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 |
|---|---|---|---|---|---|---|---|---|---|---|
decompose_symm_sum {ι'} (s : finset ι') (f : ι' → ⨁ i, ℳ i) :
(decompose ℳ).symm (∑ i in s, f i) = ∑ i in s, (decompose ℳ).symm (f i) | map_sum (decompose_add_equiv ℳ).symm f s | lemma | direct_sum.decompose_symm_sum | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_support_decompose [Π i (x : ℳ i), decidable (x ≠ 0)] (r : M) :
∑ i in (decompose ℳ r).support, (decompose ℳ r i : M) = r | begin
conv_rhs { rw [←(decompose ℳ).symm_apply_apply r,
←sum_support_of (λ i, (ℳ i)) (decompose ℳ r)] },
rw [decompose_symm_sum],
simp_rw decompose_symm_of,
end | lemma | direct_sum.sum_support_decompose | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_group_set_like [add_comm_group M] [set_like σ M] [add_subgroup_class σ M]
(ℳ : ι → σ) : add_comm_group (⨁ i, ℳ i) | by apply_instance | instance | direct_sum.add_comm_group_set_like | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"add_comm_group",
"add_subgroup_class",
"set_like"
] | The `-` in the statements below doesn't resolve without this line.
This seems to a be a problem of synthesized vs inferred typeclasses disagreeing. If we replace
the statement of `decompose_neg` with `@eq (⨁ i, ℳ i) (decompose ℳ (-x)) (-decompose ℳ x)`
instead of `decompose ℳ (-x) = -decompose ℳ x`, which forces the t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decompose_neg (x : M) : decompose ℳ (-x) = -decompose ℳ x | map_neg (decompose_add_equiv ℳ) x | lemma | direct_sum.decompose_neg | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_symm_neg (x : ⨁ i, ℳ i) :
(decompose ℳ).symm (-x) = -(decompose ℳ).symm x | map_neg (decompose_add_equiv ℳ).symm x | lemma | direct_sum.decompose_symm_neg | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_sub (x y : M) : decompose ℳ (x - y) = decompose ℳ x - decompose ℳ y | map_sub (decompose_add_equiv ℳ) x y | lemma | direct_sum.decompose_sub | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_symm_sub (x y : ⨁ i, ℳ i) :
(decompose ℳ).symm (x - y) = (decompose ℳ).symm x - (decompose ℳ).symm y | map_sub (decompose_add_equiv ℳ).symm x y | lemma | direct_sum.decompose_symm_sub | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_linear_equiv : M ≃ₗ[R] ⨁ i, ℳ i | linear_equiv.symm
{ map_smul' := map_smul (direct_sum.coe_linear_map ℳ),
..(decompose_add_equiv ℳ).symm } | def | direct_sum.decompose_linear_equiv | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.coe_linear_map",
"linear_equiv.symm"
] | If `M` is graded by `ι` with degree `i` component `ℳ i`, then it is isomorphic as
a module to a direct sum of components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decompose_smul (r : R) (x : M) : decompose ℳ (r • x) = r • decompose ℳ x | map_smul (decompose_linear_equiv ℳ) r x | lemma | direct_sum.decompose_smul | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finsupp_lequiv_direct_sum : (ι →₀ M) ≃ₗ[R] ⨁ i : ι, M | by haveI : Π m : M, decidable (m ≠ 0) := classical.dec_pred _; exact finsupp_lequiv_dfinsupp R | def | finsupp_lequiv_direct_sum | algebra.direct_sum | src/algebra/direct_sum/finsupp.lean | [
"algebra.direct_sum.module",
"data.finsupp.to_dfinsupp"
] | [
"classical.dec_pred",
"finsupp_lequiv_dfinsupp"
] | The finitely supported functions `ι →₀ M` are in linear equivalence with the direct sum of
copies of M indexed by ι. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finsupp_lequiv_direct_sum_single (i : ι) (m : M) :
finsupp_lequiv_direct_sum R M ι (finsupp.single i m) = direct_sum.lof R ι _ i m | finsupp.to_dfinsupp_single i m | theorem | finsupp_lequiv_direct_sum_single | algebra.direct_sum | src/algebra/direct_sum/finsupp.lean | [
"algebra.direct_sum.module",
"data.finsupp.to_dfinsupp"
] | [
"direct_sum.lof",
"finsupp.single",
"finsupp.to_dfinsupp_single",
"finsupp_lequiv_direct_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finsupp_lequiv_direct_sum_symm_lof (i : ι) (m : M) :
(finsupp_lequiv_direct_sum R M ι).symm (direct_sum.lof R ι _ i m) = finsupp.single i m | begin
letI : Π m : M, decidable (m ≠ 0) := classical.dec_pred _,
exact (dfinsupp.to_finsupp_single i m),
end | theorem | finsupp_lequiv_direct_sum_symm_lof | algebra.direct_sum | src/algebra/direct_sum/finsupp.lean | [
"algebra.direct_sum.module",
