁,
suffices : derived_series_of_ideal R L k₂ I ≤ ⊥, { exact eq_bot_iff.mpr this, },
change deriv... | lemma | lie_algebra.derived_series_of_derived_length_succ | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_lie_abelian",
"lie_ideal",
"nat.Inf_upward_closed_eq_succ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_length_eq_derived_length_of_ideal (I : lie_ideal R L) :
derived_length R I = derived_length_of_ideal R L I | begin
let s₁ := {k | derived_series R I k = ⊥},
let s₂ := {k | derived_series_of_ideal R L k I = ⊥},
change Inf s₁ = Inf s₂,
congr, ext k, exact I.derived_series_eq_bot_iff k,
end | lemma | lie_algebra.derived_length_eq_derived_length_of_ideal | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"derived_series",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_abelian_of_ideal (I : lie_ideal R L) : lie_ideal R L | match derived_length_of_ideal R L I with
| 0 := ⊥
| k + 1 := derived_series_of_ideal R L k I
end | def | lie_algebra.derived_abelian_of_ideal | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"lie_ideal"
] | Given a solvable Lie ideal `I` with derived series `I = D₀ ≥ D₁ ≥ ⋯ ≥ Dₖ = ⊥`, this is the
`k-1`th term in the derived series (and is therefore an Abelian ideal contained in `I`).
For a non-solvable ideal, this is the zero ideal, `⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abelian_derived_abelian_of_ideal (I : lie_ideal R L) :
is_lie_abelian (derived_abelian_of_ideal I) | begin
dunfold derived_abelian_of_ideal,
cases h : derived_length_of_ideal R L I with k,
{ exact is_lie_abelian_bot R L, },
{ rw derived_series_of_derived_length_succ at h, exact h.1, },
end | lemma | lie_algebra.abelian_derived_abelian_of_ideal | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_lie_abelian",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_length_zero (I : lie_ideal R L) [hI : is_solvable R I] :
derived_length_of_ideal R L I = 0 ↔ I = ⊥ | begin
let s := {k | derived_series_of_ideal R L k I = ⊥}, change Inf s = 0 ↔ _,
have hne : s ≠ ∅,
{ obtain ⟨k, hk⟩ := id hI,
refine set.nonempty.ne_empty ⟨k, _⟩,
rw [derived_series_def, lie_ideal.derived_series_eq_bot_iff] at hk, exact hk, },
simp [hne],
end | lemma | lie_algebra.derived_length_zero | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_solvable",
"lie_ideal",
"lie_ideal.derived_series_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abelian_of_solvable_ideal_eq_bot_iff (I : lie_ideal R L) [h : is_solvable R I] :
derived_abelian_of_ideal I = ⊥ ↔ I = ⊥ | begin
dunfold derived_abelian_of_ideal,
cases h : derived_length_of_ideal R L I with k,
{ rw derived_length_zero at h, rw h, refl, },
{ obtain ⟨h₁, h₂⟩ := (derived_series_of_derived_length_succ R L I k).mp h,
have h₃ : I ≠ ⊥, { intros contra, apply h₂, rw contra, apply derived_series_of_bot_eq_bot, },
c... | lemma | lie_algebra.abelian_of_solvable_ideal_eq_bot_iff | algebra.lie | src/algebra/lie/solvable.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"order.hom.basic"
] | [
"is_solvable",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_subalgebra extends submodule R L | (lie_mem' : ∀ {x y}, x ∈ carrier → y ∈ carrier → ⁅x, y⁆ ∈ carrier) | structure | lie_subalgebra | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule"
] | A Lie subalgebra of a Lie algebra is submodule that is closed under the Lie bracket.
