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alg_hom_ext {φ ψ : unitization R A →ₐ[S] B} (h : ∀ a : A, φ a = ψ a)
(h' : ∀ r, φ (algebra_map R (unitization R A) r) = ψ (algebra_map R (unitization R A) r)) :
φ = ψ | begin
ext,
induction x using unitization.ind,
simp only [map_add, ←algebra_map_eq_inl, h, h'],
end | lemma | unitization.alg_hom_ext | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"algebra_map",
"unitization",
"unitization.ind"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom_ext' {φ ψ : unitization R A →ₐ[R] C}
(h : φ.to_non_unital_alg_hom.comp (coe_non_unital_alg_hom R A) =
ψ.to_non_unital_alg_hom.comp (coe_non_unital_alg_hom R A)) :
φ = ψ | alg_hom_ext (non_unital_alg_hom.congr_fun h) (by simp [alg_hom.commutes]) | lemma | unitization.alg_hom_ext' | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"alg_hom.commutes",
"non_unital_alg_hom.congr_fun",
"unitization"
] | See note [partially-applied ext lemmas] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : (A →ₙₐ[R] C) ≃ (unitization R A →ₐ[R] C) | { to_fun := λ φ,
{ to_fun := λ x, algebra_map R C x.fst + φ x.snd,
map_one' := by simp only [fst_one, map_one, snd_one, φ.map_zero, add_zero],
map_mul' := λ x y,
begin
induction x using unitization.ind,
induction y using unitization.ind,
simp only [mul_add, add_mul, coe_mul, fst_add, fst... | def | unitization.lift | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"inv_fun",
"lift",
"map_mul",
"map_one",
"mul_zero",
"smul_one_mul",
"smul_zero",
"unitization",
"unitization.alg_hom_ext'",
"unitization.ind",
"zero_mul",
"zero_smul"
] | Non-unital algebra homomorphisms from `A` into a unital `R`-algebra `C` lift uniquely to
`unitization R A →ₐ[R] C`. This is the universal property of the unitization. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_symm_apply (φ : unitization R A →ₐ[R] C) (a : A) :
unitization.lift.symm φ a = φ a | rfl | lemma | unitization.lift_symm_apply | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra (R : Type u) (A : Type v)
[comm_semiring R] [semiring A] [algebra R A] extends subsemiring A : Type v | (algebra_map_mem' : ∀ r, algebra_map R A r ∈ carrier)
(zero_mem' := (algebra_map R A).map_zero ▸ algebra_map_mem' 0)
(one_mem' := (algebra_map R A).map_one ▸ algebra_map_mem' 1) | structure | subalgebra | 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",
"comm_semiring",
"map_one",
"semiring",
"subsemiring"
] | A subalgebra is a sub(semi)ring that includes the range of `algebra_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_carrier {s : subalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s | iff.rfl | lemma | subalgebra.mem_carrier | 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 | |
ext {S T : subalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T | set_like.ext h | theorem | subalgebra.ext | 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.ext",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_subsemiring {S : subalgebra R A} {x} : x ∈ S.to_subsemiring ↔ x ∈ S | iff.rfl | lemma | subalgebra.mem_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_to_subsemiring (S : subalgebra R A) : (↑S.to_subsemiring : set A) = S | rfl | lemma | subalgebra.coe_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 | |
to_subsemiring_injective :
function.injective (to_subsemiring : subalgebra R A → subsemiring A) | λ S T h, ext $ λ x, by rw [← mem_to_subsemiring, ← mem_to_subsemiring, h] | theorem | subalgebra.to_subsemiring_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",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subsemiring_inj {S U : subalgebra R A} : S.to_subsemiring = U.to_subsemiring ↔ S = U | to_subsemiring_injective.eq_iff | theorem | subalgebra.to_subsemiring_inj | 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 | |
copy (S : subalgebra R A) (s : set A) (hs : s = ↑S) : subalgebra R A | { carrier := s,
add_mem' := λ _ _, hs.symm ▸ S.add_mem',
mul_mem' := λ _ _, hs.symm ▸ S.mul_mem',
algebra_map_mem' := hs.symm ▸ S.algebra_map_mem' } | def | subalgebra.copy | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra"
] | Copy of a subalgebra with a new `carrier` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (S : subalgebra R A) (s : set A) (hs : s = ↑S) :
(S.copy s hs : set A) = s | rfl | lemma | subalgebra.coe_copy | 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 | |
copy_eq (S : subalgebra R A) (s : set A) (hs : s = ↑S) : S.copy s hs = S | set_like.coe_injective hs | lemma | subalgebra.copy_eq | 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 | |
