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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
map_smul (r : R) (x : A) : φ (r • x) = r • φ x | map_smul _ _ _ | lemma | alg_hom.map_smul | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sum {ι : Type*} (f : ι → A) (s : finset ι) :
φ (∑ x in s, f x) = ∑ x in s, φ (f x) | map_sum _ _ _ | lemma | alg_hom.map_sum | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_finsupp_sum {α : Type*} [has_zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :
φ (f.sum g) = f.sum (λ i a, φ (g i a)) | map_finsupp_sum _ _ _ | lemma | alg_hom.map_finsupp_sum | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bit0 (x) : φ (bit0 x) = bit0 (φ x) | map_bit0 _ _ | lemma | alg_hom.map_bit0 | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bit1 (x) : φ (bit1 x) = bit1 (φ x) | map_bit1 _ _ | lemma | alg_hom.map_bit1 | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_bit1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (f : A →+* B) (h : ∀ (c : R) x, f (c • x) = c • f x) : A →ₐ[R] B | { to_fun := f,
commutes' := λ c, by simp only [algebra.algebra_map_eq_smul_one, h, f.map_one],
.. f } | def | alg_hom.mk' | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra.algebra_map_eq_smul_one",
"mk'"
] | If a `ring_hom` is `R`-linear, then it is an `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk' (f : A →+* B) (h : ∀ (c : R) x, f (c • x) = c • f x) : ⇑(mk' f h) = f | rfl | lemma | alg_hom.coe_mk' | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : A →ₐ[R] A | { commutes' := λ _, rfl,
..ring_hom.id A } | def | alg_hom.id | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"ring_hom.id"
] | Identity map as an `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(alg_hom.id R A) = id | rfl | lemma | alg_hom.coe_id | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_to_ring_hom : (alg_hom.id R A : A →+* A) = ring_hom.id _ | rfl | lemma | alg_hom.id_to_ring_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (p : A) : alg_hom.id R A p = p | rfl | lemma | alg_hom.id_apply | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C | { commutes' := λ r : R, by rw [← φ₁.commutes, ← φ₂.commutes]; refl,
.. φ₁.to_ring_hom.comp ↑φ₂ } | def | alg_hom.comp | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | Composition of algebra homeomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ | rfl | lemma | alg_hom.coe_comp | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) | rfl | lemma | alg_hom.comp_apply | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_to_ring_hom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :
(φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ | rfl | lemma | alg_hom.comp_to_ring_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id : φ.comp (alg_hom.id R A) = φ | ext $ λ x, rfl | theorem | alg_hom.comp_id | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp : (alg_hom.id R B).comp φ = φ | ext $ λ x, rfl | theorem | alg_hom.id_comp | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :
(φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) | ext $ λ x, rfl | theorem | alg_hom.comp_assoc | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map : A →ₗ[R] B | { to_fun := φ,
map_add' := map_add _,
map_smul' := map_smul _ } | def | alg_hom.to_linear_map | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | R-Alg ⥤ R-Mod | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_map_apply (p : A) : φ.to_linear_map p = φ p | rfl | lemma | alg_hom.to_linear_map_apply | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_injective : function.injective (to_linear_map : _ → (A →ₗ[R] B)) | λ φ₁ φ₂ h, ext $ linear_map.congr_fun h | theorem | alg_hom.to_linear_map_injective | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"linear_map.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_to_linear_map (f : A →ₐ[R] B) (g : B →ₐ[R] C) :
(g.comp f).to_linear_map = g.to_linear_map.comp f.to_linear_map | rfl | lemma | alg_hom.comp_to_linear_map | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_id : to_linear_map (alg_hom.id R A) = linear_map.id | linear_map.ext $ λ _, rfl | lemma | alg_hom.to_linear_map_id | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.id",
"linear_map.ext",
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_linear_map (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :
A →ₐ[R] B | { to_fun := f,
map_one' := map_one,
map_mul' := map_mul,
