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 |
|---|---|---|---|---|---|---|---|---|---|---|
lmul' : S ⊗[R] S →ₐ[R] S | alg_hom_of_linear_map_tensor_product (linear_map.mul' R S)
(λ a₁ a₂ b₁ b₂, by simp only [linear_map.mul'_apply, mul_mul_mul_comm])
(λ r, by simp only [linear_map.mul'_apply, _root_.mul_one]) | def | algebra.tensor_product.lmul' | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"linear_map.mul'",
"linear_map.mul'_apply",
"mul_mul_mul_comm"
] | `linear_map.mul'` is an alg_hom on commutative rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lmul'_to_linear_map : (lmul' R : _ →ₐ[R] S).to_linear_map = linear_map.mul' R S | rfl | lemma | algebra.tensor_product.lmul'_to_linear_map | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"linear_map.mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lmul'_apply_tmul (a b : S) : lmul' R (a ⊗ₜ[R] b) = a * b | rfl | lemma | algebra.tensor_product.lmul'_apply_tmul | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lmul'_comp_include_left : (lmul' R : _ →ₐ[R] S).comp include_left = alg_hom.id R S | alg_hom.ext $ _root_.mul_one | lemma | algebra.tensor_product.lmul'_comp_include_left | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"alg_hom.ext",
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lmul'_comp_include_right : (lmul' R : _ →ₐ[R] S).comp include_right = alg_hom.id R S | alg_hom.ext $ _root_.one_mul | lemma | algebra.tensor_product.lmul'_comp_include_right | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"alg_hom.ext",
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_map : A ⊗[R] B →ₐ[R] S | (lmul' R).comp (tensor_product.map f g) | def | algebra.tensor_product.product_map | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"tensor_product.map"
] | If `S` is commutative, for a pair of morphisms `f : A →ₐ[R] S`, `g : B →ₐ[R] S`,
We obtain a map `A ⊗[R] B →ₐ[R] S` that commutes with `f`, `g` via `a ⊗ b ↦ f(a) * g(b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_map_apply_tmul (a : A) (b : B) : product_map f g (a ⊗ₜ b) = f a * g b | by { unfold product_map lmul', simp } | lemma | algebra.tensor_product.product_map_apply_tmul | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_map_left_apply (a : A) :
product_map f g ((include_left : A →ₐ[R] A ⊗ B) a) = f a | by simp | lemma | algebra.tensor_product.product_map_left_apply | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_map_left : (product_map f g).comp include_left = f | alg_hom.ext $ by simp | lemma | algebra.tensor_product.product_map_left | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"alg_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_map_right_apply (b : B) : product_map f g (include_right b) = g b | by simp | lemma | algebra.tensor_product.product_map_right_apply | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_map_right : (product_map f g).comp include_right = g | alg_hom.ext $ by simp | lemma | algebra.tensor_product.product_map_right | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"alg_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_map_range : (product_map f g).range = f.range ⊔ g.range | by rw [product_map, alg_hom.range_comp, map_range, map_sup, ←alg_hom.range_comp,
←alg_hom.range_comp, ←alg_hom.comp_assoc, ←alg_hom.comp_assoc, lmul'_comp_include_left,
lmul'_comp_include_right, alg_hom.id_comp, alg_hom.id_comp] | lemma | algebra.tensor_product.product_map_range | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"alg_hom.id_comp",
"alg_hom.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_left_alg_hom (f : A' →ₐ[A] S) (g : B →ₐ[R] S) : A' ⊗[R] B →ₐ[A] S | { commutes' := λ r, by { dsimp, simp },
..(product_map (f.restrict_scalars R) g).to_ring_hom } | def | algebra.tensor_product.product_left_alg_hom | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [] | If `A`, `B` are `R`-algebras, `A'` is an `A`-algebra, then the product map of `f : A' →ₐ[A] S`
and `g : B →ₐ[R] S` is an `A`-algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_aux : R ⊗[k] M ≃ₗ[k] (ι →₀ R) | (_root_.tensor_product.congr (finsupp.linear_equiv.finsupp_unique k R punit).symm b.repr) ≪≫ₗ
(finsupp_tensor_finsupp k R k punit ι).trans (finsupp.lcongr (equiv.unique_prod ι punit)
(_root_.tensor_product.rid k R)) | def | algebra.tensor_product.basis_aux | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"equiv.unique_prod",
"finsupp.lcongr",
"finsupp.linear_equiv.finsupp_unique",
"finsupp_tensor_finsupp"
] | Given a `k`-algebra `R` and a `k`-basis of `M,` this is a `k`-linear isomorphism
`R ⊗[k] M ≃ (ι →₀ R)` (which is in fact `R`-linear). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_aux_tmul (r : R) (m : M) :
basis_aux R b (r ⊗ₜ m) = r • (finsupp.map_range (algebra_map k R)
(map_zero _) (b.repr m)) | begin
ext,
simp [basis_aux, ←algebra.commutes, algebra.smul_def],
end | lemma | algebra.tensor_product.basis_aux_tmul | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"algebra.smul_def",
"algebra_map",
"finsupp.map_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_aux_map_smul (r : R) (x : R ⊗[k] M) :
basis_aux R b (r • x) = r • basis_aux R b x | tensor_product.induction_on x (by simp) (λ x y, by simp only [tensor_product.smul_tmul',
basis_aux_tmul, smul_assoc]) (λ x y hx hy, by simp [hx, hy]) | lemma | algebra.tensor_product.basis_aux_map_smul | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"smul_assoc",
"tensor_product.induction_on",
"tensor_product.smul_tmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis : basis ι R (R ⊗[k] M) | { repr := { map_smul' := basis_aux_map_smul b, .. basis_aux R b } } | def | algebra.tensor_product.basis | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"basis"
] | Given a `k`-algebra `R`, this is the `R`-basis of `R ⊗[k] M` induced by a `k`-basis of `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_repr_tmul (r : R) (m : M) :
(basis R b).repr (r ⊗ₜ m) = r • (finsupp.map_range (algebra_map k R) (map_zero _) (b.repr m)) | basis_aux_tmul _ _ _ | lemma | algebra.tensor_product.basis_repr_tmul | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"algebra_map",
"basis",
"finsupp.map_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_repr_symm_apply (r : R) (i : ι) :
(basis R b).repr.symm (finsupp.single i r) = r ⊗ₜ b.repr.symm (finsupp.single i 1) | by simp [basis, equiv.unique_prod_symm_apply, basis_aux] | lemma | algebra.tensor_product.basis_repr_symm_apply | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"basis",
"equiv.unique_prod_symm_apply",
