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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to_subalgebra_to_submodule (p : submodule R A) (h_one h_mul) :
(p.to_subalgebra h_one h_mul).to_submodule = p | set_like.coe_injective rfl | lemma | submodule.to_subalgebra_to_submodule | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set_like.coe_injective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.subalgebra.to_submodule_to_subalgebra (S : subalgebra R A) :
S.to_submodule.to_subalgebra S.one_mem (λ _ _, S.mul_mem) = S | set_like.coe_injective rfl | lemma | subalgebra.to_submodule_to_subalgebra | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set_like.coe_injective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range (φ : A →ₐ[R] B) : subalgebra R B | { algebra_map_mem' := λ r, ⟨algebra_map R A r, φ.commutes r⟩,
.. φ.to_ring_hom.srange } | def | alg_hom.range | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | Range of an `alg_hom` as a subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_range (φ : A →ₐ[R] B) {y : B} :
y ∈ φ.range ↔ ∃ x, φ x = y | ring_hom.mem_srange | lemma | alg_hom.mem_range | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"ring_hom.mem_srange"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range_self (φ : A →ₐ[R] B) (x : A) : φ x ∈ φ.range | φ.mem_range.2 ⟨x, rfl⟩ | theorem | alg_hom.mem_range_self | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_range (φ : A →ₐ[R] B) : (φ.range : set B) = set.range φ | by { ext, rw [set_like.mem_coe, mem_range], refl } | lemma | alg_hom.coe_range | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.range",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_comp (f : A →ₐ[R] B) (g : B →ₐ[R] C) : (g.comp f).range = f.range.map g | set_like.coe_injective (set.range_comp g f) | theorem | alg_hom.range_comp | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.range_comp",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_comp_le_range (f : A →ₐ[R] B) (g : B →ₐ[R] C) : (g.comp f).range ≤ g.range | set_like.coe_mono (set.range_comp_subset_range f g) | theorem | alg_hom.range_comp_le_range | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.range_comp_subset_range",
"set_like.coe_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cod_restrict (f : A →ₐ[R] B) (S : subalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₐ[R] S | { commutes' := λ r, subtype.eq $ f.commutes r,
.. ring_hom.cod_restrict (f : A →+* B) S hf } | def | alg_hom.cod_restrict | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"ring_hom.cod_restrict",
"subalgebra"
] | Restrict the codomain of an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
val_comp_cod_restrict (f : A →ₐ[R] B) (S : subalgebra R B) (hf : ∀ x, f x ∈ S) :
S.val.comp (f.cod_restrict S hf) = f | alg_hom.ext $ λ _, rfl | lemma | alg_hom.val_comp_cod_restrict | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_hom.ext",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_cod_restrict (f : A →ₐ[R] B) (S : subalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :
↑(f.cod_restrict S hf x) = f x | rfl | lemma | alg_hom.coe_cod_restrict | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_cod_restrict (f : A →ₐ[R] B) (S : subalgebra R B) (hf : ∀ x, f x ∈ S) :
function.injective (f.cod_restrict S hf) ↔ function.injective f | ⟨λ H x y hxy, H $ subtype.eq hxy, λ H x y hxy, H (congr_arg subtype.val hxy : _)⟩ | theorem | alg_hom.injective_cod_restrict | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_restrict (f : A →ₐ[R] B) : A →ₐ[R] f.range | f.cod_restrict f.range f.mem_range_self | def | alg_hom.range_restrict | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | Restrict the codomain of a alg_hom `f` to `f.range`.