algebra_map_mem (r : R) : algebra_map R A r ∈ S | S.algebra_map_mem' r | theorem | subalgebra.algebra_map_mem | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_le : (algebra_map R A).srange ≤ S.to_subsemiring | λ x ⟨r, hr⟩, hr ▸ S.algebra_map_mem r | theorem | subalgebra.srange_le | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_subset : set.range (algebra_map R A) ⊆ S | λ x ⟨r, hr⟩, hr ▸ S.algebra_map_mem r | theorem | subalgebra.range_subset | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_le : set.range (algebra_map R A) ≤ S | S.range_subset | theorem | subalgebra.range_le | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem {x : A} (hx : x ∈ S) (r : R) : r • x ∈ S | (algebra.smul_def r x).symm ▸ mul_mem (S.algebra_map_mem r) hx | theorem | subalgebra.smul_mem | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem : (1 : A) ∈ S | one_mem S | theorem | subalgebra.one_mem | 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 | |
mul_mem {x y : A} (hx : x ∈ S) (hy : y ∈ S) : x * y ∈ S | mul_mem hx hy | theorem | subalgebra.mul_mem | 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 | |
pow_mem {x : A} (hx : x ∈ S) (n : ℕ) : x ^ n ∈ S | pow_mem hx n | theorem | subalgebra.pow_mem | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"pow_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem : (0 : A) ∈ S | zero_mem S | theorem | subalgebra.zero_mem | 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 | |
add_mem {x y : A} (hx : x ∈ S) (hy : y ∈ S) : x + y ∈ S | add_mem hx hy | theorem | subalgebra.add_mem | 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 | |
nsmul_mem {x : A} (hx : x ∈ S) (n : ℕ) : n • x ∈ S | nsmul_mem hx n | theorem | subalgebra.nsmul_mem | 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_nat_mem (n : ℕ) : (n : A) ∈ S | coe_nat_mem S n | theorem | subalgebra.coe_nat_mem | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"coe_nat_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_prod_mem {L : list A} (h : ∀ x ∈ L, x ∈ S) : L.prod ∈ S | list_prod_mem h | theorem | subalgebra.list_prod_mem | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"list_prod_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_sum_mem {L : list A} (h : ∀ x ∈ L, x ∈ S) : L.sum ∈ S | list_sum_mem h | theorem | subalgebra.list_sum_mem | 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 | |
multiset_sum_mem {m : multiset A} (h : ∀ x ∈ m, x ∈ S) : m.sum ∈ S | multiset_sum_mem m h | theorem | subalgebra.multiset_sum_mem | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mem {ι : Type w} {t : finset ι} {f : ι → A} (h : ∀ x ∈ t, f x ∈ S) :
∑ x in t, f x ∈ S | sum_mem h | theorem | subalgebra.sum_mem | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_prod_mem {R : Type u} {A : Type v} [comm_semiring R] [comm_semiring A]
[algebra R A] (S : subalgebra R A) {m : multiset A} (h : ∀ x ∈ m, x ∈ S) : m.prod ∈ S | multiset_prod_mem m h | theorem | subalgebra.multiset_prod_mem | 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_semiring",
"multiset",
"multiset_prod_mem",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mem {R : Type u} {A : Type v} [comm_semiring R] [comm_semiring A]
[algebra R A] (S : subalgebra R A) {ι : Type w} {t : finset ι} {f : ι → A}
(h : ∀ x ∈ t, f x ∈ S) : ∏ x in t, f x ∈ S | prod_mem h | theorem | subalgebra.prod_mem | 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_semiring",
"finset",
"prod_mem",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mem {R : Type u} {A : Type v} [comm_ring R] [ring A]
[algebra R A] (S : subalgebra R A) {x : A} (hx : x ∈ S) : -x ∈ S | neg_mem hx | theorem | subalgebra.neg_mem | 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 | |
sub_mem {R : Type u} {A : Type v} [comm_ring R] [ring A]
[algebra R A] (S : subalgebra R A) {x y : A} (hx : x ∈ S) (hy : y ∈ S) : x - y ∈ S | sub_mem hx hy | theorem | subalgebra.sub_mem | 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 | |
zsmul_mem {R : Type u} {A : Type v} [comm_ring R] [ring A]
[algebra R A] (S : subalgebra R A) {x : A} (hx : x ∈ S) (n : ℤ) : n • x ∈ S | zsmul_mem hx n | theorem | subalgebra.zsmul_mem | 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 | |
coe_int_mem {R : Type u} {A : Type v} [comm_ring R] [ring A]
[algebra R A] (S : subalgebra R A) (n : ℤ) : (n : A) ∈ S | coe_int_mem S n | theorem | subalgebra.coe_int_mem | 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_int_mem",