"finsupp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End_tensor_End_alg_hom : (End R M) ⊗[R] (End R N) →ₐ[R] End R (M ⊗[R] N) | begin
refine algebra.tensor_product.alg_hom_of_linear_map_tensor_product
(hom_tensor_hom_map R M N M N) _ _,
{ intros f₁ f₂ g₁ g₂,
simp only [hom_tensor_hom_map_apply, tensor_product.map_mul] },
{ intro r,
simp only [hom_tensor_hom_map_apply],
ext m n, simp [smul_tmul] }
end | def | module.End_tensor_End_alg_hom | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"algebra.tensor_product.alg_hom_of_linear_map_tensor_product",
"tensor_product.map_mul"
] | The algebra homomorphism from `End M ⊗ End N` to `End (M ⊗ N)` sending `f ⊗ₜ g` to
the `tensor_product.map f g`, the tensor product of the two maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
End_tensor_End_alg_hom_apply (f : End R M) (g : End R N) :
End_tensor_End_alg_hom (f ⊗ₜ[R] g) = tensor_product.map f g | by simp only [End_tensor_End_alg_hom,
algebra.tensor_product.alg_hom_of_linear_map_tensor_product_apply, hom_tensor_hom_map_apply] | lemma | module.End_tensor_End_alg_hom_apply | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"algebra.tensor_product.alg_hom_of_linear_map_tensor_product_apply",
"tensor_product.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.finite_dimensional_sup {K L : Type*} [field K] [comm_ring L] [algebra K L]
(E1 E2 : subalgebra K L) [finite_dimensional K E1] [finite_dimensional K E2] :
finite_dimensional K ↥(E1 ⊔ E2) | begin
rw [←E1.range_val, ←E2.range_val, ←algebra.tensor_product.product_map_range],
exact (algebra.tensor_product.product_map E1.val E2.val).to_linear_map.finite_dimensional_range,
end | lemma | subalgebra.finite_dimensional_sup | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"algebra",
"algebra.tensor_product.product_map",
"comm_ring",
"field",
"finite_dimensional",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_aux : A ⊗[R] B →ₗ[R] M →ₗ[R] M | tensor_product.lift
{ to_fun := λ a, a • (algebra.lsmul R M : B →ₐ[R] module.End R M).to_linear_map,
map_add' := λ r t, by { ext, simp only [add_smul, linear_map.add_apply] },
map_smul' := λ n r, by { ext, simp only [ring_hom.id_apply, linear_map.smul_apply, smul_assoc] } } | def | tensor_product.algebra.module_aux | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"add_smul",
"algebra.lsmul",
"linear_map.add_apply",
"linear_map.smul_apply",
"module.End",
"ring_hom.id_apply",
"smul_assoc",
"tensor_product.lift"
] | An auxiliary definition, used for constructing the `module (A ⊗[R] B) M` in
`tensor_product.algebra.module` below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module_aux_apply (a : A) (b : B) (m : M) :
module_aux (a ⊗ₜ[R] b) m = a • b • m | rfl | lemma | tensor_product.algebra.module_aux_apply | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module : module (A ⊗[R] B) M | { smul := λ x m, module_aux x m,
zero_smul := λ m, by simp only [map_zero, linear_map.zero_apply],
smul_zero := λ x, by simp only [map_zero],
smul_add := λ x m₁ m₂, by simp only [map_add],
add_smul := λ x y m, by simp only [map_add, linear_map.add_apply],
one_smul := λ m, by simp only [module_aux_apply, algeb... | def | tensor_product.algebra.module | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [
"add_smul",
"algebra.tensor_product.one_def",
"algebra.tensor_product.tmul_mul_tmul",
"linear_map.add_apply",
"linear_map.mul_apply",
"linear_map.zero_apply",
"module",
"mul_zero",
"one_smul",
"smul_add",
"smul_zero",
"tensor_product.induction_on",
"zero_mul",
"zero_smul"
] | If `M` is a representation of two different `R`-algebras `A` and `B` whose actions commute,
then it is a representation the `R`-algebra `A ⊗[R] B`.
An important example arises from a semiring `S`; allowing `S` to act on itself via left and right
multiplication, the roles of `R`, `A`, `B`, `M` are played by `ℕ`, `S`, `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def (a : A) (b : B) (m : M) : (a ⊗ₜ[R] b) • m = a • b • m | rfl | lemma | tensor_product.algebra.smul_def | ring_theory | src/ring_theory/tensor_product.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.adjoin.basic",
"linear_algebra.direct_sum.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace : S →ₗ[R] R | (linear_map.trace R S).comp (lmul R S).to_linear_map | def | algebra.trace | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"linear_map.trace"
] | The trace of an element `s` of an `R`-algebra is the trace of `(*) s`,
as an `R`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trace_apply (x) : trace R S x = linear_map.trace R S (lmul R S x) | rfl | lemma | algebra.trace_apply | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"linear_map.trace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_eq_zero_of_not_exists_basis
(h : ¬ ∃ (s : finset S), nonempty (basis s R S)) : trace R S = 0 | by { ext s, simp [trace_apply, linear_map.trace, h] } | lemma | algebra.trace_eq_zero_of_not_exists_basis | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"basis",
"finset",
"linear_map.trace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_eq_matrix_trace [decidable_eq ι] (b : basis ι R S) (s : S) :
trace R S s = matrix.trace (algebra.left_mul_matrix b s) | by { rw [trace_apply, linear_map.trace_eq_matrix_trace _ b, ←to_matrix_lmul_eq], refl } | lemma | algebra.trace_eq_matrix_trace | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra.left_mul_matrix",
"basis",
"linear_map.trace_eq_matrix_trace",
"matrix.trace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_algebra_map_of_basis (x : R) :
trace R S (algebra_map R S x) = fintype.card ι • x | begin
haveI := classical.dec_eq ι,
rw [trace_apply, linear_map.trace_eq_matrix_trace R b, matrix.trace],
convert finset.sum_const _,
ext i,
simp [-coe_lmul_eq_mul],
end | lemma | algebra.trace_algebra_map_of_basis | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra_map",
"classical.dec_eq",
"fintype.card",
"linear_map.trace_eq_matrix_trace",
"matrix.trace"
] | If `x` is in the base field `K`, then the trace is `[L : K] * x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trace_algebra_map (x : K) : trace K L (algebra_map K L x) = finrank K L • x | begin
by_cases H : ∃ (s : finset L), nonempty (basis s K L),
{ rw [trace_algebra_map_of_basis H.some_spec.some, finrank_eq_card_basis H.some_spec.some] },
{ simp [trace_eq_zero_of_not_exists_basis K H, finrank_eq_zero_of_not_exists_basis_finset H] }
end | lemma | algebra.trace_algebra_map | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra_map",
"basis",
"finrank_eq_zero_of_not_exists_basis_finset",
"finset"
] | If `x` is in the base field `K`, then the trace is `[L : K] * x`.