This is the bundled version of `set.range_factorization`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer (ϕ ψ : A →ₐ[R] B) : subalgebra R A | { carrier := {a | ϕ a = ψ a},
add_mem' := λ x y (hx : ϕ x = ψ x) (hy : ϕ y = ψ y),
by rw [set.mem_set_of_eq, ϕ.map_add, ψ.map_add, hx, hy],
mul_mem' := λ x y (hx : ϕ x = ψ x) (hy : ϕ y = ψ y),
by rw [set.mem_set_of_eq, ϕ.map_mul, ψ.map_mul, hx, hy],
algebra_map_mem' := λ x,
by rw [set.mem_set_of_eq, a... | def | alg_hom.equalizer | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_hom.commutes",
"subalgebra"
] | The equalizer of two R-algebra homomorphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_equalizer (ϕ ψ : A →ₐ[R] B) (x : A) :
x ∈ ϕ.equalizer ψ ↔ ϕ x = ψ x | iff.rfl | lemma | alg_hom.mem_equalizer | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype_range [fintype A] [decidable_eq B] (φ : A →ₐ[R] B) : fintype φ.range | set.fintype_range φ | instance | alg_hom.fintype_range | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"fintype",
"set.fintype_range"
] | The range of a morphism of algebras is a fintype, if the domain is a fintype.
Note that this instance can cause a diamond with `subtype.fintype` if `B` is also a fintype. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_left_inverse
{g : B → A} {f : A →ₐ[R] B} (h : function.left_inverse g f) :
A ≃ₐ[R] f.range | { to_fun := f.range_restrict,
inv_fun := g ∘ f.range.val,
left_inv := h,
right_inv := λ x, subtype.ext $
let ⟨x', hx'⟩ := f.mem_range.mp x.prop in
show f (g x) = x, by rw [←hx', h x'],
..f.range_restrict } | def | alg_equiv.of_left_inverse | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"inv_fun",
"subtype.ext"
] | Restrict an algebra homomorphism with a left inverse to an algebra isomorphism to its range.
This is a computable alternative to `alg_equiv.of_injective`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_left_inverse_apply
{g : B → A} {f : A →ₐ[R] B} (h : function.left_inverse g f) (x : A) :
↑(of_left_inverse h x) = f x | rfl | lemma | alg_equiv.of_left_inverse_apply | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_left_inverse_symm_apply
{g : B → A} {f : A →ₐ[R] B} (h : function.left_inverse g f) (x : f.range) :
(of_left_inverse h).symm x = g x | rfl | lemma | alg_equiv.of_left_inverse_symm_apply | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_injective (f : A →ₐ[R] B) (hf : function.injective f) :
A ≃ₐ[R] f.range | of_left_inverse (classical.some_spec hf.has_left_inverse) | def | alg_equiv.of_injective | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | Restrict an injective algebra homomorphism to an algebra isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_injective_apply (f : A →ₐ[R] B) (hf : function.injective f) (x : A) :
↑(of_injective f hf x) = f x | rfl | lemma | alg_equiv.of_injective_apply | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_injective_field {E F : Type*} [division_ring E] [semiring F]
[nontrivial F] [algebra R E] [algebra R F] (f : E →ₐ[R] F) : E ≃ₐ[R] f.range | of_injective f f.to_ring_hom.injective | def | alg_equiv.of_injective_field | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"division_ring",
"nontrivial",
"semiring"
] | Restrict an algebra homomorphism between fields to an algebra isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra_map (e : A ≃ₐ[R] B) (S : subalgebra R A) :
S ≃ₐ[R] (S.map e.to_alg_hom) | { commutes' := λ r, by { ext, simp },
..e.to_ring_equiv.subsemiring_map S.to_subsemiring } | def | alg_equiv.subalgebra_map | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | Given an equivalence `e : A ≃ₐ[R] B` of `R`-algebras and a subalgebra `S` of `A`,
`subalgebra_map` is the induced equivalence between `S` and `S.map e` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin (s : set A) : subalgebra R A | { algebra_map_mem' := λ r, subsemiring.subset_closure $ or.inl ⟨r, rfl⟩,
.. subsemiring.closure (set.range (algebra_map R A) ∪ s) } | def | algebra.adjoin | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra_map",