"comm_ring",
"ring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]
(S : subalgebra R A) : add_submonoid A | S.to_subsemiring.to_add_submonoid | def | subalgebra.to_add_submonoid | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"add_submonoid",
"algebra",
"comm_semiring",
"semiring",
"subalgebra"
] | The projection from a subalgebra of `A` to an additive submonoid of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_submonoid {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]
(S : subalgebra R A) : submonoid A | S.to_subsemiring.to_submonoid | def | subalgebra.to_submonoid | 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_semiring",
"semiring",
"subalgebra",
"submonoid"
] | The projection from a subalgebra of `A` to a submonoid of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_subring {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) :
subring A | { neg_mem' := λ _, S.neg_mem,
.. S.to_subsemiring } | def | subalgebra.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",
"subring"
] | A subalgebra over a ring is also a `subring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_to_subring {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A]
{S : subalgebra R A} {x} : x ∈ S.to_subring ↔ x ∈ S | iff.rfl | lemma | subalgebra.mem_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 | |
coe_to_subring {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A]
(S : subalgebra R A) : (↑S.to_subring : set A) = S | rfl | lemma | subalgebra.coe_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_subring_injective {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] :
function.injective (to_subring : subalgebra R A → subring A) | λ S T h, ext $ λ x, by rw [← mem_to_subring, ← mem_to_subring, h] | theorem | subalgebra.to_subring_injective | 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",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subring_inj {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A]
{S U : subalgebra R A} : S.to_subring = U.to_subring ↔ S = U | to_subring_injective.eq_iff | theorem | subalgebra.to_subring_inj | 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_semiring {R A}
[comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) :
semiring S | S.to_subsemiring.to_semiring | instance | subalgebra.to_semiring | 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_semiring",
"semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_comm_semiring {R A}
[comm_semiring R] [comm_semiring A] [algebra R A] (S : subalgebra R A) :
comm_semiring S | S.to_subsemiring.to_comm_semiring | instance | subalgebra.to_comm_semiring | 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_semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring {R A}
[comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) :
ring S | S.to_subring.to_ring | instance | subalgebra.to_ring | 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_comm_ring {R A}
[comm_ring R] [comm_ring A] [algebra R A] (S : subalgebra R A) :
comm_ring S | S.to_subring.to_comm_ring | instance | subalgebra.to_comm_ring | 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",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordered_semiring {R A}
[comm_semiring R] [ordered_semiring A] [algebra R A] (S : subalgebra R A) :
ordered_semiring S | S.to_subsemiring.to_ordered_semiring | instance | subalgebra.to_ordered_semiring | 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_semiring",
"ordered_semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_strict_ordered_semiring {R A}
[comm_semiring R] [strict_ordered_semiring A] [algebra R A] (S : subalgebra R A) :
strict_ordered_semiring S | S.to_subsemiring.to_strict_ordered_semiring | instance | subalgebra.to_strict_ordered_semiring | 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_semiring",
"strict_ordered_semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordered_comm_semiring {R A}
[comm_semiring R] [ordered_comm_semiring A] [algebra R A] (S : subalgebra R A) :
ordered_comm_semiring S | S.to_subsemiring.to_ordered_comm_semiring | instance | subalgebra.to_ordered_comm_semiring | 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_semiring",
"ordered_comm_semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_strict_ordered_comm_semiring {R A}
[comm_semiring R] [strict_ordered_comm_semiring A] [algebra R A] (S : subalgebra R A) :
strict_ordered_comm_semiring S | S.to_subsemiring.to_strict_ordered_comm_semiring | instance | subalgebra.to_strict_ordered_comm_semiring | 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_semiring",