(If `L` is not finite-dimensional over `K`, then `trace` and `finrank` return `0`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trace_trace_of_basis [algebra S T] [is_scalar_tower R S T] {ι κ : Type*} [finite ι] [finite κ]
(b : basis ι R S) (c : basis κ S T) (x : T) :
trace R S (trace S T x) = trace R T x | begin
haveI := classical.dec_eq ι,
haveI := classical.dec_eq κ,
casesI nonempty_fintype ι,
casesI nonempty_fintype κ,
rw [trace_eq_matrix_trace (b.smul c), trace_eq_matrix_trace b, trace_eq_matrix_trace c,
matrix.trace, matrix.trace, matrix.trace,
← finset.univ_product_univ, finset.sum_product],
... | lemma | algebra.trace_trace_of_basis | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"alg_hom.map_sum",
"algebra",
"basis",
"classical.dec_eq",
"finite",
"finset.univ_product_univ",
"is_scalar_tower",
"matrix.diag",
"matrix.trace",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_comp_trace_of_basis [algebra S T] [is_scalar_tower R S T] {ι κ : Type*} [finite ι]
[fintype κ] (b : basis ι R S) (c : basis κ S T) :
(trace R S).comp ((trace S T).restrict_scalars R) = trace R T | by { ext, rw [linear_map.comp_apply, linear_map.restrict_scalars_apply, trace_trace_of_basis b c] } | lemma | algebra.trace_comp_trace_of_basis | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra",
"basis",
"finite",
"fintype",
"is_scalar_tower",
"linear_map.comp_apply",
"linear_map.restrict_scalars_apply",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_trace [algebra K T] [algebra L T] [is_scalar_tower K L T]
[finite_dimensional K L] [finite_dimensional L T] (x : T) :
trace K L (trace L T x) = trace K T x | trace_trace_of_basis (basis.of_vector_space K L) (basis.of_vector_space L T) x | lemma | algebra.trace_trace | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra",
"basis.of_vector_space",
"finite_dimensional",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_comp_trace [algebra K T] [algebra L T] [is_scalar_tower K L T]
[finite_dimensional K L] [finite_dimensional L T] :
(trace K L).comp ((trace L T).restrict_scalars K) = trace K T | by { ext, rw [linear_map.comp_apply, linear_map.restrict_scalars_apply, trace_trace] } | lemma | algebra.trace_comp_trace | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra",
"finite_dimensional",
"is_scalar_tower",
"linear_map.comp_apply",
"linear_map.restrict_scalars_apply",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_prod_apply
[module.free R S] [module.free R T] [module.finite R S] [module.finite R T]
(x : S × T) : trace R (S × T) x = trace R S x.fst + trace R T x.snd | begin
nontriviality R,
let f := (lmul R S).to_linear_map.prod_map (lmul R T).to_linear_map,
have : (lmul R (S × T)).to_linear_map = (prod_map_linear R S T S T R).comp f :=
linear_map.ext₂ prod.mul_def,
simp_rw [trace, this],
exact trace_prod_map' _ _,
end | lemma | algebra.trace_prod_apply | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"linear_map.ext₂",
"module.finite",
"module.free",
"prod.mul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_prod
[module.free R S] [module.free R T] [module.finite R S] [module.finite R T] :
trace R (S × T) = (trace R S).coprod (trace R T) | linear_map.ext $ λ p, by rw [coprod_apply, trace_prod_apply] | lemma | algebra.trace_prod | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"linear_map.ext",
"module.finite",
"module.free"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_form : bilin_form R S | (linear_map.compr₂ (lmul R S).to_linear_map (trace R S)).to_bilin | def | algebra.trace_form | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"bilin_form",
"linear_map.compr₂"
] | The `trace_form` maps `x y : S` to the trace of `x * y`.
It is a symmetric bilinear form and is nondegenerate if the extension is separable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trace_form_apply (x y : S) : trace_form R S x y = trace R S (x * y) | rfl | lemma | algebra.trace_form_apply | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_form_is_symm : (trace_form R S).is_symm | λ x y, congr_arg (trace R S) (mul_comm _ _) | lemma | algebra.trace_form_is_symm | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_form_to_matrix [decidable_eq ι] (i j) :
bilin_form.to_matrix b (trace_form R S) i j = trace R S (b i * b j) | by rw [bilin_form.to_matrix_apply, trace_form_apply] | lemma | algebra.trace_form_to_matrix | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"bilin_form.to_matrix",
"bilin_form.to_matrix_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_form_to_matrix_power_basis (h : power_basis R S) :
bilin_form.to_matrix h.basis (trace_form R S) = of (λ i j, trace R S (h.gen ^ (↑i + ↑j : ℕ))) | by { ext, rw [trace_form_to_matrix, of_apply, pow_add, h.basis_eq_pow, h.basis_eq_pow] } | lemma | algebra.trace_form_to_matrix_power_basis | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"bilin_form.to_matrix",
"pow_add",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_basis.trace_gen_eq_next_coeff_minpoly [nontrivial S] (pb : power_basis K S) :
algebra.trace K S pb.gen = -(minpoly K pb.gen).next_coeff | begin
have d_pos : 0 < pb.dim := power_basis.dim_pos pb,
have d_pos' : 0 < (minpoly K pb.gen).nat_degree, { simpa },
haveI : nonempty (fin pb.dim) := ⟨⟨0, d_pos⟩⟩,
rw [trace_eq_matrix_trace pb.basis, trace_eq_neg_charpoly_coeff, charpoly_left_mul_matrix,
← pb.nat_degree_minpoly, fintype.card_fin, ← next_c... | lemma | power_basis.trace_gen_eq_next_coeff_minpoly | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra.trace",
"charpoly_left_mul_matrix",
"fintype.card_fin",
"minpoly",
"nontrivial",
"power_basis",
"power_basis.dim_pos"
] | Given `pb : power_basis K S`, the trace of `pb.gen` is `-(minpoly K pb.gen).next_coeff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_basis.trace_gen_eq_sum_roots [nontrivial S] (pb : power_basis K S)
(hf : (minpoly K pb.gen).splits (algebra_map K F)) :
algebra_map K F (trace K S pb.gen) =
((minpoly K pb.gen).map (algebra_map K F)).roots.sum | begin
rw [power_basis.trace_gen_eq_next_coeff_minpoly, ring_hom.map_neg, ← next_coeff_map
(algebra_map K F).injective, sum_roots_eq_next_coeff_of_monic_of_split
((minpoly.monic (power_basis.is_integral_gen _)).map _)
((splits_id_iff_splits _).2 hf), neg_neg]
end | lemma | power_basis.trace_gen_eq_sum_roots | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra_map",
"minpoly",
"minpoly.monic",
"nontrivial",
"power_basis",
"power_basis.is_integral_gen",
"power_basis.trace_gen_eq_next_coeff_minpoly",
"ring_hom.map_neg"
] | Given `pb : power_basis K S`, then the trace of `pb.gen` is
`((minpoly K pb.gen).map (algebra_map K F)).roots.sum`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trace_gen_eq_zero {x : L} (hx : ¬ is_integral K x) :
algebra.trace K K⟮x⟯ (adjoin_simple.gen K x) = 0 | begin
rw [trace_eq_zero_of_not_exists_basis, linear_map.zero_apply],
contrapose! hx,
obtain ⟨s, ⟨b⟩⟩ := hx,
refine is_integral_of_mem_of_fg (K⟮x⟯).to_subalgebra _ x _,
{ exact (submodule.fg_iff_finite_dimensional _).mpr (finite_dimensional.of_fintype_basis b) },
{ exact subset_adjoin K _ (set.mem_singleton ... | lemma | intermediate_field.adjoin_simple.trace_gen_eq_zero | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra.trace",
