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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|---|---|---|---|---|---|---|---|---|---|---|
zero_eq [nontrivial A] : σ (0 : A) = {0} | begin
refine set.subset.antisymm _ (by simp [algebra.algebra_map_eq_smul_one, mem_iff]),
rw [spectrum, set.compl_subset_comm],
intros k hk,
rw set.mem_compl_singleton_iff at hk,
have : is_unit (units.mk0 k hk • (1 : A)) := is_unit.smul (units.mk0 k hk) is_unit_one,
simpa [mem_resolvent_set_iff, algebra.alge... | lemma | spectrum.zero_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra.algebra_map_eq_smul_one",
"is_unit",
"is_unit.smul",
"is_unit_one",
"nontrivial",
"set.compl_subset_comm",
"set.mem_compl_singleton_iff",
"set.subset.antisymm",
"spectrum",
"units.mk0"
] | Without the assumption `nontrivial A`, then `0 : A` would be invertible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
scalar_eq [nontrivial A] (k : 𝕜) : σ (↑ₐk) = {k} | by rw [←add_zero (↑ₐk), ←singleton_add_eq, zero_eq, set.singleton_add_singleton, add_zero] | theorem | spectrum.scalar_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq [nontrivial A] : σ (1 : A) = {1} | calc σ (1 : A) = σ (↑ₐ1) : by rw [algebra.algebra_map_eq_smul_one, one_smul]
... = {1} : scalar_eq 1 | lemma | spectrum.one_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra.algebra_map_eq_smul_one",
"nontrivial",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq_smul [nontrivial A] (k : 𝕜) (a : A) (ha : (σ a).nonempty) :
σ (k • a) = k • (σ a) | begin
rcases eq_or_ne k 0 with rfl | h,
{ simpa [ha, zero_smul_set] },
{ exact unit_smul_eq_smul a (units.mk0 k h) },
end | theorem | spectrum.smul_eq_smul | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"eq_or_ne",
"nontrivial",
"units.mk0"
] | the assumption `(σ a).nonempty` is necessary and cannot be removed without
further conditions on the algebra `A` and scalar field `𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonzero_mul_eq_swap_mul (a b : A) : σ (a * b) \ {0} = σ (b * a) \ {0} | begin
suffices h : ∀ (x y : A), σ (x * y) \ {0} ⊆ σ (y * x) \ {0},
{ exact set.eq_of_subset_of_subset (h a b) (h b a) },
{ rintros _ _ k ⟨k_mem, k_neq⟩,
change k with ↑(units.mk0 k k_neq) at k_mem,
exact ⟨unit_mem_mul_iff_mem_swap_mul.mp k_mem, k_neq⟩ },
end | theorem | spectrum.nonzero_mul_eq_swap_mul | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"set.eq_of_subset_of_subset",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv (a : Aˣ) : (σ (a : A))⁻¹ = σ (↑a⁻¹ : A) | begin
refine set.eq_of_subset_of_subset (λ k hk, _) (λ k hk, _),
{ rw set.mem_inv at hk,
have : k ≠ 0,
{ simpa only [inv_inv] using inv_ne_zero (ne_zero_of_mem_of_unit hk), },
lift k to 𝕜ˣ using is_unit_iff_ne_zero.mpr this,
rw ←units.coe_inv k at hk,
exact inv_mem_iff.mp hk },
{ lift k to 𝕜... | lemma | spectrum.map_inv | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"inv_inv",
"inv_ne_zero",
"lift",
"map_inv",
"set.eq_of_subset_of_subset",
"set.mem_inv",
"units.coe_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_resolvent_set_apply (φ : F) {a : A} {r : R} (h : r ∈ resolvent_set R a) :
r ∈ resolvent_set R ((φ : A → B) a) | by simpa only [map_sub, alg_hom_class.commutes] using h.map φ | lemma | alg_hom.mem_resolvent_set_apply | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"resolvent_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectrum_apply_subset (φ : F) (a : A) : σ ((φ : A → B) a) ⊆ σ a | λ _, mt (mem_resolvent_set_apply φ) | lemma | alg_hom.spectrum_apply_subset | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_mem_spectrum [nontrivial R] (φ : F) (a : A) : φ a ∈ σ a | begin
have h : ↑ₐ(φ a) - a ∈ (φ : A →+* R).ker,
{ simp only [ring_hom.mem_ker, map_sub, ring_hom.coe_coe, alg_hom_class.commutes,
algebra.id.map_eq_id, ring_hom.id_apply, sub_self], },
simp only [spectrum.mem_iff, ←mem_nonunits_iff, coe_subset_nonunits ((φ : A →+* R).ker_ne_top) h],
end | lemma | alg_hom.apply_mem_spectrum | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra.id.map_eq_id",