"strict_ordered_comm_semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordered_ring {R A}
[comm_ring R] [ordered_ring A] [algebra R A] (S : subalgebra R A) :
ordered_ring S | S.to_subring.to_ordered_ring | instance | subalgebra.to_ordered_ring | 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",
"ordered_ring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordered_comm_ring {R A}
[comm_ring R] [ordered_comm_ring A] [algebra R A] (S : subalgebra R A) :
ordered_comm_ring S | S.to_subring.to_ordered_comm_ring | instance | subalgebra.to_ordered_comm_ring | 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",
"ordered_comm_ring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_ordered_semiring {R A}
[comm_semiring R] [linear_ordered_semiring A] [algebra R A] (S : subalgebra R A) :
linear_ordered_semiring S | S.to_subsemiring.to_linear_ordered_semiring | instance | subalgebra.to_linear_ordered_semiring | 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_semiring",
"linear_ordered_semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_ordered_comm_semiring {R A}
[comm_semiring R] [linear_ordered_comm_semiring A] [algebra R A] (S : subalgebra R A) :
linear_ordered_comm_semiring S | S.to_subsemiring.to_linear_ordered_comm_semiring | instance | subalgebra.to_linear_ordered_comm_semiring | 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_semiring",
"linear_ordered_comm_semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_ordered_ring {R A}
[comm_ring R] [linear_ordered_ring A] [algebra R A] (S : subalgebra R A) :
linear_ordered_ring S | S.to_subring.to_linear_ordered_ring | instance | subalgebra.to_linear_ordered_ring | 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",
"linear_ordered_ring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_ordered_comm_ring {R A}
[comm_ring R] [linear_ordered_comm_ring A] [algebra R A] (S : subalgebra R A) :
linear_ordered_comm_ring S | S.to_subring.to_linear_ordered_comm_ring | instance | subalgebra.to_linear_ordered_comm_ring | 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",
"linear_ordered_comm_ring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submodule : subalgebra R A ↪o submodule R A | { to_embedding :=
{ to_fun := λ S,
{ carrier := S,
zero_mem' := (0:S).2,
add_mem' := λ x y hx hy, (⟨x, hx⟩ + ⟨y, hy⟩ : S).2,
smul_mem' := λ c x hx, (algebra.smul_def c x).symm ▸
(⟨algebra_map R A c, S.range_le ⟨c, rfl⟩⟩ * ⟨x, hx⟩:S).2 },
inj' := λ S T h, ext $ by apply set_like.ext_i... | def | 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"
] | [
"algebra.smul_def",
"set_like.coe_subset_coe",
"subalgebra",
"submodule"
] | The forgetful map from `subalgebra` to `submodule` as an `order_embedding` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_to_submodule {x} : x ∈ S.to_submodule ↔ x ∈ S | iff.rfl | lemma | subalgebra.mem_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 | |
coe_to_submodule (S : subalgebra R A) : (↑S.to_submodule : set A) = S | rfl | lemma | subalgebra.coe_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 | |
module' [semiring R'] [has_smul R' R] [module R' A] [is_scalar_tower R' R A] :
module R' S | S.to_submodule.module' | instance | subalgebra.module' | 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",
"module",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra' [comm_semiring R'] [has_smul R' R] [algebra R' A]
[is_scalar_tower R' R A] : algebra R' S | { commutes' := λ c x, subtype.eq $ algebra.commutes _ _,
smul_def' := λ c x, subtype.eq $ algebra.smul_def _ _,
.. (algebra_map R' A).cod_restrict S $ λ x, begin
rw [algebra.algebra_map_eq_smul_one, ←smul_one_smul R x (1 : A),
←algebra.algebra_map_eq_smul_one],
exact algebra_map_mem S _,
end } | instance | subalgebra.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.algebra_map_eq_smul_one",
"algebra.commutes",
"algebra.smul_def",
"algebra_map",
"comm_semiring",
"has_smul",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_smul_divisors_bot [no_zero_smul_divisors R A] : no_zero_smul_divisors R S | ⟨λ c x h,
have c = 0 ∨ (x : A) = 0,
from eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg coe h),
this.imp_right (@subtype.ext_iff _ _ x 0).mpr⟩ | instance | subalgebra.no_zero_smul_divisors_bot | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"no_zero_smul_divisors",