"set.range",
"subalgebra",
"subsemiring.closure",
"subsemiring.subset_closure"
] | The minimal subalgebra that includes `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gc : galois_connection (adjoin R : set A → subalgebra R A) coe | λ s S, ⟨λ H, le_trans (le_trans (set.subset_union_right _ _) subsemiring.subset_closure) H,
λ H, show subsemiring.closure (set.range (algebra_map R A) ∪ s) ≤ S.to_subsemiring,
from subsemiring.closure_le.2 $ set.union_subset S.range_subset H⟩ | lemma | algebra.gc | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra_map",
"galois_connection",
"set.range",
"set.subset_union_right",
"set.union_subset",
"subalgebra",
"subsemiring.closure",
"subsemiring.subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gi : galois_insertion (adjoin R : set A → subalgebra R A) coe | { choice := λ s hs, (adjoin R s).copy s $ le_antisymm (algebra.gc.le_u_l s) hs,
gc := algebra.gc,
le_l_u := λ S, (algebra.gc (S : set A) (adjoin R S)).1 $ le_rfl,
choice_eq := λ _ _, subalgebra.copy_eq _ _ _ } | def | algebra.gi | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra.gc",
"galois_insertion",
"le_rfl",
"subalgebra",
"subalgebra.copy_eq"
] | Galois insertion between `adjoin` and `coe`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_top : (↑(⊤ : subalgebra R A) : set A) = set.univ | rfl | lemma | algebra.coe_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top {x : A} : x ∈ (⊤ : subalgebra R A) | set.mem_univ x | lemma | algebra.mem_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.mem_univ",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_to_submodule : (⊤ : subalgebra R A).to_submodule = ⊤ | rfl | lemma | algebra.top_to_submodule | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_to_subsemiring : (⊤ : subalgebra R A).to_subsemiring = ⊤ | rfl | lemma | algebra.top_to_subsemiring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_to_subring {R A : Type*} [comm_ring R] [ring A] [algebra R A] :
(⊤ : subalgebra R A).to_subring = ⊤ | rfl | lemma | algebra.top_to_subring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"comm_ring",
"ring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submodule_eq_top {S : subalgebra R A} : S.to_submodule = ⊤ ↔ S = ⊤ | subalgebra.to_submodule.injective.eq_iff' top_to_submodule | lemma | algebra.to_submodule_eq_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subsemiring_eq_top {S : subalgebra R A} : S.to_subsemiring = ⊤ ↔ S = ⊤ | subalgebra.to_subsemiring_injective.eq_iff' top_to_subsemiring | lemma | algebra.to_subsemiring_eq_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subring_eq_top {R A : Type*} [comm_ring R] [ring A] [algebra R A]
{S : subalgebra R A} : S.to_subring = ⊤ ↔ S = ⊤ | subalgebra.to_subring_injective.eq_iff' top_to_subring | lemma | algebra.to_subring_eq_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"comm_ring",
"ring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sup_left {S T : subalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T | show S ≤ S ⊔ T, from le_sup_left | lemma | algebra.mem_sup_left | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"le_sup_left",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sup_right {S T : subalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T | show T ≤ S ⊔ T, from le_sup_right | lemma | algebra.mem_sup_right | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"le_sup_right",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_sup {S T : subalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :
x * y ∈ S ⊔ T | (S ⊔ T).mul_mem (mem_sup_left hx) (mem_sup_right hy) | lemma | algebra.mul_mem_sup | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup (f : A →ₐ[R] B) (S T : subalgebra R A) : (S ⊔ T).map f = S.map f ⊔ T.map f | (subalgebra.gc_map_comap f).l_sup | lemma | algebra.map_sup | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra",