"coe_subset_nonunits",
"nontrivial",
"ring_hom.coe_coe",
"ring_hom.id_apply",
"ring_hom.mem_ker",
"spectrum.mem_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsmul : A →ₐ[R] module.End R M | { to_fun := distrib_mul_action.to_linear_map R M,
map_one' := linear_map.ext $ λ _, one_smul A _,
map_mul' := λ a b, linear_map.ext $ smul_assoc a b,
map_zero' := linear_map.ext $ λ _, zero_smul A _,
map_add' := λ a b, linear_map.ext $ λ _, add_smul _ _ _,
commutes' := λ r, linear_map.ext $ algebra_map_smul A... | def | algebra.lsmul | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"add_smul",
"algebra_map_smul",
"distrib_mul_action.to_linear_map",
"linear_map.ext",
"module.End",
"one_smul",
"smul_assoc",
"zero_smul"
] | The `R`-algebra morphism `A → End (M)` corresponding to the representation of the algebra `A`
on the `R`-module `M`.
This is a stronger version of `distrib_mul_action.to_linear_map`, and could also have been
called `algebra.to_module_End`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lsmul_coe (a : A) : (lsmul R M a : M → M) = (•) a | rfl | lemma | algebra.lsmul_coe | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_smul (r : R) (x : M) : algebra_map R A r • x = r • x | by rw [algebra.algebra_map_eq_smul_one, smul_assoc, one_smul] | theorem | is_scalar_tower.algebra_map_smul | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"algebra_map_smul",
"one_smul",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_algebra_map_eq [algebra R A]
(h : ∀ x, algebra_map R A x = algebra_map S A (algebra_map R S x)) :
is_scalar_tower R S A | ⟨λ x y z, by simp_rw [algebra.smul_def, ring_hom.map_mul, mul_assoc, h]⟩ | theorem | is_scalar_tower.of_algebra_map_eq | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"is_scalar_tower",
"mul_assoc",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_algebra_map_eq' [algebra R A]
(h : algebra_map R A = (algebra_map S A).comp (algebra_map R S)) :
is_scalar_tower R S A | of_algebra_map_eq $ ring_hom.ext_iff.1 h | theorem | is_scalar_tower.of_algebra_map_eq' | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra",
"algebra_map",
"is_scalar_tower"
] | See note [partially-applied ext lemmas]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_eq :
algebra_map R A = (algebra_map S A).comp (algebra_map R S) | ring_hom.ext $ λ x, by simp_rw [ring_hom.comp_apply, algebra.algebra_map_eq_smul_one,
smul_assoc, one_smul] | theorem | is_scalar_tower.algebra_map_eq | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"one_smul",
"ring_hom.comp_apply",
"ring_hom.ext",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply (x : R) : algebra_map R A x = algebra_map S A (algebra_map R S x) | by rw [algebra_map_eq R S A, ring_hom.comp_apply] | theorem | is_scalar_tower.algebra_map_apply | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map",
"algebra_map_apply",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.ext {S : Type u} {A : Type v} [comm_semiring S] [semiring A]
(h1 h2 : algebra S A) (h : ∀ (r : S) (x : A), (by haveI := h1; exact r • x) = r • x) : h1 = h2 | algebra.algebra_ext _ _ $ λ r, by
simpa only [@algebra.smul_def _ _ _ _ h1, @algebra.smul_def _ _ _ _ h2, mul_one] using h r 1 | lemma | is_scalar_tower.algebra.ext | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra",
"algebra.algebra_ext",
"algebra.smul_def",
"comm_semiring",
"mul_one",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_alg_hom : S →ₐ[R] A | { commutes' := λ _, (algebra_map_apply _ _ _ _).symm,
.. algebra_map S A } | def | is_scalar_tower.to_alg_hom | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map",
"algebra_map_apply"
] | In a tower, the canonical map from the middle element to the top element is an