commutes' := λ c, by simp only [algebra.algebra_map_eq_smul_one, f.map_smul, map_one],
.. f.to_add_monoid_hom } | def | alg_hom.of_linear_map | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra.algebra_map_eq_smul_one",
"map_mul",
"map_one"
] | Promote a `linear_map` to an `alg_hom` by supplying proofs about the behavior on `1` and `*`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_linear_map_to_linear_map (map_one) (map_mul) :
of_linear_map φ.to_linear_map map_one map_mul = φ | by { ext, refl } | lemma | alg_hom.of_linear_map_to_linear_map | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_mul",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_of_linear_map (f : A →ₗ[R] B) (map_one) (map_mul) :
to_linear_map (of_linear_map f map_one map_mul) = f | by { ext, refl } | lemma | alg_hom.to_linear_map_of_linear_map | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_mul",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_linear_map_id (map_one) (map_mul) :
of_linear_map linear_map.id map_one map_mul = alg_hom.id R A | ext $ λ _, rfl | lemma | alg_hom.of_linear_map_id | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.id",
"linear_map.id",
"map_mul",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul_of_tower {R'} [has_smul R' A] [has_smul R' B]
[linear_map.compatible_smul A B R' R] (r : R') (x : A) : φ (r • x) = r • φ x | φ.to_linear_map.map_smul_of_tower r x | lemma | alg_hom.map_smul_of_tower | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"has_smul",
"linear_map.compatible_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_list_prod (s : list A) :
φ s.prod = (s.map φ).prod | φ.to_ring_hom.map_list_prod s | lemma | alg_hom.map_list_prod | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_list_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End : monoid (A →ₐ[R] A) | { mul := comp,
mul_assoc := λ ϕ ψ χ, rfl,
one := alg_hom.id R A,
one_mul := λ ϕ, ext $ λ x, rfl,
mul_one := λ ϕ, ext $ λ x, rfl } | instance | alg_hom.End | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.id",
"monoid",
"mul_assoc",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (x : A) : (1 : A →ₐ[R] A) x = x | rfl | lemma | alg_hom.one_apply | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) | rfl | lemma | alg_hom.mul_apply | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebra_map R A y = x) :
algebra_map R B y = f x | h ▸ (f.commutes _).symm | lemma | alg_hom.algebra_map_eq_apply | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_multiset_prod (s : multiset A) :
φ s.prod = (s.map φ).prod | map_multiset_prod _ _ | lemma | alg_hom.map_multiset_prod | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_multiset_prod",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod {ι : Type*} (f : ι → A) (s : finset ι) :
φ (∏ x in s, f x) = ∏ x in s, φ (f x) | map_prod _ _ _ | lemma | alg_hom.map_prod | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"finset",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_finsupp_prod {α : Type*} [has_zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :
φ (f.prod g) = f.prod (λ i a, φ (g i a)) | map_finsupp_prod _ _ _ | lemma | alg_hom.map_finsupp_prod | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_finsupp_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (x) : φ (-x) = -φ x | map_neg _ _ | lemma | alg_hom.map_neg | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (x y) : φ (x - y) = φ x - φ y | map_sub _ _ _ | lemma | alg_hom.map_sub | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_alg_hom [semiring R] [semiring S] (f : R →+* S) :
R →ₐ[ℕ] S | { to_fun := f, commutes' := λ n, by simp, .. f } | def | ring_hom.to_nat_alg_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"semiring"
] | Reinterpret a `ring_hom` as an `ℕ`-algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_int_alg_hom [ring R] [ring S] [algebra ℤ R] [algebra ℤ S] (f : R →+* S) :
R →ₐ[ℤ] S | { commutes' := λ n, by simp, .. f } | def | ring_hom.to_int_alg_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra",
"ring"
] | Reinterpret a `ring_hom` as a `ℤ`-algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_rat_alg_hom [ring R] [ring S] [algebra ℚ R] [algebra ℚ S] (f : R →+* S) :
R →ₐ[ℚ] S | { commutes' := f.map_rat_algebra_map, .. f } | def | ring_hom.to_rat_alg_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra",
"ring"