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y | rfl | lemma | subalgebra.coe_add | 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_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y | rfl | lemma | subalgebra.coe_mul | 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_zero : ((0 : S) : A) = 0 | rfl | lemma | subalgebra.coe_zero | 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_one : ((1 : S) : A) = 1 | rfl | lemma | subalgebra.coe_one | 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_neg {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A]
{S : subalgebra R A} (x : S) : (↑(-x) : A) = -↑x | rfl | lemma | subalgebra.coe_neg | 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 | |
coe_sub {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A]
{S : subalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y | rfl | lemma | subalgebra.coe_sub | 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 | |
coe_smul [semiring R'] [has_smul R' R] [module R' A]
[is_scalar_tower R' R A] (r : R') (x : S) : (↑(r • x) : A) = r • ↑x | rfl | lemma | subalgebra.coe_smul | 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",
"module",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_algebra_map [comm_semiring R'] [has_smul R' R] [algebra R' A]
[is_scalar_tower R' R A] (r : R') :
↑(algebra_map R' S r) = algebra_map R' A r | rfl | lemma | subalgebra.coe_algebra_map | 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",
"comm_semiring",
"has_smul",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (x : S) (n : ℕ) : (↑(x^n) : A) = (↑x)^n | submonoid_class.coe_pow x n | lemma | subalgebra.coe_pow | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"submonoid_class.coe_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 | zero_mem_class.coe_eq_zero | lemma | subalgebra.coe_eq_zero | 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_eq_one {x : S} : (x : A) = 1 ↔ x = 1 | one_mem_class.coe_eq_one | lemma | subalgebra.coe_eq_one | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"one_mem_class.coe_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val : S →ₐ[R] A | by refine_struct { to_fun := (coe : S → A) }; intros; refl | def | subalgebra.val | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | Embedding of a subalgebra into the algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_val : (S.val : S → A) = coe | rfl | lemma | subalgebra.coe_val | 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 | |
val_apply (x : S) : S.val x = (x : A) | rfl | lemma | subalgebra.val_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 | |
to_subsemiring_subtype : S.to_subsemiring.subtype = (S.val : S →+* A) | rfl | lemma | subalgebra.to_subsemiring_subtype | 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 | |
to_subring_subtype {R A : Type*} [comm_ring R] [ring A]
[algebra R A] (S : subalgebra R A) : S.to_subring.subtype = (S.val : S →+* A) | rfl | lemma | subalgebra.to_subring_subtype | 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_equiv (S : subalgebra R A) : S.to_submodule ≃ₗ[R] S | linear_equiv.of_eq _ _ rfl | def | subalgebra.to_submodule_equiv | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"linear_equiv.of_eq",
"subalgebra"
] | Linear equivalence between `S : submodule R A` and `S`. Though these types are equal,
we define it as a `linear_equiv` to avoid type equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : A →ₐ[R] B) (S : subalgebra R A) : subalgebra R B | { algebra_map_mem' := λ r, f.commutes r ▸ set.mem_image_of_mem _ (S.algebra_map_mem r),
.. S.to_subsemiring.map (f : A →+* B) } | def | 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"
] | [
"set.mem_image_of_mem",
"subalgebra"
] | Transport a subalgebra via an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mono {S₁ S₂ : subalgebra R A} {f : A →ₐ[R] B} :
S₁ ≤ S₂ → S₁.map f ≤ S₂.map f | set.image_subset f | lemma | subalgebra.map_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.image_subset",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_injective {f : A →ₐ[R] B} (hf : function.injective f) :
function.injective (map f) | λ S₁ S₂ ih, ext $ set.ext_iff.1 $ set.image_injective.2 hf $ set.ext $ set_like.ext_iff.mp ih | lemma | subalgebra.map_injective | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"ih",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (S : subalgebra R A) : S.map (alg_hom.id R A) = S | set_like.coe_injective $ set.image_id _ | lemma | subalgebra.map_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",
"map_id",
"set.image_id",