"subalgebra.gc_map_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (S T : subalgebra R A) : (↑(S ⊓ T) : set A) = S ∩ T | rfl | lemma | algebra.coe_inf | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf {S T : subalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T | iff.rfl | lemma | algebra.mem_inf | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_to_submodule (S T : subalgebra R A) :
(S ⊓ T).to_submodule = S.to_submodule ⊓ T.to_submodule | rfl | lemma | algebra.inf_to_submodule | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_to_subsemiring (S T : subalgebra R A) :
(S ⊓ T).to_subsemiring = S.to_subsemiring ⊓ T.to_subsemiring | rfl | lemma | algebra.inf_to_subsemiring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Inf (S : set (subalgebra R A)) : (↑(Inf S) : set A) = ⋂ s ∈ S, ↑s | Inf_image | lemma | algebra.coe_Inf | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"Inf_image",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Inf {S : set (subalgebra R A)} {x : A} : x ∈ Inf S ↔ ∀ p ∈ S, x ∈ p | by simp only [← set_like.mem_coe, coe_Inf, set.mem_Inter₂] | lemma | algebra.mem_Inf | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.mem_Inter₂",
"set_like.mem_coe",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_to_submodule (S : set (subalgebra R A)) :
(Inf S).to_submodule = Inf (subalgebra.to_submodule '' S) | set_like.coe_injective $ by simp | lemma | algebra.Inf_to_submodule | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set_like.coe_injective",
"subalgebra",
"subalgebra.to_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_to_subsemiring (S : set (subalgebra R A)) :
(Inf S).to_subsemiring = Inf (subalgebra.to_subsemiring '' S) | set_like.coe_injective $ by simp | lemma | algebra.Inf_to_subsemiring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set_like.coe_injective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi {ι : Sort*} {S : ι → subalgebra R A} : (↑(⨅ i, S i) : set A) = ⋂ i, S i | by simp [infi] | lemma | algebra.coe_infi | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"infi",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi {ι : Sort*} {S : ι → subalgebra R A} {x : A} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i | by simp only [infi, mem_Inf, set.forall_range_iff] | lemma | algebra.mem_infi | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"infi",
"set.forall_range_iff",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_to_submodule {ι : Sort*} (S : ι → subalgebra R A) :
(⨅ i, S i).to_submodule = ⨅ i, (S i).to_submodule | set_like.coe_injective $ by simp | lemma | algebra.infi_to_submodule | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set_like.coe_injective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot {x : A} : x ∈ (⊥ : subalgebra R A) ↔ x ∈ set.range (algebra_map R A) | suffices (of_id R A).range = (⊥ : subalgebra R A),
by { rw [← this, ←set_like.mem_coe, alg_hom.coe_range], refl },
le_bot_iff.mp (λ x hx, subalgebra.range_le _ ((of_id R A).coe_range ▸ hx)) | theorem | algebra.mem_bot | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_hom.coe_range",
"algebra_map",
"set.range",
"subalgebra",
"subalgebra.range_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submodule_bot : (⊥ : subalgebra R A).to_submodule = R ∙ 1 | by { ext x, simp [mem_bot, -set.singleton_one, submodule.mem_span_singleton, algebra.smul_def] } | theorem | algebra.to_submodule_bot | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra.smul_def",
"set.singleton_one",
"subalgebra",
"submodule.mem_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ((⊥ : subalgebra R A) : set A) = set.range (algebra_map R A) | by simp [set.ext_iff, algebra.mem_bot] | theorem | algebra.coe_bot | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra.mem_bot",
"algebra_map",
"set.ext_iff",
"set.range",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff {S : subalgebra R A} :