This is a sufficient condition for the subset itself to form a Lie algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_mem : (0 : L) ∈ L' | zero_mem L' | lemma | lie_subalgebra.zero_mem | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem {x y : L} : x ∈ L' → y ∈ L' → (x + y : L) ∈ L' | add_mem | lemma | lie_subalgebra.add_mem | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mem {x y : L} : x ∈ L' → y ∈ L' → (x - y : L) ∈ L' | sub_mem | lemma | lie_subalgebra.sub_mem | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem (t : R) {x : L} (h : x ∈ L') : t • x ∈ L' | (L' : submodule R L).smul_mem t h | lemma | lie_subalgebra.smul_mem | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_mem {x y : L} (hx : x ∈ L') (hy : y ∈ L') : (⁅x, y⁆ : L) ∈ L' | L'.lie_mem' hx hy | lemma | lie_subalgebra.lie_mem | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_carrier {x : L} : x ∈ L'.carrier ↔ x ∈ (L' : set L) | iff.rfl | lemma | lie_subalgebra.mem_carrier | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk_iff (S : set L) (h₁ h₂ h₃ h₄) {x : L} :
x ∈ (⟨⟨S, h₁, h₂, h₃⟩, h₄⟩ : lie_subalgebra R L) ↔ x ∈ S | iff.rfl | lemma | lie_subalgebra.mem_mk_iff | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe_submodule {x : L} : x ∈ (L' : submodule R L) ↔ x ∈ L' | iff.rfl | lemma | lie_subalgebra.mem_coe_submodule | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe {x : L} : x ∈ (L' : set L) ↔ x ∈ L' | iff.rfl | lemma | lie_subalgebra.mem_coe | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bracket (x y : L') : (↑⁅x, y⁆ : L) = ⁅(↑x : L), ↑y⁆ | rfl | lemma | lie_subalgebra.coe_bracket | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff (x y : L') : x = y ↔ (x : L) = y | subtype.ext_iff | lemma | lie_subalgebra.ext_iff | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero_iff_zero (x : L') : (x : L) = 0 ↔ x = 0 | (ext_iff L' x 0).symm | lemma | lie_subalgebra.coe_zero_iff_zero | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (L₁' L₂' : lie_subalgebra R L) (h : ∀ x, x ∈ L₁' ↔ x ∈ L₂') :
L₁' = L₂' | set_like.ext h | lemma | lie_subalgebra.ext | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff' (L₁' L₂' : lie_subalgebra R L) : L₁' = L₂' ↔ ∀ x, x ∈ L₁' ↔ x ∈ L₂' | set_like.ext_iff | lemma | lie_subalgebra.ext_iff' | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"set_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (S : set L) (h₁ h₂ h₃ h₄) :
((⟨⟨S, h₁, h₂, h₃⟩, h₄⟩ : lie_subalgebra R L) : set L) = S | rfl | lemma | lie_subalgebra.mk_coe | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submodule_mk (p : submodule R L) (h) :
(({lie_mem' := h, ..p} : lie_subalgebra R L) : submodule R L) = p | by { cases p, refl, } | lemma | lie_subalgebra.coe_to_submodule_mk | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : function.injective (coe : lie_subalgebra R L → set L) | set_like.coe_injective | lemma | lie_subalgebra.coe_injective | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_set_eq (L₁' L₂' : lie_subalgebra R L) :
(L₁' : set L) = L₂' ↔ L₁' = L₂' | set_like.coe_set_eq | theorem | lie_subalgebra.coe_set_eq | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"set_like.coe_set_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submodule_injective :
function.injective (coe : lie_subalgebra R L → submodule R L) | λ L₁' L₂' h, by { rw set_like.ext'_iff at h, rw ← coe_set_eq, exact h, } | lemma | lie_subalgebra.to_submodule_injective | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"set_like.ext'_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submodule_eq_iff (L₁' L₂' : lie_subalgebra R L) :
(L₁' : submodule R L) = (L₂' : submodule R L) ↔ L₁' = L₂' | to_submodule_injective.eq_iff | lemma | lie_subalgebra.coe_to_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 | |
coe_to_submodule : ((L' : submodule R L) : set L) = L' | rfl | lemma | lie_subalgebra.coe_to_submodule | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bracket_of_module (x : L') (m : M) : ⁅x, m⁆ = ⁅(x : L), m⁆ | rfl | lemma | lie_subalgebra.coe_bracket_of_module | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.lie_module_hom.restrict_lie (f : M →ₗ⁅R,L⁆ N) (L' : lie_subalgebra R L) : M →ₗ⁅R,L'⁆ N | { map_lie' := λ x m, f.map_lie ↑x m,
.. (f : M →ₗ[R] N)} | def | lie_module_hom.restrict_lie | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | An `L`-equivariant map of Lie modules `M → N` is `L'`-equivariant for any Lie subalgebra