] | Reinterpret a `ring_hom` as a `ℚ`-algebra homomorphism. This actually yields an equivalence,
see `ring_hom.equiv_rat_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_rat_alg_hom_to_ring_hom [ring R] [ring S] [algebra ℚ R] [algebra ℚ S]
(f : R →+* S) : ↑f.to_rat_alg_hom = f | ring_hom.ext $ λ x, rfl | lemma | ring_hom.to_rat_alg_hom_to_ring_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra",
"ring",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom.to_ring_hom_to_rat_alg_hom [ring R] [ring S] [algebra ℚ R] [algebra ℚ S]
(f : R →ₐ[ℚ] S) : (f : R →+* S).to_rat_alg_hom = f | alg_hom.ext $ λ x, rfl | lemma | alg_hom.to_ring_hom_to_rat_alg_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.ext",
"algebra",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.equiv_rat_alg_hom [ring R] [ring S] [algebra ℚ R] [algebra ℚ S] :
(R →+* S) ≃ (R →ₐ[ℚ] S) | { to_fun := ring_hom.to_rat_alg_hom,
inv_fun := alg_hom.to_ring_hom,
left_inv := ring_hom.to_rat_alg_hom_to_ring_hom,
right_inv := alg_hom.to_ring_hom_to_rat_alg_hom, } | def | ring_hom.equiv_rat_alg_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom.to_ring_hom_to_rat_alg_hom",
"algebra",
"inv_fun",
"ring",
"ring_hom.to_rat_alg_hom",
"ring_hom.to_rat_alg_hom_to_ring_hom"
] | The equivalence between `ring_hom` and `ℚ`-algebra homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_id : R →ₐ[R] A | { commutes' := λ _, rfl, .. algebra_map R A } | def | algebra.of_id | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra_map"
] | `algebra_map` as an `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_id_apply (r) : of_id R A r = algebra_map R A r | rfl | theorem | algebra.of_id_apply | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_alg_hom (m : M) : A →ₐ[R] A | { to_fun := λ a, m • a,
commutes' := smul_algebra_map _,
..mul_semiring_action.to_ring_hom _ _ m } | def | mul_semiring_action.to_alg_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"mul_semiring_action.to_ring_hom",
"smul_algebra_map"
] | Each element of the monoid defines a algebra homomorphism.
This is a stronger version of `mul_semiring_action.to_ring_hom` and
`distrib_mul_action.to_linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_alg_hom_injective [has_faithful_smul M A] :
function.injective (mul_semiring_action.to_alg_hom R A : M → A →ₐ[R] A) | λ m₁ m₂ h, eq_of_smul_eq_smul $ λ r, alg_hom.ext_iff.1 h r | theorem | mul_semiring_action.to_alg_hom_injective | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"has_faithful_smul",
"mul_semiring_action.to_alg_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_mem (r : R) : algebra_map R A r ∈ (1 : sub_mul_action R A) | ⟨r, (algebra_map_eq_smul_one r).symm⟩ | lemma | sub_mul_action.algebra_map_mem | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra_map",
"sub_mul_action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_one' {x : A} : x ∈ (1 : sub_mul_action R A) ↔ ∃ y, algebra_map R A y = x | exists_congr $ λ r, by rw algebra_map_eq_smul_one | lemma | sub_mul_action.mem_one' | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra_map",
"sub_mul_action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_range :
(1 : submodule R A) = (algebra.linear_map R A).range | rfl | theorem | submodule.one_eq_range | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra.linear_map",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_to_add_submonoid :
1 ≤ (1 : submodule R A).to_add_submonoid | begin
rintros x ⟨n, rfl⟩,
exact ⟨n, map_nat_cast (algebra_map R A) n⟩,
end | lemma | submodule.le_one_to_add_submonoid | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra_map",
"map_nat_cast",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_mem (r : R) : algebra_map R A r ∈ (1 : submodule R A) | linear_map.mem_range_self _ _ | lemma | submodule.algebra_map_mem | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra_map",
"linear_map.mem_range_self",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_one {x : A} : x ∈ (1 : submodule R A) ↔ ∃ y, algebra_map R A y = x | iff.rfl | lemma | submodule.mem_one | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra_map",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sub_mul_action_one : (1 : submodule R A).to_sub_mul_action = 1 | set_like.ext $ λ x, mem_one.trans sub_mul_action.mem_one'.symm | lemma | submodule.to_sub_mul_action_one | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"set_like.ext",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_span : (1 : submodule R A) = R ∙ 1 | begin