algebra homomorphism over the bottom element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_alg_hom_apply (y : S) : to_alg_hom R S A y = algebra_map S A y | rfl | lemma | is_scalar_tower.to_alg_hom_apply | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_alg_hom : ↑(to_alg_hom R S A) = algebra_map S A | ring_hom.ext $ λ _, rfl | lemma | is_scalar_tower.coe_to_alg_hom | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_alg_hom' : (to_alg_hom R S A : S → A) = algebra_map S A | rfl | lemma | is_scalar_tower.coe_to_alg_hom' | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.alg_hom.map_algebra_map (f : A →ₐ[S] B) (r : R) :
f (algebra_map R A r) = algebra_map R B r | by rw [algebra_map_apply R S A r, f.commutes, ← algebra_map_apply R S B] | lemma | alg_hom.map_algebra_map | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map",
"algebra_map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.alg_hom.comp_algebra_map_of_tower (f : A →ₐ[S] B) :
(f : A →+* B).comp (algebra_map R A) = algebra_map R B | ring_hom.ext f.map_algebra_map | lemma | alg_hom.comp_algebra_map_of_tower | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring (U : subsemiring S) : is_scalar_tower U S A | of_algebra_map_eq $ λ x, rfl | instance | is_scalar_tower.subsemiring | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"is_scalar_tower",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_ring_hom {R A B : Type*} [comm_semiring R] [comm_semiring A] [comm_semiring B]
[algebra R A] [algebra R B] (f : A →ₐ[R] B) :
@is_scalar_tower R A B _ (f.to_ring_hom.to_algebra.to_has_smul) _ | by { letI := (f : A →+* B).to_algebra, exact of_algebra_map_eq (λ x, (f.commutes x).symm) } | instance | is_scalar_tower.of_ring_hom | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra",
"comm_semiring",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars (f : A →ₐ[S] B) : A →ₐ[R] B | { commutes' := λ r, by { rw [algebra_map_apply R S A, algebra_map_apply R S B],
exact f.commutes (algebra_map R S r) },
.. (f : A →+* B) } | def | alg_hom.restrict_scalars | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map",
"algebra_map_apply",
"restrict_scalars"
] | R ⟶ S induces S-Alg ⥤ R-Alg | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars_apply (f : A →ₐ[S] B) (x : A) : f.restrict_scalars R x = f x | rfl | lemma | alg_hom.restrict_scalars_apply | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_restrict_scalars (f : A →ₐ[S] B) : (f.restrict_scalars R : A →+* B) = f | rfl | lemma | alg_hom.coe_restrict_scalars | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_restrict_scalars' (f : A →ₐ[S] B) : (restrict_scalars R f : A → B) = f | rfl | lemma | alg_hom.coe_restrict_scalars' | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_injective :
function.injective (restrict_scalars R : (A →ₐ[S] B) → (A →ₐ[R] B)) | λ f g h, alg_hom.ext (alg_hom.congr_fun h : _) | lemma | alg_hom.restrict_scalars_injective | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"alg_hom.congr_fun",
"alg_hom.ext",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars (f : A ≃ₐ[S] B) : A ≃ₐ[R] B | { commutes' := λ r, by { rw [algebra_map_apply R S A, algebra_map_apply R S B],
exact f.commutes (algebra_map R S r) },
.. (f : A ≃+* B) } | def | alg_equiv.restrict_scalars | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map",
"algebra_map_apply",
"restrict_scalars"
] | R ⟶ S induces S-Alg ⥤ R-Alg | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars_apply (f : A ≃ₐ[S] B) (x : A) : f.restrict_scalars R x = f x | rfl | lemma | alg_equiv.restrict_scalars_apply | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_restrict_scalars (f : A ≃ₐ[S] B) : (f.restrict_scalars R : A ≃+* B) = f | rfl | lemma | alg_equiv.coe_restrict_scalars | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_restrict_scalars' (f : A ≃ₐ[S] B) : (restrict_scalars R f : A → B) = f | rfl | lemma | alg_equiv.coe_restrict_scalars' | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_injective :
function.injective (restrict_scalars R : (A ≃ₐ[S] B) → (A ≃ₐ[R] B)) | λ f g h, alg_equiv.ext (alg_equiv.congr_fun h : _) | lemma | alg_equiv.restrict_scalars_injective | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"alg_equiv.congr_fun",