S = ⊤ ↔ ∀ x : A, x ∈ S | ⟨λ h x, by rw h; exact mem_top, λ h, by ext x; exact ⟨λ _, mem_top, λ _, h x⟩⟩ | theorem | algebra.eq_top_iff | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"eq_top_iff",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_top_iff_surjective (f : A →ₐ[R] B) :
f.range = (⊤ : subalgebra R B) ↔ function.surjective f | algebra.eq_top_iff | lemma | algebra.range_top_iff_surjective | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra.eq_top_iff",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_id : (alg_hom.id R A).range = ⊤ | set_like.coe_injective set.range_id | theorem | algebra.range_id | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_hom.id",
"set.range_id",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_top (f : A →ₐ[R] B) : (⊤ : subalgebra R A).map f = f.range | set_like.coe_injective set.image_univ | theorem | algebra.map_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.image_univ",
"set_like.coe_injective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bot (f : A →ₐ[R] B) : (⊥ : subalgebra R A).map f = ⊥ | set_like.coe_injective $
by simp only [← set.range_comp, (∘), algebra.coe_bot, subalgebra.coe_map, f.commutes] | theorem | algebra.map_bot | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra.coe_bot",
"set.range_comp",
"set_like.coe_injective",
"subalgebra",
"subalgebra.coe_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_top (f : A →ₐ[R] B) : (⊤ : subalgebra R B).comap f = ⊤ | eq_top_iff.2 $ λ x, mem_top | theorem | algebra.comap_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_top : A →ₐ[R] (⊤ : subalgebra R A) | (alg_hom.id R A).cod_restrict ⊤ (λ _, mem_top) | def | algebra.to_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_hom.id",
"subalgebra",
"to_top"
] | `alg_hom` to `⊤ : subalgebra R A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective_algebra_map_iff :
function.surjective (algebra_map R A) ↔ (⊤ : subalgebra R A) = ⊥ | ⟨λ h, eq_bot_iff.2 $ λ y _, let ⟨x, hx⟩ := h y in hx ▸ subalgebra.algebra_map_mem _ _,
λ h y, algebra.mem_bot.1 $ eq_bot_iff.1 h (algebra.mem_top : y ∈ _)⟩ | theorem | algebra.surjective_algebra_map_iff | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra.mem_top",
"algebra_map",
"subalgebra",
"subalgebra.algebra_map_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijective_algebra_map_iff {R A : Type*} [field R] [semiring A] [nontrivial A]
[algebra R A] :
function.bijective (algebra_map R A) ↔ (⊤ : subalgebra R A) = ⊥ | ⟨λ h, surjective_algebra_map_iff.1 h.2,
λ h, ⟨(algebra_map R A).injective, surjective_algebra_map_iff.2 h⟩⟩ | theorem | algebra.bijective_algebra_map_iff | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"algebra_map",
"field",
"nontrivial",
"semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_equiv_of_injective (h : function.injective (algebra_map R A)) :
(⊥ : subalgebra R A) ≃ₐ[R] R | alg_equiv.symm $ alg_equiv.of_bijective (algebra.of_id R _)
⟨λ x y hxy, h (congr_arg subtype.val hxy : _),
λ ⟨y, hy⟩, let ⟨x, hx⟩ := algebra.mem_bot.1 hy in ⟨x, subtype.eq hx⟩⟩ | def | algebra.bot_equiv_of_injective | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_equiv.of_bijective",
"alg_equiv.symm",
"algebra.of_id",
"algebra_map",
"subalgebra"
] | The bottom subalgebra is isomorphic to the base ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bot_equiv (F R : Type*) [field F] [semiring R] [nontrivial R] [algebra F R] :
(⊥ : subalgebra F R) ≃ₐ[F] F | bot_equiv_of_injective (ring_hom.injective _) | def | algebra.bot_equiv | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"field",
"nontrivial",
"ring_hom.injective",
"semiring",
"subalgebra"
] | The bottom subalgebra is isomorphic to the field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
top_equiv : (⊤ : subalgebra R A) ≃ₐ[R] A | alg_equiv.of_alg_hom (subalgebra.val ⊤) to_top rfl $ alg_hom.ext $ λ _, subtype.ext rfl | def | subalgebra.top_equiv | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_equiv.of_alg_hom",
"alg_hom.ext",
"subalgebra",
"subalgebra.val",
"subtype.ext",
"to_top"
] | The top subalgebra is isomorphic to the algebra.