apply submodule.ext,
intro a,
simp only [mem_one, mem_span_singleton, algebra.smul_def, mul_one]
end | theorem | submodule.one_eq_span | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra.smul_def",
"mul_one",
"submodule",
"submodule.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_span_one_set : (1 : submodule R A) = span R 1 | one_eq_span | theorem | submodule.one_eq_span_one_set | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le : (1 : submodule R A) ≤ P ↔ (1 : A) ∈ P | by simpa only [one_eq_span, span_le, set.singleton_subset_iff] | theorem | submodule.one_le | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"one_le",
"set.singleton_subset_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one {A'} [semiring A'] [algebra R A'] (f : A →ₐ[R] A') :
map f.to_linear_map (1 : submodule R A) = 1 | by { ext, simp } | lemma | submodule.map_one | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra",
"map_one",
"semiring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_op_one :
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) (1 : submodule R A) = 1 | by { ext, induction x using mul_opposite.rec, simp } | lemma | submodule.map_op_one | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_opposite.rec",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_op_one :
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) (1 : submodule R Aᵐᵒᵖ) = 1 | by { ext, simp } | lemma | submodule.comap_op_one | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_unop_one :
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) (1 : submodule R Aᵐᵒᵖ) = 1 | by rw [←comap_equiv_eq_map_symm, comap_op_one] | lemma | submodule.map_unop_one | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_unop_one :
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) (1 : submodule R A) = 1 | by rw [←map_equiv_eq_comap_symm, map_op_one] | lemma | submodule.comap_unop_one | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_mul (hm : m ∈ M) (hn : n ∈ N) : m * n ∈ M * N | apply_mem_map₂ _ hm hn | theorem | submodule.mul_mem_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le : M * N ≤ P ↔ ∀ (m ∈ M) (n ∈ N), m * n ∈ P | map₂_le | theorem | submodule.mul_le | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_to_add_submonoid (M N : submodule R A) :
(M * N).to_add_submonoid = M.to_add_submonoid * N.to_add_submonoid | begin
dsimp [has_mul.mul],
simp_rw [←linear_map.mul_left_to_add_monoid_hom R, linear_map.mul_left, ←map_to_add_submonoid _ N,
map₂],
rw supr_to_add_submonoid,
refl,
end | lemma | submodule.mul_to_add_submonoid | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"linear_map.mul_left",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_induction_on
{C : A → Prop} {r : A} (hr : r ∈ M * N)
(hm : ∀ (m ∈ M) (n ∈ N), C (m * n)) (ha : ∀ x y, C x → C y → C (x + y)) : C r | begin
rw [←mem_to_add_submonoid, mul_to_add_submonoid] at hr,
exact add_submonoid.mul_induction_on hr hm ha,
end | theorem | submodule.mul_induction_on | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"add_submonoid.mul_induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_induction_on'
{C : Π r, r ∈ M * N → Prop}
(hm : ∀ (m ∈ M) (n ∈ N), C (m * n) (mul_mem_mul ‹_› ‹_›))
(ha : ∀ x hx y hy, C x hx → C y hy → C (x + y) (add_mem ‹_› ‹_›))
{r : A} (hr : r ∈ M * N) : C r hr | begin
refine exists.elim _ (λ (hr : r ∈ M * N) (hc : C r hr), hc),
exact submodule.mul_induction_on hr
(λ x hx y hy, ⟨_, hm _ hx _ hy⟩) (λ x y ⟨_, hx⟩ ⟨_, hy⟩, ⟨_, ha _ _ _ _ hx hy⟩),
end | theorem | submodule.mul_induction_on' | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule.mul_induction_on"
] | A dependent version of `mul_induction_on`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_mul_span : span R S * span R T = span R (S * T) | map₂_span_span _ _ _ _ | theorem | submodule.span_mul_span | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bot : M * ⊥ = ⊥ | map₂_bot_right _ _ | theorem | submodule.mul_bot | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_mul : ⊥ * M = ⊥ | map₂_bot_left _ _ | theorem | submodule.bot_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mul : (1 : submodule R A) * M = M | by { conv_lhs { rw [one_eq_span, ← span_eq M] }, erw [span_mul_span, one_mul, span_eq] } | theorem | submodule.one_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"one_mul",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one : M * 1 = M | by { conv_lhs { rw [one_eq_span, ← span_eq M] }, erw [span_mul_span, mul_one, span_eq] } | theorem | submodule.mul_one | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul (hmp : M ≤ P) (hnq : N ≤ Q) : M * N ≤ P * Q | map₂_le_map₂ hmp hnq | theorem | submodule.mul_le_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_left (h : M ≤ N) : M * P ≤ N * P | map₂_le_map₂_left h | theorem | submodule.mul_le_mul_left | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_le_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_right (h : N ≤ P) : M * N ≤ M * P | map₂_le_map₂_right h | theorem | submodule.mul_le_mul_right | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sup : M * (N ⊔ P) = M * N ⊔ M * P | map₂_sup_right _ _ _ _ | theorem | submodule.mul_sup | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mul : (M ⊔ N) * P = M * P ⊔ N * P | map₂_sup_left _ _ _ _ | theorem | submodule.sup_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"sup_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_subset_mul : (↑M : set A) * (↑N : set A) ⊆ (↑(M * N) : set A) | image2_subset_map₂ (algebra.lmul R A).to_linear_map M N | lemma | submodule.mul_subset_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra.lmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul {A'} [semiring A'] [algebra R A'] (f : A →ₐ[R] A') :
map f.to_linear_map (M * N) = map f.to_linear_map M * map f.to_linear_map N | calc map f.to_linear_map (M * N)
= ⨆ (i : M), (N.map (linear_map.mul R A i)).map f.to_linear_map : map_supr _ _
... = map f.to_linear_map M * map f.to_linear_map N :
begin
apply congr_arg Sup,
ext S,
split; rintros ⟨y, hy⟩,
{ use [f y, mem_map.mpr ⟨y.1, y.2, rfl⟩],
refine trans _ hy,
... | lemma | submodule.map_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra",
"linear_map.mul",
"map_mul",
"map_supr",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_op_mul :
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) (M * N) =
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) N *
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) M | begin
apply le_antisymm,
{ simp_rw map_le_iff_le_comap,
refine mul_le.2 (λ m hm n hn, _),
rw [mem_comap, map_equiv_eq_comap_symm, map_equiv_eq_comap_symm],
show op n * op m ∈ _,
exact mul_mem_mul hn hm },
{ refine mul_le.2 (mul_opposite.rec $ λ m hm, mul_opposite.rec $ λ n hn, _),
rw submodule... | lemma | submodule.map_op_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_opposite.rec",
"submodule.mem_map_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_unop_mul :
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) (M * N) =
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) N *
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) M | by simp_rw [←map_equiv_eq_comap_symm, map_op_mul] | lemma | submodule.comap_unop_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_unop_mul (M N : submodule R Aᵐᵒᵖ) :
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) (M * N) =
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) N *
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) M | have function.injective (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) :=
linear_equiv.injective _,
map_injective_of_injective this $
by rw [← map_comp, map_op_mul, ←map_comp, ←map_comp, linear_equiv.comp_coe,
linear_equiv.symm_trans_self, linear_equiv.refl_to_linear_map, map_id, map_id, map_id] | lemma | submodule.map_unop_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"linear_equiv.comp_coe",
"linear_equiv.injective",
"linear_equiv.refl_to_linear_map",
"linear_equiv.symm_trans_self",
"map_comp",
"map_id",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_op_mul (M N : submodule R Aᵐᵒᵖ) :
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) (M * N) =
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) N *
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) M | by simp_rw [comap_equiv_eq_map_symm, map_unop_mul] | lemma | submodule.comap_op_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_distrib_pointwise_neg {A} [ring A] [algebra R A] :
has_distrib_neg (submodule R A) | to_add_submonoid_injective.has_distrib_neg _ neg_to_add_submonoid mul_to_add_submonoid | def | submodule.has_distrib_pointwise_neg | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra",
"has_distrib_neg",
"ring",
"submodule"
] | `submodule.has_pointwise_neg` distributes over multiplication.