`L' ⊆ L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.lie_module_hom.coe_restrict_lie (f : M →ₗ⁅R,L⁆ N) :
⇑(f.restrict_lie L') = f | rfl | lemma | lie_module_hom.coe_restrict_lie | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl : L' →ₗ⁅R⁆ L | { map_lie' := λ x y, by { simp only [linear_map.to_fun_eq_coe, submodule.subtype_apply], refl, },
.. (L' : submodule R L).subtype, } | def | lie_subalgebra.incl | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"linear_map.to_fun_eq_coe",
"submodule",
"submodule.subtype_apply"
] | The embedding of a Lie subalgebra into the ambient space as a morphism of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_incl : ⇑L'.incl = coe | rfl | lemma | lie_subalgebra.coe_incl | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl' : L' →ₗ⁅R,L'⁆ L | { map_lie' := λ x y, by simp only [coe_bracket_of_module, linear_map.to_fun_eq_coe,
submodule.subtype_apply, coe_bracket],
.. (L' : submodule R L).subtype, } | def | lie_subalgebra.incl' | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"linear_map.to_fun_eq_coe",
"submodule",
"submodule.subtype_apply"
] | The embedding of a Lie subalgebra into the ambient space as a morphism of Lie modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_incl' : ⇑L'.incl' = coe | rfl | lemma | lie_subalgebra.coe_incl' | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range : lie_subalgebra R L₂ | { lie_mem' := λ x y,
show x ∈ f.to_linear_map.range → y ∈ f.to_linear_map.range → ⁅x, y⁆ ∈ f.to_linear_map.range,
by { repeat { rw linear_map.mem_range }, rintros ⟨x', hx⟩ ⟨y', hy⟩, refine ⟨⁅x', y'⁆, _⟩,
rw [←hx, ←hy], change f ⁅x', y'⁆ = ⁅f x', f y'⁆, rw map_lie, },
..(f : L →ₗ[R] L₂).range } | def | lie_hom.range | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"linear_map.mem_range"
] | The range of a morphism of Lie algebras is a Lie subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
range_coe : (f.range : set L₂) = set.range f | linear_map.range_coe ↑f | lemma | lie_hom.range_coe | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"linear_map.range_coe",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range (x : L₂) : x ∈ f.range ↔ ∃ (y : L), f y = x | linear_map.mem_range | lemma | lie_hom.mem_range | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"linear_map.mem_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range_self (x : L) : f x ∈ f.range | linear_map.mem_range_self f x | lemma | lie_hom.mem_range_self | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"linear_map.mem_range_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_restrict : L →ₗ⁅R⁆ f.range | { map_lie' := λ x y, by { apply subtype.ext, exact f.map_lie x y, },
..(f : L →ₗ[R] L₂).range_restrict, } | def | lie_hom.range_restrict | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"subtype.ext"
] | We can restrict a morphism to a (surjective) map to its range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
range_restrict_apply (x : L) : f.range_restrict x = ⟨f x, f.mem_range_self x⟩ | rfl | lemma | lie_hom.range_restrict_apply | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_range_restrict : function.surjective (f.range_restrict) | begin
rintros ⟨y, hy⟩,
erw mem_range at hy, obtain ⟨x, rfl⟩ := hy,
use x,
simp only [subtype.mk_eq_mk, range_restrict_apply],
end | lemma | lie_hom.surjective_range_restrict | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_range_of_injective (h : function.injective f) : L ≃ₗ⁅R⁆ f.range | lie_equiv.of_bijective f.range_restrict ⟨λ x y hxy,
begin
simp only [subtype.mk_eq_mk, range_restrict_apply] at hxy,
exact h hxy,
end, f.surjective_range_restrict⟩ | def | lie_hom.equiv_range_of_injective | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_equiv.of_bijective",
"subtype.mk_eq_mk"
] | A Lie algebra is equivalent to its range under an injective Lie algebra morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_range_of_injective_apply (h : function.injective f) (x : L) :
f.equiv_range_of_injective h x = ⟨f x, mem_range_self f x⟩ | rfl | lemma | lie_hom.equiv_range_of_injective_apply | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.exists_lie_subalgebra_coe_eq_iff (p : submodule R L) :
(∃ (K : lie_subalgebra R L), ↑K = p) ↔ ∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p | begin