This is the algebra version of `submodule.top_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_of_subsingleton [subsingleton A] : subsingleton (subalgebra R A) | ⟨λ B C, ext (λ x, by { simp only [subsingleton.elim x 0, zero_mem B, zero_mem C] })⟩ | instance | subalgebra.subsingleton_of_subsingleton | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.alg_hom.subsingleton [subsingleton (subalgebra R A)] : subsingleton (A →ₐ[R] B) | ⟨λ f g, alg_hom.ext $ λ a,
have a ∈ (⊥ : subalgebra R A) := subsingleton.elim (⊤ : subalgebra R A) ⊥ ▸ mem_top,
let ⟨x, hx⟩ := set.mem_range.mp (mem_bot.mp this) in
hx ▸ (f.commutes _).trans (g.commutes _).symm⟩ | instance | alg_hom.subsingleton | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_hom.ext",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.alg_equiv.subsingleton_left [subsingleton (subalgebra R A)] :
subsingleton (A ≃ₐ[R] B) | ⟨λ f g, alg_equiv.ext (λ x, alg_hom.ext_iff.mp (subsingleton.elim f.to_alg_hom g.to_alg_hom) x)⟩ | instance | alg_equiv.subsingleton_left | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_equiv.ext",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.alg_equiv.subsingleton_right [subsingleton (subalgebra R B)] :
subsingleton (A ≃ₐ[R] B) | ⟨λ f g, by rw [← f.symm_symm, subsingleton.elim f.symm g.symm, g.symm_symm]⟩ | instance | alg_equiv.subsingleton_right | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_val : S.val.range = S | ext $ set.ext_iff.1 $ S.val.coe_range.trans subtype.range_val | lemma | subalgebra.range_val | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subtype.range_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion {S T : subalgebra R A} (h : S ≤ T) : S →ₐ[R] T | { to_fun := set.inclusion h,
map_one' := rfl,
map_add' := λ _ _, rfl,
map_mul' := λ _ _, rfl,
map_zero' := rfl,
commutes' := λ _, rfl } | def | subalgebra.inclusion | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.inclusion",
"subalgebra"
] | The map `S → T` when `S` is a subalgebra contained in the subalgebra `T`.
This is the subalgebra version of `submodule.of_le`, or `subring.inclusion` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_injective {S T : subalgebra R A} (h : S ≤ T) :
function.injective (inclusion h) | λ _ _, subtype.ext ∘ subtype.mk.inj | lemma | subalgebra.inclusion_injective | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_self {S : subalgebra R A}:
inclusion (le_refl S) = alg_hom.id R S | alg_hom.ext $ λ x, subtype.ext rfl | lemma | subalgebra.inclusion_self | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_hom.ext",
"alg_hom.id",
"subalgebra",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_mk {S T : subalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :
inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ | rfl | lemma | subalgebra.inclusion_mk | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_right {S T : subalgebra R A} (h : S ≤ T) (x : T)
(m : (x : A) ∈ S) : inclusion h ⟨x, m⟩ = x | subtype.ext rfl | lemma | subalgebra.inclusion_right | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_inclusion {S T U : subalgebra R A} (hst : S ≤ T) (htu : T ≤ U)
(x : S) : inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x | subtype.ext rfl | lemma | subalgebra.inclusion_inclusion | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inclusion {S T : subalgebra R A} (h : S ≤ T) (s : S) :
(inclusion h s : A) = s | rfl | lemma | subalgebra.coe_inclusion | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_eq (S T : subalgebra R A) (h : S = T) : S ≃ₐ[R] T | { to_fun := λ x, ⟨x, h ▸ x.2⟩,
inv_fun := λ x, ⟨x, h.symm ▸ x.2⟩,
map_mul' := λ _ _, rfl,
commutes' := λ _, rfl,
.. linear_equiv.of_eq _ _ (congr_arg to_submodule h) } | def | subalgebra.equiv_of_eq | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"inv_fun",
"linear_equiv.of_eq",
"subalgebra"
] | Two subalgebras that are equal are also equivalent as algebras.