"finite_dimensional.of_fintype_basis",
"is_integral",
"is_integral_of_mem_of_fg",
"linear_map.zero_apply",
"set.mem_singleton",
"submodule.fg_iff_finite_dimensional"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_gen_eq_sum_roots (x : L)
(hf : (minpoly K x).splits (algebra_map K F)) :
algebra_map K F (trace K K⟮x⟯ (adjoin_simple.gen K x)) =
((minpoly K x).map (algebra_map K F)).roots.sum | begin
have injKxL := (algebra_map K⟮x⟯ L).injective,
by_cases hx : is_integral K x, swap,
{ simp [minpoly.eq_zero hx, trace_gen_eq_zero hx], },
have hx' : is_integral K (adjoin_simple.gen K x),
{ rwa [← is_integral_algebra_map_iff injKxL, adjoin_simple.algebra_map_gen],
apply_instance },
rw [← adjoin.po... | lemma | intermediate_field.adjoin_simple.trace_gen_eq_sum_roots | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra_map",
"is_integral",
"is_integral_algebra_map_iff",
"minpoly",
"minpoly.eq_of_algebra_map_eq",
"minpoly.eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_eq_trace_adjoin [finite_dimensional K L] (x : L) :
algebra.trace K L x = finrank K⟮x⟯ L • trace K K⟮x⟯ (adjoin_simple.gen K x) | begin
rw ← @trace_trace _ _ K K⟮x⟯ _ _ _ _ _ _ _ _ x,
conv in x { rw ← intermediate_field.adjoin_simple.algebra_map_gen K x },
rw [trace_algebra_map, linear_map.map_smul_of_tower],
end | lemma | trace_eq_trace_adjoin | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra.trace",
"finite_dimensional",
"intermediate_field.adjoin_simple.algebra_map_gen",
"linear_map.map_smul_of_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_eq_sum_roots [finite_dimensional K L]
{x : L} (hF : (minpoly K x).splits (algebra_map K F)) :
algebra_map K F (algebra.trace K L x) =
finrank K⟮x⟯ L • ((minpoly K x).map (algebra_map K _)).roots.sum | by rw [trace_eq_trace_adjoin K x, algebra.smul_def, ring_hom.map_mul, ← algebra.smul_def,
intermediate_field.adjoin_simple.trace_gen_eq_sum_roots _ hF, is_scalar_tower.algebra_map_smul] | lemma | trace_eq_sum_roots | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra.smul_def",
"algebra.trace",
"algebra_map",
"finite_dimensional",
"intermediate_field.adjoin_simple.trace_gen_eq_sum_roots",
"is_scalar_tower.algebra_map_smul",
"minpoly",
"ring_hom.map_mul",
"trace_eq_trace_adjoin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.is_integral_trace [finite_dimensional L F] {x : F} (hx : is_integral R x) :
is_integral R (algebra.trace L F x) | begin
have hx' : is_integral L x := is_integral_of_is_scalar_tower hx,
rw [← is_integral_algebra_map_iff (algebra_map L (algebraic_closure F)).injective,
trace_eq_sum_roots],
{ refine (is_integral.multiset_sum _).nsmul _,
intros y hy,
rw mem_roots_map (minpoly.ne_zero hx') at hy,
use [minpoly R ... | lemma | algebra.is_integral_trace | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra.trace",
"algebra_map",
"algebraic_closure",
"finite_dimensional",
"is_alg_closed.splits_codomain",
"is_integral",
"is_integral.multiset_sum",
"is_integral_algebra_map_iff",
"is_integral_of_is_scalar_tower",
"minpoly",
"minpoly.aeval_of_is_scalar_tower",
"minpoly.monic",
"minpoly.ne_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_eq_sum_embeddings_gen
(pb : power_basis K L)
(hE : (minpoly K pb.gen).splits (algebra_map K E)) (hfx : (minpoly K pb.gen).separable) :
algebra_map K E (algebra.trace K L pb.gen) =
(@@finset.univ (power_basis.alg_hom.fintype pb)).sum (λ σ, σ pb.gen) | begin
letI := classical.dec_eq E,
rw [pb.trace_gen_eq_sum_roots hE, fintype.sum_equiv pb.lift_equiv', finset.sum_mem_multiset,
finset.sum_eq_multiset_sum, multiset.to_finset_val,
multiset.dedup_eq_self.mpr _, multiset.map_id],
{ exact nodup_roots ((separable_map _).mpr hfx) },
{ intro x, refl },
{... | lemma | trace_eq_sum_embeddings_gen | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra.trace",
"algebra_map",
"classical.dec_eq",
"finset.univ",
"minpoly",
"multiset.map_id",
"multiset.to_finset_val",
"power_basis",
"power_basis.alg_hom.fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_embeddings_eq_finrank_mul [finite_dimensional K F] [is_separable K F]
(pb : power_basis K L) :
∑ σ : F →ₐ[K] E, σ (algebra_map L F pb.gen) =
finrank L F • (@@finset.univ (power_basis.alg_hom.fintype pb)).sum
(λ σ : L →ₐ[K] E, σ pb.gen) | begin
haveI : finite_dimensional L F := finite_dimensional.right K L F,
haveI : is_separable L F := is_separable_tower_top_of_is_separable K L F,
letI : fintype (L →ₐ[K] E) := power_basis.alg_hom.fintype pb,
letI : ∀ (f : L →ₐ[K] E), fintype (@@alg_hom L F E _ _ _ _ f.to_ring_hom.to_algebra) :=
_, -- will b... | lemma | sum_embeddings_eq_finrank_mul | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"alg_hom",
"alg_hom.card",
"alg_hom.comp_apply",
"alg_hom.restrict_domain",
"alg_hom_equiv_sigma",
"algebra",
"algebra_map",
"equiv.coe_fn_mk",
"finite_dimensional",
"finite_dimensional.right",
"finset.card_univ",
"finset.univ",
"finset.univ_sigma_univ",
"fintype",
"is_scalar_tower.coe_t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_eq_sum_embeddings [finite_dimensional K L] [is_separable K L]
{x : L} : algebra_map K E (algebra.trace K L x) = ∑ σ : L →ₐ[K] E, σ x | begin
have hx := is_separable.is_integral K x,
rw [trace_eq_trace_adjoin K x, algebra.smul_def, ring_hom.map_mul, ← adjoin.power_basis_gen hx,
trace_eq_sum_embeddings_gen E (adjoin.power_basis hx) (is_alg_closed.splits_codomain _),
← algebra.smul_def, algebra_map_smul],
{ exact (sum_embeddings_eq_finr... | lemma | trace_eq_sum_embeddings | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra.smul_def",
"algebra.trace",
"algebra_map",
"algebra_map_smul",
"finite_dimensional",
"is_alg_closed.splits_codomain",
"is_separable",
"is_separable.is_integral",
"is_separable.separable",
"is_separable_tower_bot_of_is_separable",
"ring_hom.map_mul",
"sum_embeddings_eq_finrank_mul",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_eq_sum_automorphisms (x : L) [finite_dimensional K L] [is_galois K L] :
algebra_map K L (algebra.trace K L x) = ∑ (σ : L ≃ₐ[K] L), σ x | begin
apply no_zero_smul_divisors.algebra_map_injective L (algebraic_closure L),
rw map_sum (algebra_map L (algebraic_closure L)),
rw ← fintype.sum_equiv (normal.alg_hom_equiv_aut K (algebraic_closure L) L),
{ rw ←trace_eq_sum_embeddings (algebraic_closure L),
{ simp only [algebra_map_eq_smul_one, smul_one_... | lemma | trace_eq_sum_automorphisms | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"alg_equiv.coe_of_bijective",
"alg_hom.restrict_normal'",
"alg_hom.restrict_normal_commutes",
"algebra.trace",
"algebra_map",
"algebraic_closure",
"equiv.coe_fn_mk",
"finite_dimensional",
"is_galois",
"no_zero_smul_divisors.algebra_map_injective",
"normal.alg_hom_equiv_aut",
"ring_hom.id_apply... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_matrix (b : κ → B) : matrix κ κ A | of $ λ i j, trace_form A B (b i) (b j) | def | algebra.trace_matrix | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"matrix"
] | Given an `A`-algebra `B` and `b`, an `κ`-indexed family of elements of `B`, we define