"alg_equiv.ext",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_span (hsur : function.surjective (algebra_map R A)) (X : set M) :
restrict_scalars R (span A X) = span R X | begin
refine ((span_le_restrict_scalars R A X).antisymm (λ m hm, _)).symm,
refine span_induction hm subset_span (zero_mem _) (λ _ _, add_mem) (λ a m hm, _),
obtain ⟨r, rfl⟩ := hsur a,
simpa [algebra_map_smul] using smul_mem _ r hm
end | lemma | submodule.restrict_scalars_span | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map",
"algebra_map_smul",
"restrict_scalars"
] | If `A` is an `R`-algebra such that the induced morphism `R →+* A` is surjective, then the
`R`-module generated by a set `X` equals the `A`-module generated by `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_span_eq_span_of_surjective
(h : function.surjective (algebra_map R A)) (s : set M) :
(submodule.span A s : set M) = submodule.span R s | congr_arg coe (submodule.restrict_scalars_span R A h s) | lemma | submodule.coe_span_eq_span_of_surjective | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra_map",
"submodule.restrict_scalars_span",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_span_smul_of_mem {s : set S} {t : set A} {k : S} (hks : k ∈ span R s)
{x : A} (hx : x ∈ t) : k • x ∈ span R (s • t) | span_induction hks (λ c hc, subset_span $ set.mem_smul.2 ⟨c, x, hc, hx, rfl⟩)
(by { rw zero_smul, exact zero_mem _ })
(λ c₁ c₂ ih₁ ih₂, by { rw add_smul, exact add_mem ih₁ ih₂ })
(λ b c hc, by { rw is_scalar_tower.smul_assoc, exact smul_mem _ _ hc }) | theorem | submodule.smul_mem_span_smul_of_mem | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"add_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_span_smul {s : set S} (hs : span R s = ⊤) {t : set A} {k : S}
{x : A} (hx : x ∈ span R t) :
k • x ∈ span R (s • t) | span_induction hx (λ x hx, smul_mem_span_smul_of_mem (hs.symm ▸ mem_top) hx)
(by { rw smul_zero, exact zero_mem _ })
(λ x y ihx ihy, by { rw smul_add, exact add_mem ihx ihy })
(λ c x hx, smul_comm c k x ▸ smul_mem _ _ hx) | theorem | submodule.smul_mem_span_smul | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"smul_add",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_span_smul' {s : set S} (hs : span R s = ⊤) {t : set A} {k : S}
{x : A} (hx : x ∈ span R (s • t)) :
k • x ∈ span R (s • t) | span_induction hx (λ x hx, let ⟨p, q, hp, hq, hpq⟩ := set.mem_smul.1 hx in
by { rw [← hpq, smul_smul], exact smul_mem_span_smul_of_mem (hs.symm ▸ mem_top) hq })
(by { rw smul_zero, exact zero_mem _ })
(λ x y ihx ihy, by { rw smul_add, exact add_mem ihx ihy })
(λ c x hx, smul_comm c k x ▸ smul_mem _ _ hx) | theorem | submodule.smul_mem_span_smul' | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"smul_add",
"smul_smul",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_smul_of_span_eq_top {s : set S} (hs : span R s = ⊤) (t : set A) :
span R (s • t) = (span S t).restrict_scalars R | le_antisymm (span_le.2 $ λ x hx, let ⟨p, q, hps, hqt, hpqx⟩ := set.mem_smul.1 hx in
hpqx ▸ (span S t).smul_mem p (subset_span hqt)) $
λ p hp, span_induction hp (λ x hx, one_smul S x ▸ smul_mem_span_smul hs (subset_span hx))
(zero_mem _)
(λ _ _, add_mem)
(λ k x hx, smul_mem_span_smul' hs hx) | theorem | submodule.span_smul_of_span_eq_top | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"one_smul",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_algebra_map_image (a : set R) :
submodule.span R (algebra_map R S '' a) =
(submodule.span R a).map (algebra.linear_map R S) | (submodule.span_image $ algebra.linear_map R S).trans rfl | lemma | submodule.span_algebra_map_image | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra.linear_map",
"algebra_map",
"submodule.span",
"submodule.span_image"
] | A variant of `submodule.span_image` for `algebra_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_algebra_map_image_of_tower {S T : Type*} [comm_semiring S] [semiring T]
[module R S] [is_scalar_tower R S S] [algebra R T] [algebra S T] [is_scalar_tower R S T]