This is the `subalgebra` version of `linear_equiv.of_eq` and `equiv.set.of_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_eq_symm (S T : subalgebra R A) (h : S = T) :
(equiv_of_eq S T h).symm = equiv_of_eq T S h.symm | rfl | lemma | subalgebra.equiv_of_eq_symm | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_eq_rfl (S : subalgebra R A) :
equiv_of_eq S S rfl = alg_equiv.refl | by { ext, refl } | lemma | subalgebra.equiv_of_eq_rfl | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_equiv.refl",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_eq_trans (S T U : subalgebra R A) (hST : S = T) (hTU : T = U) :
(equiv_of_eq S T hST).trans (equiv_of_eq T U hTU) = equiv_of_eq S U (trans hST hTU) | rfl | lemma | subalgebra.equiv_of_eq_trans | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod : subalgebra R (A × B) | { carrier := S ×ˢ S₁,
algebra_map_mem' := λ r, ⟨algebra_map_mem _ _, algebra_map_mem _ _⟩,
.. S.to_subsemiring.prod S₁.to_subsemiring } | def | subalgebra.prod | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | The product of two subalgebras is a subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod : (prod S S₁ : set (A × B)) = S ×ˢ S₁ | rfl | lemma | subalgebra.coe_prod | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_to_submodule :
(S.prod S₁).to_submodule = S.to_submodule.prod S₁.to_submodule | rfl | lemma | subalgebra.prod_to_submodule | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_prod {S : subalgebra R A} {S₁ : subalgebra R B} {x : A × B} :
x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ | set.mem_prod | lemma | subalgebra.mem_prod | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.mem_prod",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_top : (prod ⊤ ⊤ : subalgebra R (A × B)) = ⊤ | by ext; simp | lemma | subalgebra.prod_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono {S T : subalgebra R A} {S₁ T₁ : subalgebra R B} :
S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ | set.prod_mono | lemma | subalgebra.prod_mono | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.prod_mono",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_inf_prod {S T : subalgebra R A} {S₁ T₁ : subalgebra R B} :
S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) | set_like.coe_injective set.prod_inter_prod | lemma | subalgebra.prod_inf_prod | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.prod_inter_prod",
"set_like.coe_injective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr_of_directed [nonempty ι] {S : ι → subalgebra R A}
(dir : directed (≤) S) : ↑(supr S) = ⋃ i, (S i : set A) | let K : subalgebra R A :=
{ carrier := ⋃ i, (S i),
mul_mem' := λ x y hx hy,
let ⟨i, hi⟩ := set.mem_Union.1 hx in
let ⟨j, hj⟩ := set.mem_Union.1 hy in
let ⟨k, hik, hjk⟩ := dir i j in
set.mem_Union.2 ⟨k, subalgebra.mul_mem (S k) (hik hi) (hjk hj)⟩ ,
add_mem' := λ x y hx hy,
let ⟨i,... | lemma | subalgebra.coe_supr_of_directed | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"directed",
"le_supr",
"nonempty.some",
"set.Union_subset",
"set.subset_Union",
"subalgebra",
"subalgebra.add_mem",
"subalgebra.algebra_map_mem",
"subalgebra.mul_mem",
"supr",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_lift [nonempty ι]
(K : ι → subalgebra R A)
(dir : directed (≤) K)
(f : Π i, K i →ₐ[R] B)
(hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))
(T : subalgebra R A) (hT : T = supr K) :
↥T →ₐ[R] B | by subst hT; exact
{ to_fun := set.Union_lift (λ i, ↑(K i)) (λ i x, f i x)
(λ i j x hxi hxj,
let ⟨k, hik, hjk⟩ := dir i j in
begin
rw [hf i k hik, hf j k hjk],
refl
end) ↑(supr K)
(by rw coe_supr_of_directed dir; refl),
map_one' := set.Union_lift_const _ (λ _, 1) (λ _, rfl) _... | def | subalgebra.supr_lift | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"alg_hom.commutes",
"algebra_map",
"directed",
"set.Union_lift",
"set.Union_lift_binary",
"set.Union_lift_const",
"subalgebra",
"supr"
] | Define an algebra homomorphism on a directed supremum of subalgebras by defining