This is available as an instance in the `pointwise` locale. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_span_mul_finite_of_mem_span_mul
{R A} [semiring R] [add_comm_monoid A] [has_mul A] [module R A]
{S : set A} {S' : set A} {x : A} (hx : x ∈ span R (S * S')) :
∃ (T T' : finset A), ↑T ⊆ S ∧ ↑T' ⊆ S' ∧ x ∈ span R (T * T' : set A) | begin
obtain ⟨U, h, hU⟩ := mem_span_finite_of_mem_span hx,
obtain ⟨T, T', hS, hS', h⟩ := finset.subset_mul h,
use [T, T', hS, hS'],
have h' : (U : set A) ⊆ T * T', { assumption_mod_cast, },
have h'' := span_mono h' hU,
assumption,
end | lemma | submodule.mem_span_mul_finite_of_mem_span_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"add_comm_monoid",
"finset",
"finset.subset_mul",
"module",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_span_mul_set (s t : submodule R A) : s * t = span R ((s : set A) * (t : set A)) | map₂_eq_span_image2 _ s t | lemma | submodule.mul_eq_span_mul_set | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_mul (s : ι → submodule R A) (t : submodule R A) : (⨆ i, s i) * t = ⨆ i, s i * t | map₂_supr_left _ s t | lemma | submodule.supr_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_supr (t : submodule R A) (s : ι → submodule R A) : t * (⨆ i, s i) = ⨆ i, t * s i | map₂_supr_right _ t s | lemma | submodule.mul_supr | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_mul_finite_of_mem_mul {P Q : submodule R A} {x : A} (hx : x ∈ P * Q) :
∃ (T T' : finset A), (T : set A) ⊆ P ∧ (T' : set A) ⊆ Q ∧ x ∈ span R (T * T' : set A) | submodule.mem_span_mul_finite_of_mem_span_mul
(by rwa [← submodule.span_eq P, ← submodule.span_eq Q, submodule.span_mul_span] at hx) | lemma | submodule.mem_span_mul_finite_of_mem_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"finset",
"submodule",
"submodule.mem_span_mul_finite_of_mem_span_mul",
"submodule.span_eq",
"submodule.span_mul_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_singleton_mul {x y : A} : x ∈ span R {y} * P ↔ ∃ z ∈ P, y * z = x | by { simp_rw [(*), map₂_span_singleton_eq_map, exists_prop], refl } | lemma | submodule.mem_span_singleton_mul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mul_span_singleton {x y : A} : x ∈ P * span R {y} ↔ ∃ z ∈ P, z * y = x | by { simp_rw [(*), map₂_span_singleton_eq_map_flip, exists_prop], refl } | lemma | submodule.mem_mul_span_singleton | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_pow (s : set A) : ∀ n : ℕ, span R s ^ n = span R (s ^ n) | | 0 := by rw [pow_zero, pow_zero, one_eq_span_one_set]