`trace_matrix A b` as the matrix whose `(i j)`-th element is the trace of `b i * b j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trace_matrix_apply (b : κ → B) (i j) :
trace_matrix A b i j = trace_form A B (b i) (b j) | rfl | lemma | algebra.trace_matrix_apply | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_matrix_reindex {κ' : Type*} (b : basis κ A B) (f : κ ≃ κ') :
trace_matrix A (b.reindex f) = reindex f f (trace_matrix A b) | by {ext x y, simp} | lemma | algebra.trace_matrix_reindex | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"basis",
"reindex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_matrix_of_matrix_vec_mul [fintype κ] (b : κ → B) (P : matrix κ κ A) :
trace_matrix A ((P.map (algebra_map A B)).vec_mul b) = Pᵀ ⬝ (trace_matrix A b) ⬝ P | begin
ext α β,
rw [trace_matrix_apply, vec_mul, dot_product, vec_mul, dot_product, matrix.mul_apply,
bilin_form.sum_left, fintype.sum_congr _ _ (λ (i : κ), @bilin_form.sum_right _ _ _ _ _ _ _ _
(b i * P.map (algebra_map A B) i α) (λ (y : κ), b y * P.map (algebra_map A B) y β)), sum_comm],
congr, ext x,
... | lemma | algebra.trace_matrix_of_matrix_vec_mul | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra.smul_mul_assoc",
"algebra_map",
"bilin_form.sum_left",
"bilin_form.sum_right",
"fintype",
"linear_map.map_smulₛₗ",
"matrix",
"matrix.mul_apply",
"mul_comm",
"ring_hom.id_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_matrix_of_matrix_mul_vec [fintype κ] (b : κ → B) (P : matrix κ κ A) :
trace_matrix A ((P.map (algebra_map A B)).mul_vec b) = P ⬝ (trace_matrix A b) ⬝ Pᵀ | begin
refine add_equiv.injective (transpose_add_equiv _ _ _) _,
rw [transpose_add_equiv_apply, transpose_add_equiv_apply, ← vec_mul_transpose,
← transpose_map, trace_matrix_of_matrix_vec_mul, transpose_transpose, transpose_mul,
transpose_transpose, transpose_mul]
end | lemma | algebra.trace_matrix_of_matrix_mul_vec | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra_map",
"fintype",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_matrix_of_basis [fintype κ] [decidable_eq κ] (b : basis κ A B) :
trace_matrix A b = bilin_form.to_matrix b (trace_form A B) | begin
ext i j,
rw [trace_matrix_apply, trace_form_apply, trace_form_to_matrix]
end | lemma | algebra.trace_matrix_of_basis | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"basis",
"bilin_form.to_matrix",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_matrix_of_basis_mul_vec (b : basis ι A B) (z : B) :
(trace_matrix A b).mul_vec (b.equiv_fun z) = (λ i, trace A B (z * (b i))) | begin
ext i,
rw [← col_apply ((trace_matrix A b).mul_vec (b.equiv_fun z)) i unit.star, col_mul_vec,
matrix.mul_apply, trace_matrix],
simp only [col_apply, trace_form_apply],
conv_lhs
{ congr, skip, funext,
rw [mul_comm _ (b.equiv_fun z _), ← smul_eq_mul, of_apply, ← linear_map.map_smul] },
rw [← l... | lemma | algebra.trace_matrix_of_basis_mul_vec | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"basis",
"finset.mul_sum",
"linear_map.map_smul",
"linear_map.map_sum",
"matrix.mul_apply",
"mul_comm",
"mul_smul_comm",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embeddings_matrix (b : κ → B) : matrix κ (B →ₐ[A] C) C | of $ λ i (σ : B →ₐ[A] C), σ (b i) | def | algebra.embeddings_matrix | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"matrix"
] | `embeddings_matrix A C b : matrix κ (B →ₐ[A] C) C` is the matrix whose `(i, σ)` coefficient is
`σ (b i)`. It is mostly useful for fields when `fintype.card κ = finrank A B` and `C` is
algebraically closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embeddings_matrix_apply (b : κ → B) (i) (σ : B →ₐ[A] C) :
embeddings_matrix A C b i σ = σ (b i) | rfl | lemma | algebra.embeddings_matrix_apply | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embeddings_matrix_reindex (b : κ → B) (e : κ ≃ (B →ₐ[A] C)) | reindex (equiv.refl κ) e.symm (embeddings_matrix A C b) | def | algebra.embeddings_matrix_reindex | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"equiv.refl",
"reindex"
] | `embeddings_matrix_reindex A C b e : matrix κ κ C` is the matrix whose `(i, j)` coefficient
is `σⱼ (b i)`, where `σⱼ : B →ₐ[A] C` is the embedding corresponding to `j : κ` given by a
bijection `e : κ ≃ (B →ₐ[A] C)`. It is mostly useful for fields and `C` is algebraically closed.
In this case, in presence of `h : ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embeddings_matrix_reindex_eq_vandermonde (pb : power_basis A B)
(e : fin pb.dim ≃ (B →ₐ[A] C)) :
embeddings_matrix_reindex A C pb.basis e = (vandermonde (λ i, e i pb.gen))ᵀ | by { ext i j, simp [embeddings_matrix_reindex, embeddings_matrix] } | lemma | algebra.embeddings_matrix_reindex_eq_vandermonde | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_matrix_eq_embeddings_matrix_mul_trans :
(trace_matrix K b).map (algebra_map K E) =
(embeddings_matrix K E b) ⬝ (embeddings_matrix K E b)ᵀ | by { ext i j, simp [trace_eq_sum_embeddings, embeddings_matrix, matrix.mul_apply] } | lemma | algebra.trace_matrix_eq_embeddings_matrix_mul_trans | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra_map",
"matrix.mul_apply",
"trace_eq_sum_embeddings"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_matrix_eq_embeddings_matrix_reindex_mul_trans [fintype κ]
(e : κ ≃ (L →ₐ[K] E)) : (trace_matrix K b).map (algebra_map K E) =
(embeddings_matrix_reindex K E b e) ⬝ (embeddings_matrix_reindex K E b e)ᵀ | by rw [trace_matrix_eq_embeddings_matrix_mul_trans, embeddings_matrix_reindex, reindex_apply,
transpose_submatrix, ← submatrix_mul_transpose_submatrix, ← equiv.coe_refl, equiv.refl_symm] | lemma | algebra.trace_matrix_eq_embeddings_matrix_reindex_mul_trans | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"algebra_map",
"equiv.coe_refl",
"equiv.refl_symm",
"fintype",
"reindex_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
det_trace_matrix_ne_zero' [is_separable K L] :
det (trace_matrix K pb.basis) ≠ 0 | begin
suffices : algebra_map K (algebraic_closure L) (det (trace_matrix K pb.basis)) ≠ 0,
{ refine mt (λ ht, _) this,
rw [ht, ring_hom.map_zero] },
haveI : finite_dimensional K L := pb.finite_dimensional,
let e : fin pb.dim ≃ (L →ₐ[K] algebraic_closure L) := (fintype.equiv_fin_of_card_eq _).symm,
rw [ring... | lemma | det_trace_matrix_ne_zero' | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"alg_hom.card",
"algebra_map",
"algebraic_closure",
"finite_dimensional",
"finset.prod_eq_zero_iff",
"fintype.equiv_fin_of_card_eq",
"is_separable",
"not_exists",
"ring_hom.map_det",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
det_trace_form_ne_zero [is_separable K L] [decidable_eq ι] (b : basis ι K L) :
det (bilin_form.to_matrix b (trace_form K L)) ≠ 0 | begin
haveI : finite_dimensional K L := finite_dimensional.of_fintype_basis b,
let pb : power_basis K L := field.power_basis_of_finite_of_separable _ _,
rw [← bilin_form.to_matrix_mul_basis_to_matrix pb.basis b,
← det_comm' (pb.basis.to_matrix_mul_to_matrix_flip b) _,
← matrix.mul_assoc, det_mul],
s... | lemma | det_trace_form_ne_zero | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"basis",
"basis.to_matrix_mul_to_matrix_flip",
"bilin_form.to_matrix",
"bilin_form.to_matrix_mul_basis_to_matrix",
"det_trace_matrix_ne_zero'",