(a : set S) :
submodule.span R (algebra_map S T '' a) =
(submodule.span R a).map ((algebra.linear_map S T).restrict_scalars R) | (submodule.span_image $ (algebra.linear_map S T).restrict_scalars R).trans rfl | lemma | submodule.span_algebra_map_image_of_tower | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra",
"algebra.linear_map",
"algebra_map",
"comm_semiring",
"is_scalar_tower",
"module",
"restrict_scalars",
"semiring",
"submodule.span",
"submodule.span_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mem_span_algebra_map_image {S T : Type*} [comm_semiring S] [semiring T]
[algebra R S] [algebra R T] [algebra S T] [is_scalar_tower R S T]
(x : S) (a : set S) (hx : x ∈ submodule.span R a) :
algebra_map S T x ∈ submodule.span R (algebra_map S T '' a) | by { rw [span_algebra_map_image_of_tower, mem_map], exact ⟨x, hx, rfl⟩ } | lemma | submodule.map_mem_span_algebra_map_image | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"is_scalar_tower",
"mem_map",
"semiring",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsmul_injective [no_zero_smul_divisors A M] {x : A} (hx : x ≠ 0) :
function.injective (lsmul R M x) | smul_right_injective _ hx | lemma | algebra.lsmul_injective | algebra.algebra | src/algebra/algebra/tower.lean | [
"algebra.algebra.equiv",
"linear_algebra.span"
] | [
"no_zero_smul_divisors",
"smul_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unitization (R A : Type*) | R × A | def | unitization | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [] | The minimal unitization of a non-unital `R`-algebra `A`. This is just a type synonym for
`R × A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl [has_zero A] (r : R) : unitization R A | (r, 0) | def | unitization.inl | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"unitization"
] | The canonical inclusion `R → unitization R A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst (x : unitization R A) : R | x.1 | def | unitization.fst | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"unitization"
] | The canonical projection `unitization R A → R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd (x : unitization R A) : A | x.2 | def | unitization.snd | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"unitization"
] | The canonical projection `unitization R A → A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {x y : unitization R A} (h1 : x.fst = y.fst) (h2 : x.snd = y.snd) : x = y | prod.ext h1 h2 | lemma | unitization.ext | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"prod.ext",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_inl [has_zero A] (r : R) : (inl r : unitization R A).fst = r | rfl | lemma | unitization.fst_inl | 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 | |
snd_inl [has_zero A] (r : R) : (inl r : unitization R A).snd = 0 | rfl | lemma | unitization.snd_inl | 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 | |
fst_coe [has_zero R] (a : A) : (a : unitization R A).fst = 0 | rfl | lemma | unitization.fst_coe | 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 | |
snd_coe [has_zero R] (a : A) : (a : unitization R A).snd = a | rfl | lemma | unitization.snd_coe | 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 | |
inl_injective [has_zero A] : function.injective (inl : R → unitization R A) | function.left_inverse.injective $ fst_inl _ | lemma | unitization.inl_injective | 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 | |
coe_injective [has_zero R] : function.injective (coe : A → unitization R A) | function.left_inverse.injective $ snd_coe _ | lemma | unitization.coe_injective | 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 | |
fst_zero [has_zero R] [has_zero A] : (0 : unitization R A).fst = 0 | rfl | lemma | unitization.fst_zero | 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 | |
snd_zero [has_zero R] [has_zero A] : (0 : unitization R A).snd = 0 | rfl | lemma | unitization.snd_zero | 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 | |