| (n + 1) := by rw [pow_succ, pow_succ, span_pow, span_mul_span] | lemma | submodule.span_pow | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_eq_span_pow_set (n : ℕ) : M ^ n = span R ((M : set A) ^ n) | by rw [←span_pow, span_eq] | lemma | submodule.pow_eq_span_pow_set | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_subset_pow {n : ℕ} : (↑M : set A)^n ⊆ ↑(M^n : submodule R A) | (pow_eq_span_pow_set M n).symm ▸ subset_span | lemma | submodule.pow_subset_pow | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mem_pow {x : A} (hx : x ∈ M) (n : ℕ) : x ^ n ∈ M ^ n | pow_subset_pow _ $ set.pow_mem_pow hx _ | lemma | submodule.pow_mem_pow | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"set.pow_mem_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_to_add_submonoid {n : ℕ} (h : n ≠ 0) :
(M ^ n).to_add_submonoid = M.to_add_submonoid ^ n | begin
induction n with n ih,
{ exact (h rfl).elim },
{ rw [pow_succ, pow_succ, mul_to_add_submonoid],
cases n,
{ rw [pow_zero, pow_zero, mul_one, ←mul_to_add_submonoid, mul_one] },
{ rw ih n.succ_ne_zero } },
end | lemma | submodule.pow_to_add_submonoid | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"ih",
"mul_one",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_pow_to_add_submonoid {n : ℕ} :
M.to_add_submonoid ^ n ≤ (M ^ n).to_add_submonoid | begin
obtain rfl | hn := decidable.eq_or_ne n 0,
{ rw [pow_zero, pow_zero], exact le_one_to_add_submonoid },
{ exact (pow_to_add_submonoid M hn).ge }
end | lemma | submodule.le_pow_to_add_submonoid | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"decidable.eq_or_ne",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_induction_on_left'
{C : Π (n : ℕ) x, x ∈ M ^ n → Prop}
(hr : ∀ r : R, C 0 (algebra_map _ _ r) (algebra_map_mem r))
(hadd : ∀ x y i hx hy, C i x hx → C i y hy → C i (x + y) (add_mem ‹_› ‹_›))
(hmul : ∀ (m ∈ M) i x hx, C i x hx → C (i.succ) (m * x) (mul_mem_mul H hx))
{x : A} {n : ℕ} (hx : x ∈ M ^ n) : C n ... | begin
induction n with n n_ih generalizing x,
{ rw pow_zero at hx,
obtain ⟨r, rfl⟩ := hx,
exact hr r, },
exact submodule.mul_induction_on'
(λ m hm x ih, hmul _ hm _ _ _ (n_ih ih))
(λ x hx y hy Cx Cy, hadd _ _ _ _ _ Cx Cy) hx,
end | theorem | submodule.pow_induction_on_left' | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra_map",
"ih",
"pow_zero",
"submodule.mul_induction_on'"
] | Dependent version of `submodule.pow_induction_on_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_induction_on_right'
{C : Π (n : ℕ) x, x ∈ M ^ n → Prop}
(hr : ∀ r : R, C 0 (algebra_map _ _ r) (algebra_map_mem r))
(hadd : ∀ x y i hx hy, C i x hx → C i y hy → C i (x + y) (add_mem ‹_› ‹_›))
(hmul : ∀ i x hx, C i x hx → ∀ m ∈ M,
C (i.succ) (x * m) ((pow_succ' M i).symm ▸ mul_mem_mul hx H))
{x : A} {n... | begin
induction n with n n_ih generalizing x,
{ rw pow_zero at hx,
obtain ⟨r, rfl⟩ := hx,
exact hr r, },
revert hx,
simp_rw pow_succ',
intro hx,
exact submodule.mul_induction_on'
(λ m hm x ih, hmul _ _ hm (n_ih _) _ ih)
(λ x hx y hy Cx Cy, hadd _ _ _ _ _ Cx Cy) hx,
end | theorem | submodule.pow_induction_on_right' | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra_map",
"ih",
"pow_succ'",
"pow_zero",
"submodule.mul_induction_on'"
] | Dependent version of `submodule.pow_induction_on_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.