it on each subalgebra, and proving that it agrees on the intersection of subalgebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
supr_lift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :
supr_lift K dir f hf T hT (inclusion h x) = f i x | by subst T; exact set.Union_lift_inclusion _ _ | lemma | subalgebra.supr_lift_inclusion | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.Union_lift_inclusion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_lift_comp_inclusion {i : ι} (h : K i ≤ T) :
(supr_lift K dir f hf T hT).comp (inclusion h) = f i | by ext; simp | lemma | subalgebra.supr_lift_comp_inclusion | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_lift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :
supr_lift K dir f hf T hT ⟨x, hx⟩ = f i x | by subst hT; exact set.Union_lift_mk x hx | lemma | subalgebra.supr_lift_mk | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.Union_lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_lift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :
supr_lift K dir f hf T hT x = f i ⟨x, hx⟩ | by subst hT; exact set.Union_lift_of_mem x hx | lemma | subalgebra.supr_lift_of_mem | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.Union_lift_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_def [has_smul A α] {S : subalgebra R A} (g : S) (m : α) : g • m = (g : A) • m | rfl | lemma | subalgebra.smul_def | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"has_smul",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_left
[has_smul A β] [has_smul α β] [smul_comm_class A α β] (S : subalgebra R A) :
smul_comm_class S α β | S.to_subsemiring.smul_comm_class_left | instance | subalgebra.smul_comm_class_left | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"has_smul",
"smul_comm_class",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_right
[has_smul α β] [has_smul A β] [smul_comm_class α A β] (S : subalgebra R A) :
smul_comm_class α S β | S.to_subsemiring.smul_comm_class_right | instance | subalgebra.smul_comm_class_right | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"has_smul",
"smul_comm_class",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_left
[has_smul α β] [has_smul A α] [has_smul A β] [is_scalar_tower A α β] (S : subalgebra R A) :
is_scalar_tower S α β | S.to_subsemiring.is_scalar_tower | instance | subalgebra.is_scalar_tower_left | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"has_smul",
"is_scalar_tower",
"subalgebra"
] | Note that this provides `is_scalar_tower S R R` which is needed by `smul_mul_assoc`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_scalar_tower_mid {R S T : Type*} [comm_semiring R] [semiring S] [add_comm_monoid T]
[algebra R S] [module R T] [module S T] [is_scalar_tower R S T] (S' : subalgebra R S) :
is_scalar_tower R S' T | ⟨λ x y z, (smul_assoc _ (y : S) _ : _)⟩ | instance | subalgebra.is_scalar_tower_mid | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"algebra",
"comm_semiring",
"is_scalar_tower",
"module",
"semiring",
"smul_assoc",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_left [add_comm_monoid α] [module A α] (S : subalgebra R A) : module S α | S.to_subsemiring.module | instance | subalgebra.module_left | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"module",
"subalgebra"
] | The action by a subalgebra is the action by the underlying algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_algebra {R A : Type*} [comm_semiring R] [comm_semiring A] [semiring α]
[algebra R A] [algebra A α] (S : subalgebra R A) : algebra S α | algebra.of_subsemiring S.to_subsemiring | instance | subalgebra.to_algebra | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"algebra.of_subsemiring",
"comm_semiring",
"semiring",
"subalgebra"
] | The action by a subalgebra is the action by the underlying algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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