fst_add [has_add R] [has_add A] (x₁ x₂ : unitization R A) :
(x₁ + x₂).fst = x₁.fst + x₂.fst | rfl | lemma | unitization.fst_add | 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 | |
snd_add [has_add R] [has_add A] (x₁ x₂ : unitization R A) :
(x₁ + x₂).snd = x₁.snd + x₂.snd | rfl | lemma | unitization.snd_add | 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 | |
fst_neg [has_neg R] [has_neg A] (x : unitization R A) : (-x).fst = -x.fst | rfl | lemma | unitization.fst_neg | 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 | |
snd_neg [has_neg R] [has_neg A] (x : unitization R A) : (-x).snd = -x.snd | rfl | lemma | unitization.snd_neg | 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 | |
fst_smul [has_smul S R] [has_smul S A] (s : S) (x : unitization R A) :
(s • x).fst = s • x.fst | rfl | lemma | unitization.fst_smul | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"has_smul",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_smul [has_smul S R] [has_smul S A] (s : S) (x : unitization R A) :
(s • x).snd = s • x.snd | rfl | lemma | unitization.snd_smul | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"has_smul",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_zero [has_zero R] [has_zero A] : (inl 0 : unitization R A) = 0 | rfl | lemma | unitization.inl_zero | 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 | |
inl_add [has_add R] [add_zero_class A] (r₁ r₂ : R) :
(inl (r₁ + r₂) : unitization R A) = inl r₁ + inl r₂ | ext rfl (add_zero 0).symm | lemma | unitization.inl_add | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_zero_class",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_neg [has_neg R] [add_group A] (r : R) :
(inl (-r) : unitization R A) = -inl r | ext rfl neg_zero.symm | lemma | unitization.inl_neg | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_group",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_smul [monoid S] [add_monoid A] [has_smul S R] [distrib_mul_action S A]
(s : S) (r : R) : (inl (s • r) : unitization R A) = s • inl r | ext rfl (smul_zero s).symm | lemma | unitization.inl_smul | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_monoid",
"distrib_mul_action",
"has_smul",
"monoid",
"smul_zero",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero [has_zero R] [has_zero A] : ↑(0 : A) = (0 : unitization R A) | rfl | lemma | unitization.coe_zero | 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 | |
coe_add [add_zero_class R] [has_add A] (m₁ m₂ : A) :
(↑(m₁ + m₂) : unitization R A) = m₁ + m₂ | ext (add_zero 0).symm rfl | lemma | unitization.coe_add | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_zero_class",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg [add_group R] [has_neg A] (m : A) :
(↑(-m) : unitization R A) = -m | ext neg_zero.symm rfl | lemma | unitization.coe_neg | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_group",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul [has_zero R] [has_zero S] [smul_with_zero S R] [has_smul S A]
(r : S) (m : A) : (↑(r • m) : unitization R A) = r • m | ext (smul_zero _).symm rfl | lemma | unitization.coe_smul | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"has_smul",
"smul_with_zero",
"smul_zero",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_fst_add_coe_snd_eq [add_zero_class R] [add_zero_class A] (x : unitization R A) :
inl x.fst + ↑x.snd = x | ext (add_zero x.1) (zero_add x.2) | lemma | unitization.inl_fst_add_coe_snd_eq | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_zero_class",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ind {R A} [add_zero_class R] [add_zero_class A] {P : unitization R A → Prop}
(h : ∀ (r : R) (a : A), P (inl r + a)) (x) : P x | inl_fst_add_coe_snd_eq x ▸ h x.1 x.2 | lemma | unitization.ind | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_zero_class",
"unitization"
] | To show a property hold on all `unitization R A` it suffices to show it holds
on terms of the form `inl r + a`.