"field.power_basis_of_finite_of_separable",
"finite_dimensional",
"finite_dimensional.of_fintype_basis",
"is_separable",
"is_unit_of_mul_eq_one",
"matri... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trace_form_nondegenerate [finite_dimensional K L] [is_separable K L] :
(trace_form K L).nondegenerate | bilin_form.nondegenerate_of_det_ne_zero (trace_form K L) _
(det_trace_form_ne_zero (finite_dimensional.fin_basis K L)) | theorem | trace_form_nondegenerate | ring_theory | src/ring_theory/trace.lean | [
"linear_algebra.matrix.bilinear_form",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.vandermonde",
"linear_algebra.trace",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.primitive_element",
"field_theory.ga... | [
"bilin_form.nondegenerate_of_det_ne_zero",
"det_trace_form_ne_zero",
"finite_dimensional",
"finite_dimensional.fin_basis",
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wf_dvd_monoid (α : Type*) [comm_monoid_with_zero α] : Prop | (well_founded_dvd_not_unit : well_founded (@dvd_not_unit α _)) | class | wf_dvd_monoid | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"comm_monoid_with_zero",
"dvd_not_unit"
] | Well-foundedness of the strict version of |, which is equivalent to the descending chain
condition on divisibility and to the ascending chain condition on
principal ideals in an integral domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_ring.wf_dvd_monoid [comm_ring α] [is_domain α] [is_noetherian_ring α] :
wf_dvd_monoid α | ⟨by { convert inv_image.wf (λ a, ideal.span ({a} : set α)) (well_founded_submodule_gt _ _),
ext,
exact ideal.span_singleton_lt_span_singleton.symm }⟩ | instance | is_noetherian_ring.wf_dvd_monoid | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"comm_ring",
"ideal.span",
"is_domain",
"is_noetherian_ring",
"well_founded_submodule_gt",
"wf_dvd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_wf_dvd_monoid_associates (h : wf_dvd_monoid (associates α)): wf_dvd_monoid α | ⟨begin
haveI := h,
refine (surjective.well_founded_iff mk_surjective _).2 well_founded_dvd_not_unit,
intros, rw mk_dvd_not_unit_mk_iff
end⟩ | theorem | wf_dvd_monoid.of_wf_dvd_monoid_associates | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associates",
"surjective.well_founded_iff",
"wf_dvd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wf_dvd_monoid_associates : wf_dvd_monoid (associates α) | ⟨begin
refine (surjective.well_founded_iff mk_surjective _).1 well_founded_dvd_not_unit,
intros, rw mk_dvd_not_unit_mk_iff
end⟩ | instance | wf_dvd_monoid.wf_dvd_monoid_associates | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associates",
"surjective.well_founded_iff",
"wf_dvd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
well_founded_associates : well_founded ((<) : associates α → associates α → Prop) | subrelation.wf (λ x y, dvd_not_unit_of_lt) well_founded_dvd_not_unit | theorem | wf_dvd_monoid.well_founded_associates | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associates"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_irreducible_factor {a : α} (ha : ¬ is_unit a) (ha0 : a ≠ 0) :
∃ i, irreducible i ∧ i ∣ a | let ⟨b, hs, hr⟩ := well_founded_dvd_not_unit.has_min {b | b ∣ a ∧ ¬ is_unit b} ⟨a, dvd_rfl, ha⟩ in
⟨b, ⟨hs.2, λ c d he, let h := dvd_trans ⟨d, he⟩ hs.1 in or_iff_not_imp_left.2 $
λ hc, of_not_not $ λ hd, hr c ⟨h, hc⟩ ⟨ne_zero_of_dvd_ne_zero ha0 h, d, hd, he⟩⟩, hs.1⟩ | lemma | wf_dvd_monoid.exists_irreducible_factor | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"dvd_rfl",
"dvd_trans",
"irreducible",
"is_unit",
"of_not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on_irreducible {P : α → Prop} (a : α)
(h0 : P 0) (hu : ∀ u : α, is_unit u → P u)
(hi : ∀ a i : α, a ≠ 0 → irreducible i → P a → P (i * a)) :
P a | by haveI := classical.dec; exact
well_founded_dvd_not_unit.fix
(λ a ih, if ha0 : a = 0 then ha0.substr h0
else if hau : is_unit a then hu a hau
else let ⟨i, hii, b, hb⟩ := exists_irreducible_factor hau ha0,
hb0 : b ≠ 0 := ne_zero_of_dvd_ne_zero ha0 ⟨i, mul_comm i b ▸ hb⟩ in
hb.symm ▸ hi b i hb0 hi... | lemma | wf_dvd_monoid.induction_on_irreducible | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"classical.dec",
"ih",
"irreducible",
"is_unit",
"mul_comm",
"ne_zero_of_dvd_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_factors (a : α) : a ≠ 0 →
∃ f : multiset α, (∀ b ∈ f, irreducible b) ∧ associated f.prod a | induction_on_irreducible a
(λ h, (h rfl).elim)
(λ u hu _, ⟨0, λ _ h, h.elim, hu.unit, one_mul _⟩)
(λ a i ha0 hi ih _,
let ⟨s, hs⟩ := ih ha0 in
⟨i ::ₘ s, λ b H, (multiset.mem_cons.1 H).elim (λ h, h.symm ▸ hi) (hs.1 b),
by { rw s.prod_cons i, exact hs.2.mul_left i }⟩) | lemma | wf_dvd_monoid.exists_factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associated",
"ih",
"irreducible",
"multiset",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_unit_iff_exists_factors_eq (a : α) (hn0 : a ≠ 0) :
¬ is_unit a ↔ ∃ f : multiset α, (∀ b ∈ f, irreducible b) ∧ f.prod = a ∧ f ≠ ∅ | ⟨λ hnu, begin
obtain ⟨f, hi, u, rfl⟩ := exists_factors a hn0,
obtain ⟨b, h⟩ := multiset.exists_mem_of_ne_zero (λ h : f = 0, hnu $ by simp [h]),
classical, refine ⟨(f.erase b).cons (b * u), λ a ha, _, _, multiset.cons_ne_zero⟩,
{ obtain (rfl|ha) := multiset.mem_cons.1 ha,
exacts [associated.irreducible ⟨u,rf... | lemma | wf_dvd_monoid.not_unit_iff_exists_factors_eq | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associated.irreducible",
"irreducible",
"is_unit",
"mul_assoc",
"mul_comm",
"multiset",
"multiset.dvd_prod",
"multiset.exists_mem_of_ne_zero",
"multiset.mem_of_mem_erase",
"multiset.prod_cons",
"multiset.prod_erase",
"not_is_unit_of_not_is_unit_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wf_dvd_monoid.of_well_founded_associates [cancel_comm_monoid_with_zero α]
(h : well_founded ((<) : associates α → associates α → Prop)) : wf_dvd_monoid α | wf_dvd_monoid.of_wf_dvd_monoid_associates
⟨by { convert h, ext, exact associates.dvd_not_unit_iff_lt }⟩ | theorem | wf_dvd_monoid.of_well_founded_associates | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associates",
"associates.dvd_not_unit_iff_lt",
"cancel_comm_monoid_with_zero",
"wf_dvd_monoid",
"wf_dvd_monoid.of_wf_dvd_monoid_associates"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wf_dvd_monoid.iff_well_founded_associates [cancel_comm_monoid_with_zero α] :
wf_dvd_monoid α ↔ well_founded ((<) : associates α → associates α → Prop) | ⟨by apply wf_dvd_monoid.well_founded_associates, wf_dvd_monoid.of_well_founded_associates⟩ | theorem | wf_dvd_monoid.iff_well_founded_associates | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associates",
"cancel_comm_monoid_with_zero",
"wf_dvd_monoid",
"wf_dvd_monoid.well_founded_associates"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_factorization_monoid (α : Type*) [cancel_comm_monoid_with_zero α]
extends wf_dvd_monoid α : Prop | (irreducible_iff_prime : ∀ {a : α}, irreducible a ↔ prime a) | class | unique_factorization_monoid | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"cancel_comm_monoid_with_zero",
"irreducible",
"prime",
"wf_dvd_monoid"
] | unique factorization monoids.