split,
{ rintros ⟨K, rfl⟩ _ _, exact K.lie_mem', },
{ intros h, use { lie_mem' := h, ..p }, exact lie_subalgebra.coe_to_submodule_mk p _, },
end | lemma | submodule.exists_lie_subalgebra_coe_eq_iff | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"lie_subalgebra.coe_to_submodule_mk",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl_range : K.incl.range = K | by { rw ← coe_to_submodule_eq_iff, exact (K : submodule R L).range_subtype, } | lemma | lie_subalgebra.incl_range | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map : lie_subalgebra R L₂ | { lie_mem' := λ x y hx hy, by
{ erw submodule.mem_map at hx, rcases hx with ⟨x', hx', hx⟩, rw ←hx,
erw submodule.mem_map at hy, rcases hy with ⟨y', hy', hy⟩, rw ←hy,
erw submodule.mem_map,
exact ⟨⁅x', y'⁆, K.lie_mem hx' hy', f.map_lie x' y'⟩, },
..((K : submodule R L).map (f : L →ₗ[R] L₂)) } | def | lie_subalgebra.map | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"submodule",
"submodule.mem_map"
] | The image of a Lie subalgebra under a Lie algebra morphism is a Lie subalgebra of the
codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_map (x : L₂) : x ∈ K.map f ↔ ∃ (y : L), y ∈ K ∧ f y = x | submodule.mem_map | lemma | lie_subalgebra.mem_map | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"mem_map",
"submodule.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map_submodule (e : L ≃ₗ⁅R⁆ L₂) (x : L₂) :
x ∈ K.map (e : L →ₗ⁅R⁆ L₂) ↔ x ∈ (K : submodule R L).map (e : L →ₗ[R] L₂) | iff.rfl | lemma | lie_subalgebra.mem_map_submodule | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap : lie_subalgebra R L | { lie_mem' := λ x y hx hy, by
{ suffices : ⁅f x, f y⁆ ∈ K₂, by { simp [this], }, exact K₂.lie_mem hx hy, },
..((K₂ : submodule R L₂).comap (f : L →ₗ[R] L₂)), } | def | lie_subalgebra.comap | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"submodule"
] | The preimage of a Lie subalgebra under a Lie algebra morphism is a Lie subalgebra of the
domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_def : K ≤ K' ↔ (K : set L) ⊆ K' | iff.rfl | lemma | lie_subalgebra.le_def | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_le_coe_submodule : (K : submodule R L) ≤ K' ↔ K ≤ K' | iff.rfl | lemma | lie_subalgebra.coe_submodule_le_coe_submodule | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_coe : ((⊥ : lie_subalgebra R L) : set L) = {0} | rfl | lemma | lie_subalgebra.bot_coe | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_coe_submodule : ((⊥ : lie_subalgebra R L) : submodule R L) = ⊥ | rfl | lemma | lie_subalgebra.bot_coe_submodule | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot (x : L) : x ∈ (⊥ : lie_subalgebra R L) ↔ x = 0 | mem_singleton_iff | lemma | lie_subalgebra.mem_bot | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_coe : ((⊤ : lie_subalgebra R L) : set L) = univ | rfl | lemma | lie_subalgebra.top_coe | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_coe_submodule : ((⊤ : lie_subalgebra R L) : submodule R L) = ⊤ | rfl | lemma | lie_subalgebra.top_coe_submodule | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top (x : L) : x ∈ (⊤ : lie_subalgebra R L) | mem_univ x | lemma | lie_subalgebra.mem_top | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.lie_hom.range_eq_map : f.range = map f ⊤ | by { ext, simp } | lemma | lie_hom.range_eq_map | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_coe : (↑(K ⊓ K') : set L) = K ∩ K' | rfl | theorem | lie_subalgebra.inf_coe | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_coe_to_submodule (S : set (lie_subalgebra R L)) :
(↑(Inf S) : submodule R L) = Inf {(s : submodule R L) | s ∈ S} | rfl | lemma | lie_subalgebra.Inf_coe_to_submodule | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_coe (S : set (lie_subalgebra R L)) : (↑(Inf S) : set L) = ⋂ s ∈ S, (s : set L) | begin
rw [← coe_to_submodule, Inf_coe_to_submodule, submodule.Inf_coe],
ext x,
simpa only [mem_Inter, mem_set_of_eq, forall_apply_eq_imp_iff₂, exists_imp_distrib],
end | lemma | lie_subalgebra.Inf_coe | algebra.lie | src/algebra/lie/subalgebra.lean | [
"algebra.lie.basic",
"ring_theory.noetherian"
] | [
"exists_imp_distrib",
"forall_apply_eq_imp_iff₂",
"lie_subalgebra",
"submodule.Inf_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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