This can be used as `induction x using unitization.ind`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map_ext {N} [semiring S] [add_comm_monoid R] [add_comm_monoid A] [add_comm_monoid N]
[module S R] [module S A] [module S N] ⦃f g : unitization R A →ₗ[S] N⦄
(hl : ∀ r, f (inl r) = g (inl r)) (hr : ∀ a : A, f a = g a) :
f = g | linear_map.prod_ext (linear_map.ext hl) (linear_map.ext hr) | lemma | unitization.linear_map_ext | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_comm_monoid",
"linear_map.ext",
"linear_map.prod_ext",
"module",
"semiring",
"unitization"
] | This cannot be marked `@[ext]` as it ends up being used instead of `linear_map.prod_ext` when
working with `R × A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_hom [semiring R] [add_comm_monoid A] [module R A] : A →ₗ[R] unitization R A | { to_fun := coe, ..linear_map.inr R R A } | def | unitization.coe_hom | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_comm_monoid",
"linear_map.inr",
"module",
"semiring",
"unitization"
] | The canonical `R`-linear inclusion `A → unitization R A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_hom [semiring R] [add_comm_monoid A] [module R A] : unitization R A →ₗ[R] A | { to_fun := snd, ..linear_map.snd _ _ _ } | def | unitization.snd_hom | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_comm_monoid",
"linear_map.snd",
"module",
"semiring",
"unitization"
] | The canonical `R`-linear projection `unitization R A → A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_one [has_one R] [has_zero A] : (1 : unitization R A).fst = 1 | rfl | lemma | unitization.fst_one | 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 | |
snd_one [has_one R] [has_zero A] : (1 : unitization R A).snd = 0 | rfl | lemma | unitization.snd_one | 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 | |
fst_mul [has_mul R] [has_add A] [has_mul A] [has_smul R A]
(x₁ x₂ : unitization R A) : (x₁ * x₂).fst = x₁.fst * x₂.fst | rfl | lemma | unitization.fst_mul | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"has_smul",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_mul [has_mul R] [has_add A] [has_mul A] [has_smul R A]
(x₁ x₂ : unitization R A) : (x₁ * x₂).snd = x₁.fst • x₂.snd + x₂.fst • x₁.snd + x₁.snd * x₂.snd | rfl | lemma | unitization.snd_mul | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"has_smul",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_one [has_one R] [has_zero A] : (inl 1 : unitization R A) = 1 | rfl | lemma | unitization.inl_one | 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 | |
inl_mul [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A]
(r₁ r₂ : R) : (inl (r₁ * r₂) : unitization R A) = inl r₁ * inl r₂ | ext rfl $ show (0 : A) = r₁ • (0 : A) + r₂ • 0 + 0 * 0, by simp only [smul_zero, add_zero, mul_zero] | lemma | unitization.inl_mul | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"distrib_mul_action",
"monoid",
"mul_zero",
"non_unital_non_assoc_semiring",
"smul_zero",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_mul_inl [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A]
(r₁ r₂ : R) : (inl r₁ * inl r₂ : unitization R A) = inl (r₁ * r₂) | (inl_mul A r₁ r₂).symm | lemma | unitization.inl_mul_inl | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"distrib_mul_action",
"monoid",
"non_unital_non_assoc_semiring",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul [semiring R] [add_comm_monoid A] [has_mul A] [smul_with_zero R A]
(a₁ a₂ : A) : (↑(a₁ * a₂) : unitization R A) = a₁ * a₂ | ext (mul_zero _).symm $ show a₁ * a₂ = (0 : R) • a₂ + (0 : R) • a₁ + a₁ * a₂,
by simp only [zero_smul, zero_add] | lemma | unitization.coe_mul | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_comm_monoid",
"mul_zero",
"semiring",
"smul_with_zero",
"unitization",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_mul_coe [semiring R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A]
(r : R) (a : A) : (inl r * a : unitization R A) = ↑(r • a) | ext (mul_zero r) $ show r • a + (0 : R) • 0 + 0 * a = r • a,
by rw [smul_zero, add_zero, zero_mul, add_zero] | lemma | unitization.inl_mul_coe | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"distrib_mul_action",
"mul_zero",
"non_unital_non_assoc_semiring",
"semiring",
"smul_zero",
"unitization",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_inl [semiring R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A]
(r : R) (a : A) : (a * inl r : unitization R A) = ↑(r • a) | ext (zero_mul r) $ show (0 : R) • 0 + r • a + a * 0 = r • a,
by rw [smul_zero, zero_add, mul_zero, add_zero] | lemma | unitization.coe_mul_inl | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"distrib_mul_action",
"mul_zero",
"non_unital_non_assoc_semiring",
"semiring",
"smul_zero",
"unitization",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_class [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] :
mul_one_class (unitization R A) | { one_mul := λ x, ext (one_mul x.1) $ show (1 : R) • x.2 + x.1 • 0 + 0 * x.2 = x.2,