"set_like.coe_injective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (S : subalgebra R A) (g : B →ₐ[R] C) (f : A →ₐ[R] B) :
(S.map f).map g = S.map (g.comp f) | set_like.coe_injective $ set.image_image _ _ _ | lemma | subalgebra.map_map | 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_image",
"set_like.coe_injective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {S : subalgebra R A} {f : A →ₐ[R] B} {y : B} :
y ∈ map f S ↔ ∃ x ∈ S, f x = y | subsemiring.mem_map | lemma | subalgebra.mem_map | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"mem_map",
"subalgebra",
"subsemiring.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_to_submodule {S : subalgebra R A} {f : A →ₐ[R] B} :
(S.map f).to_submodule = S.to_submodule.map f.to_linear_map | set_like.coe_injective rfl | lemma | subalgebra.map_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 | |
map_to_subsemiring {S : subalgebra R A} {f : A →ₐ[R] B} :
(S.map f).to_subsemiring = S.to_subsemiring.map f.to_ring_hom | set_like.coe_injective rfl | lemma | subalgebra.map_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_map (S : subalgebra R A) (f : A →ₐ[R] B) :
(S.map f : set B) = f '' S | rfl | lemma | subalgebra.coe_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : A →ₐ[R] B) (S : subalgebra R B) : subalgebra R A | { algebra_map_mem' := λ r, show f (algebra_map R A r) ∈ S,
from (f.commutes r).symm ▸ S.algebra_map_mem r,
.. S.to_subsemiring.comap (f : A →+* B) } | def | subalgebra.comap | 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",
"subalgebra"
] | Preimage of a subalgebra under an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_le {S : subalgebra R A} {f : A →ₐ[R] B} {U : subalgebra R B} :
map f S ≤ U ↔ S ≤ comap f U | set.image_subset_iff | theorem | subalgebra.map_le | 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_subset_iff",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_map_comap (f : A →ₐ[R] B) : galois_connection (map f) (comap f) | λ S U, map_le | lemma | subalgebra.gc_map_comap | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap (S : subalgebra R B) (f : A →ₐ[R] B) (x : A) :
x ∈ S.comap f ↔ f x ∈ S | iff.rfl | lemma | subalgebra.mem_comap | 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_comap (S : subalgebra R B) (f : A →ₐ[R] B) :
(S.comap f : set A) = f ⁻¹' (S : set B) | rfl | lemma | subalgebra.coe_comap | 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 | |
no_zero_divisors {R A : Type*} [comm_semiring R] [semiring A] [no_zero_divisors A]
[algebra R A] (S : subalgebra R A) : no_zero_divisors S | S.to_subsemiring.no_zero_divisors | instance | subalgebra.no_zero_divisors | 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_semiring",
"no_zero_divisors",
"semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_domain {R A : Type*} [comm_ring R] [ring A] [is_domain A] [algebra R A]
(S : subalgebra R A) : is_domain S | subring.is_domain S.to_subring | instance | subalgebra.is_domain | 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",
"is_domain",
"ring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subalgebra (p : submodule R A) (h_one : (1 : A) ∈ p)
(h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) : subalgebra R A | { mul_mem' := h_mul,
algebra_map_mem' := λ r, begin
rw algebra.algebra_map_eq_smul_one,
exact p.smul_mem _ h_one,
end,
..p} | def | 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"
] | [
"algebra.algebra_map_eq_smul_one",
"subalgebra",
"submodule"
] | A submodule containing `1` and closed under multiplication is a subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_to_subalgebra {p : submodule R A} {h_one h_mul} {x} :
x ∈ p.to_subalgebra h_one h_mul ↔ x ∈ p | iff.rfl | lemma | submodule.mem_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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_subalgebra (p : submodule R A) (h_one h_mul) :
(p.to_subalgebra h_one h_mul : set A) = p | rfl | lemma | submodule.coe_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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subalgebra_mk (s : set A) (h0 hadd hsmul h1 hmul) :
(submodule.mk s hadd h0 hsmul : submodule R A).to_subalgebra h1 hmul =
subalgebra.mk s @hmul h1 @hadd h0
(λ r, by { rw algebra.algebra_map_eq_smul_one, exact hsmul r h1 }) | rfl | lemma | submodule.to_subalgebra_mk | 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_eq_smul_one",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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