These are defined as `cancel_comm_monoid_with_zero`s with well-founded strict divisibility
relations, but this is equivalent to more familiar definitions:
Each element (except zero) is uniquely represented as a multiset of irreducible factors.
Uniqueness is only up to associated elements... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ufm_of_gcd_of_wf_dvd_monoid [cancel_comm_monoid_with_zero α]
[wf_dvd_monoid α] [gcd_monoid α] : unique_factorization_monoid α | { irreducible_iff_prime := λ _, gcd_monoid.irreducible_iff_prime
.. ‹wf_dvd_monoid α› } | lemma | ufm_of_gcd_of_wf_dvd_monoid | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"cancel_comm_monoid_with_zero",
"gcd_monoid",
"gcd_monoid.irreducible_iff_prime",
"unique_factorization_monoid",
"wf_dvd_monoid"
] | Can't be an instance because it would cause a loop `ufm → wf_dvd_monoid → ufm → ...`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associates.ufm [cancel_comm_monoid_with_zero α]
[unique_factorization_monoid α] : unique_factorization_monoid (associates α) | { irreducible_iff_prime := by { rw ← associates.irreducible_iff_prime_iff,
apply unique_factorization_monoid.irreducible_iff_prime, }
.. (wf_dvd_monoid.wf_dvd_monoid_associates : wf_dvd_monoid (associates α)) } | instance | associates.ufm | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associates",
"associates.irreducible_iff_prime_iff",
"cancel_comm_monoid_with_zero",
"unique_factorization_monoid",
"wf_dvd_monoid",
"wf_dvd_monoid.wf_dvd_monoid_associates"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_prime_factors (a : α) : a ≠ 0 →
∃ f : multiset α, (∀b ∈ f, prime b) ∧ f.prod ~ᵤ a | by { simp_rw ← unique_factorization_monoid.irreducible_iff_prime,
apply wf_dvd_monoid.exists_factors a } | theorem | unique_factorization_monoid.exists_prime_factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"multiset",
"prime",
"wf_dvd_monoid.exists_factors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on_prime {P : α → Prop}
(a : α) (h₁ : P 0) (h₂ : ∀ x : α, is_unit x → P x)
(h₃ : ∀ a p : α, a ≠ 0 → prime p → P a → P (p * a)) : P a | begin
simp_rw ← unique_factorization_monoid.irreducible_iff_prime at h₃,
exact wf_dvd_monoid.induction_on_irreducible a h₁ h₂ h₃,
end | lemma | unique_factorization_monoid.induction_on_prime | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"is_unit",
"prime",
"wf_dvd_monoid.induction_on_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_factors_unique [cancel_comm_monoid_with_zero α] : ∀ {f g : multiset α},
(∀ x ∈ f, prime x) → (∀ x ∈ g, prime x) → f.prod ~ᵤ g.prod →
multiset.rel associated f g | by haveI := classical.dec_eq α; exact
λ f, multiset.induction_on f
(λ g _ hg h,
multiset.rel_zero_left.2 $
multiset.eq_zero_of_forall_not_mem $ λ x hx,
have is_unit g.prod, by simpa [associated_one_iff_is_unit] using h.symm,
(hg x hx).not_unit $ is_unit_iff_dvd_one.2 $
(multiset.dvd_prod hx).trans... | lemma | prime_factors_unique | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associated",
"associated.of_mul_left",
"associated_one_iff_is_unit",
"cancel_comm_monoid_with_zero",
"classical.dec_eq",
"exists_associated_mem_of_dvd_prod",
"ih",
"is_unit",
"multiset",
"multiset.cons_erase",
"multiset.dvd_prod",
"multiset.eq_zero_of_forall_not_mem",
"multiset.induction_on... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factors_unique {f g : multiset α} (hf : ∀ x ∈ f, irreducible x) (hg : ∀ x ∈ g, irreducible x)
(h : f.prod ~ᵤ g.prod) : multiset.rel associated f g | prime_factors_unique
(λ x hx, irreducible_iff_prime.mp (hf x hx))
(λ x hx, irreducible_iff_prime.mp (hg x hx))
h | lemma | unique_factorization_monoid.factors_unique | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associated",
"irreducible",
"multiset",
"multiset.rel",
"prime_factors_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_factors_irreducible [cancel_comm_monoid_with_zero α] {a : α} {f : multiset α}
(ha : irreducible a) (pfa : (∀ b ∈ f, prime b) ∧ f.prod ~ᵤ a) :
∃ p, a ~ᵤ p ∧ f = {p} | begin
haveI := classical.dec_eq α,
refine multiset.induction_on f (λ h, (ha.not_unit
(associated_one_iff_is_unit.1 (associated.symm h))).elim) _ pfa.2 pfa.1,
rintros p s _ ⟨u, hu⟩ hs,
use p,
have hs0 : s = 0,
{ by_contra hs0,
obtain ⟨q, hq⟩ := multiset.exists_mem_of_ne_zero hs0,
apply (hs q (by ... | lemma | prime_factors_irreducible | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associated.symm",
"by_contra",
"cancel_comm_monoid_with_zero",
"classical.dec_eq",
"irreducible",
"is_unit_of_mul_is_unit_left",
"mul_assoc",
"mul_comm",
"mul_one",
"mul_right_comm",
"multiset",
"multiset.cons_erase",
"multiset.exists_mem_of_ne_zero",
"multiset.induction_on",
"multiset.... | If an irreducible has a prime factorization,
then it is an associate of one of its prime factors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wf_dvd_monoid.of_exists_prime_factors : wf_dvd_monoid α | ⟨begin
classical,
refine rel_hom_class.well_founded
(rel_hom.mk _ _ : (dvd_not_unit : α → α → Prop) →r ((<) : ℕ∞ → ℕ∞ → Prop))
(with_top.well_founded_lt nat.lt_wf),
{ intro a,
by_cases h : a = 0, { exact ⊤ },
exact (classical.some (pf a h)).card },
rintros a b ⟨ane0, ⟨c, hc, b_eq⟩⟩,
rw dif_ne... | lemma | wf_dvd_monoid.of_exists_prime_factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associated.mul_mul",
"con",
"dvd_not_unit",
"lt_top_iff_ne_top",
"mul_zero",
"multiset.card",
"multiset.card_add",
"multiset.card_eq_card_of_rel",
"multiset.mem_add",
"multiset.prod_add",
"multiset.prod_zero",
"prime_factors_unique",
"rel_hom_class.well_founded",
"wf_dvd_monoid",
"with_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_iff_prime_of_exists_prime_factors {p : α} : irreducible p ↔ prime p | begin
by_cases hp0 : p = 0,
{ simp [hp0] },