by rw [one_smul, smul_zero, add_zero, zero_mul, add_zero],
mul_one := λ x, ext (mul_one x.1) $ show (x.1 • 0 : A) + (1 : R) • x.2 + x.2 * 0 = x.2,
by rw [smul_zero, zero_add, one_smul, mul_zero, add_zero],
.. unitization.has... | instance | unitization.mul_one_class | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"distrib_mul_action",
"monoid",
"mul_one",
"mul_one_class",
"mul_zero",
"non_unital_non_assoc_semiring",
"one_mul",
"one_smul",
"smul_zero",
"unitization",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_ring_hom [semiring R] [non_unital_semiring A] [module R A] : R →+* unitization R A | { to_fun := inl,
map_one' := inl_one A,
map_mul' := inl_mul A,
map_zero' := inl_zero A,
map_add' := inl_add A } | def | unitization.inl_ring_hom | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"module",
"non_unital_semiring",
"semiring",
"unitization"
] | The canonical inclusion of rings `R →+* unitization R A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_star [has_star R] [has_star A] (x : unitization R A) :
(star x).fst = star x.fst | rfl | lemma | unitization.fst_star | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"has_star",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_star [has_star R] [has_star A] (x : unitization R A) :
(star x).snd = star x.snd | rfl | lemma | unitization.snd_star | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"has_star",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_star [has_star R] [add_monoid A] [star_add_monoid A] (r : R) :
inl (star r) = star (inl r : unitization R A) | ext rfl (by simp only [snd_star, star_zero, snd_inl]) | lemma | unitization.inl_star | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_monoid",
"has_star",
"star_add_monoid",
"star_zero",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_star [add_monoid R] [star_add_monoid R] [has_star A] (a : A) :
↑(star a) = star (a : unitization R A) | ext (by simp only [fst_star, star_zero, fst_coe]) rfl | lemma | unitization.coe_star | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"add_monoid",
"has_star",
"star_add_monoid",
"star_zero",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra : algebra S (unitization R A) | { commutes' := λ r x,
begin
induction x using unitization.ind,
simp only [mul_add, add_mul, ring_hom.to_fun_eq_coe, ring_hom.coe_comp, function.comp_app,
inl_ring_hom_apply, inl_mul_inl],
rw [inl_mul_coe, coe_mul_inl, mul_comm]
end,
smul_def' := λ s x,
begin
induction x using unitization.i... | instance | unitization.algebra | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"algebra",
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"mul_comm",
"ring_hom.coe_comp",
"ring_hom.to_fun_eq_coe",
"smul_add",
"smul_one_mul",
"smul_one_smul",
"unitization",
"unitization.ind",
"unitization.inl_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_inl_comp : ⇑(algebra_map S (unitization R A)) = inl ∘ algebra_map S R | rfl | lemma | unitization.algebra_map_eq_inl_comp | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"algebra_map",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_inl_ring_hom_comp :
algebra_map S (unitization R A) = (inl_ring_hom R A).comp (algebra_map S R) | rfl | lemma | unitization.algebra_map_eq_inl_ring_hom_comp | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"algebra_map",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_inl : ⇑(algebra_map R (unitization R A)) = inl | rfl | lemma | unitization.algebra_map_eq_inl | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"algebra_map",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_inl_hom : algebra_map R (unitization R A) = inl_ring_hom R A | rfl | lemma | unitization.algebra_map_eq_inl_hom | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"algebra_map",
"unitization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_hom : unitization R A →ₐ[R] R | { to_fun := fst,
map_one' := fst_one,
map_mul' := fst_mul,
map_zero' := fst_zero,
map_add' := fst_add,
commutes' := fst_inl A } | def | unitization.fst_hom | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"unitization"
] | The canonical `R`-algebra projection `unitization R A → R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_non_unital_alg_hom (R A : Type*) [comm_semiring R] [non_unital_semiring A] [module R A] :
A →ₙₐ[R] unitization R A | { to_fun := coe,
map_smul' := coe_smul R,
map_zero' := coe_zero R,
map_add' := coe_add R,
map_mul' := coe_mul R } | def | unitization.coe_non_unital_alg_hom | algebra.algebra | src/algebra/algebra/unitization.lean | [
"algebra.algebra.basic",
"linear_algebra.prod",
"algebra.hom.non_unital_alg"
] | [
"comm_semiring",
"module",
"non_unital_semiring",
"unitization"
] | The coercion from a non-unital `R`-algebra `A` to its unitization `unitization R A`
realized as a non-unital algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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