refine ⟨λ h, _, prime.irreducible⟩,
obtain ⟨f, hf⟩ := pf p hp0,
obtain ⟨q, hq, rfl⟩ := prime_factors_irreducible h hf,
rw hq.prime_iff,
exact hf.1 q (multiset.mem_singleton_self _)
end | lemma | irreducible_iff_prime_of_exists_prime_factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"irreducible",
"multiset.mem_singleton_self",
"prime",
"prime_factors_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_factorization_monoid.of_exists_prime_factors :
unique_factorization_monoid α | { irreducible_iff_prime := λ _, irreducible_iff_prime_of_exists_prime_factors pf,
.. wf_dvd_monoid.of_exists_prime_factors pf } | theorem | unique_factorization_monoid.of_exists_prime_factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"irreducible_iff_prime_of_exists_prime_factors",
"unique_factorization_monoid",
"wf_dvd_monoid.of_exists_prime_factors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_factorization_monoid.iff_exists_prime_factors [cancel_comm_monoid_with_zero α] :
unique_factorization_monoid α ↔
(∀ (a : α), a ≠ 0 → ∃ f : multiset α, (∀b ∈ f, prime b) ∧ f.prod ~ᵤ a) | ⟨λ h, @unique_factorization_monoid.exists_prime_factors _ _ h,
unique_factorization_monoid.of_exists_prime_factors⟩ | theorem | unique_factorization_monoid.iff_exists_prime_factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"cancel_comm_monoid_with_zero",
"multiset",
"prime",
"unique_factorization_monoid",
"unique_factorization_monoid.exists_prime_factors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv.unique_factorization_monoid (e : α ≃* β)
(hα : unique_factorization_monoid α) : unique_factorization_monoid β | begin
rw unique_factorization_monoid.iff_exists_prime_factors at hα ⊢, intros a ha,
obtain ⟨w,hp,u,h⟩ := hα (e.symm a) (λ h, ha $ by { convert ← map_zero e, simp [← h] }),
exact ⟨ w.map e,
λ b hb, let ⟨c,hc,he⟩ := multiset.mem_map.1 hb in he ▸ e.prime_iff.1 (hp c hc),
units.map e.to_monoid_hom u,
by {... | lemma | mul_equiv.unique_factorization_monoid | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"multiset.prod_hom",
"unique_factorization_monoid",
"unique_factorization_monoid.iff_exists_prime_factors",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv.unique_factorization_monoid_iff (e : α ≃* β) :
unique_factorization_monoid α ↔ unique_factorization_monoid β | ⟨ e.unique_factorization_monoid, e.symm.unique_factorization_monoid ⟩ | lemma | mul_equiv.unique_factorization_monoid_iff | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"unique_factorization_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_iff_prime_of_exists_unique_irreducible_factors [cancel_comm_monoid_with_zero α]
(eif : ∀ (a : α), a ≠ 0 → ∃ f : multiset α, (∀b ∈ f, irreducible b) ∧ f.prod ~ᵤ a)
(uif : ∀ (f g : multiset α),
(∀ x ∈ f, irreducible x) → (∀ x ∈ g, irreducible x) → f.prod ~ᵤ g.prod →
multiset.rel associated f g)
(p... | ⟨by letI := classical.dec_eq α; exact λ hpi,
⟨hpi.ne_zero, hpi.1,
λ a b ⟨x, hx⟩,
if hab0 : a * b = 0
then (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim
(λ ha0, by simp [ha0])
(λ hb0, by simp [hb0])
else
have hx0 : x ≠ 0, from λ hx0, by simp * at *,
have ha0 : ... | theorem | irreducible_iff_prime_of_exists_unique_irreducible_factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associated",
"cancel_comm_monoid_with_zero",
"classical.dec_eq",
"irreducible",
"left_ne_zero_of_mul",
"multiset",
"multiset.dvd_prod",
"multiset.exists_mem_of_rel_of_mem",
"multiset.mem_cons_self",
"multiset.prod",
"multiset.prod_add",
"multiset.prod_cons",
"multiset.rel",
"prime",
"ri... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_factorization_monoid.of_exists_unique_irreducible_factors
[cancel_comm_monoid_with_zero α]
(eif : ∀ (a : α), a ≠ 0 → ∃ f : multiset α, (∀b ∈ f, irreducible b) ∧ f.prod ~ᵤ a)
(uif : ∀ (f g : multiset α),
(∀ x ∈ f, irreducible x) → (∀ x ∈ g, irreducible x) → f.prod ~ᵤ g.prod →
multiset.rel associated f... | unique_factorization_monoid.of_exists_prime_factors (by
{ convert eif,
simp_rw irreducible_iff_prime_of_exists_unique_irreducible_factors eif uif }) | theorem | unique_factorization_monoid.of_exists_unique_irreducible_factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associated",
"cancel_comm_monoid_with_zero",
"irreducible",
"irreducible_iff_prime_of_exists_unique_irreducible_factors",
"multiset",
"multiset.rel",
"unique_factorization_monoid",
"unique_factorization_monoid.of_exists_prime_factors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factors (a : α) : multiset α | if h : a = 0 then 0 else
classical.some (unique_factorization_monoid.exists_prime_factors a h) | def | unique_factorization_monoid.factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"multiset",
"unique_factorization_monoid.exists_prime_factors"
] | Noncomputably determines the multiset of prime factors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
factors_prod {a : α} (ane0 : a ≠ 0) : associated (factors a).prod a | begin
rw [factors, dif_neg ane0],
exact (classical.some_spec (exists_prime_factors a ane0)).2
end | theorem | unique_factorization_monoid.factors_prod | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"associated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_mem_factors {p a : α} (h : p ∈ factors a) : a ≠ 0 | begin
intro ha,
rw [factors, dif_pos ha] at h,
exact multiset.not_mem_zero _ h
end | lemma | unique_factorization_monoid.ne_zero_of_mem_factors | ring_theory | src/ring_theory/unique_factorization_domain.lean | [
"algebra.big_operators.associated",
"algebra.gcd_monoid.basic",
"data.finsupp.multiset",
"ring_theory.noetherian",
"ring_theory.multiplicity"
] | [
"multiset.not_mem_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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