"data.finsupp.to_dfinsupp"
] | [
"classical.dec_pred",
"dfinsupp.to_finsupp_single",
"direct_sum.lof",
"finsupp.single",
"finsupp_lequiv_direct_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_monoid.of_submonoid_on_semiring [semiring R] [set_like σ R]
[add_submonoid_class σ R] (A : ι → σ) : ∀ i, add_comm_monoid (A i) | λ i, by apply_instance | instance | add_comm_monoid.of_submonoid_on_semiring | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_comm_monoid",
"add_submonoid_class",
"semiring",
"set_like"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_group.of_subgroup_on_ring [ring R] [set_like σ R]
[add_subgroup_class σ R] (A : ι → σ) : ∀ i, add_comm_group (A i) | λ i, by apply_instance | instance | add_comm_group.of_subgroup_on_ring | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_comm_group",
"add_subgroup_class",
"ring",
"set_like"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.algebra_map_mem_graded [has_zero ι]
[comm_semiring S] [semiring R] [algebra S R]
(A : ι → submodule S R) [set_like.has_graded_one A] (s : S) : algebra_map S R s ∈ A 0 | begin
rw algebra.algebra_map_eq_smul_one,
exact ((A 0).smul_mem s $ set_like.one_mem_graded _),
end | lemma | set_like.algebra_map_mem_graded | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"algebra",
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"comm_semiring",
"semiring",
"set_like.has_graded_one",
"set_like.one_mem_graded",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.nat_cast_mem_graded [has_zero ι] [add_monoid_with_one R]
[set_like σ R] [add_submonoid_class σ R] (A : ι → σ) [set_like.has_graded_one A] (n : ℕ) :
(n : R) ∈ A 0 | begin
induction n,
{ rw nat.cast_zero,
exact zero_mem (A 0), },
{ rw nat.cast_succ,
exact add_mem n_ih (set_like.one_mem_graded _), },
end | lemma | set_like.nat_cast_mem_graded | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid_with_one",
"add_submonoid_class",
"nat.cast_succ",
"nat.cast_zero",
"set_like",
"set_like.has_graded_one",
"set_like.one_mem_graded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.int_cast_mem_graded [has_zero ι] [add_group_with_one R]
[set_like σ R] [add_subgroup_class σ R] (A : ι → σ) [set_like.has_graded_one A] (z : ℤ) :
(z : R) ∈ A 0 | begin
induction z,
{ rw int.cast_of_nat,
exact set_like.nat_cast_mem_graded _ _, },
{ rw int.cast_neg_succ_of_nat,
exact neg_mem (set_like.nat_cast_mem_graded _ _), },
end | lemma | set_like.int_cast_mem_graded | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_group_with_one",
"add_subgroup_class",
"int.cast_neg_succ_of_nat",
"int.cast_of_nat",
"set_like",
"set_like.has_graded_one",
"set_like.nat_cast_mem_graded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gnon_unital_non_assoc_semiring [has_add ι] [non_unital_non_assoc_semiring R]
[set_like σ R] [add_submonoid_class σ R]
(A : ι → σ) [set_like.has_graded_mul A] :
direct_sum.gnon_unital_non_assoc_semiring (λ i, A i) | { mul_zero := λ i j _, subtype.ext (mul_zero _),
zero_mul := λ i j _, subtype.ext (zero_mul _),
mul_add := λ i j _ _ _, subtype.ext (mul_add _ _ _),
add_mul := λ i j _ _ _, subtype.ext (add_mul _ _ _),
..set_like.ghas_mul A } | instance | set_like.gnon_unital_non_assoc_semiring | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_submonoid_class",
"direct_sum.gnon_unital_non_assoc_semiring",
"mul_zero",
"non_unital_non_assoc_semiring",
"set_like",
"set_like.ghas_mul",
"set_like.has_graded_mul",
"subtype.ext",
"zero_mul"
] | Build a `gnon_unital_non_assoc_semiring` instance for a collection of additive submonoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gsemiring [add_monoid ι] [semiring R] [set_like σ R] [add_submonoid_class σ R]
(A : ι → σ) [set_like.graded_monoid A] :
direct_sum.gsemiring (λ i, A i) | { mul_zero := λ i j _, subtype.ext (mul_zero _),
zero_mul := λ i j _, subtype.ext (zero_mul _),
mul_add := λ i j _ _ _, subtype.ext (mul_add _ _ _),
add_mul := λ i j _ _ _, subtype.ext (add_mul _ _ _),
nat_cast := λ n, ⟨n, set_like.nat_cast_mem_graded _ _⟩,
nat_cast_zero := subtype.ext nat.cast_zero,
nat_ca... | instance | set_like.gsemiring | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"add_submonoid_class",
"direct_sum.gsemiring",
"mul_zero",
"nat.cast_succ",
"nat.cast_zero",
"semiring",
"set_like",
"set_like.gmonoid",
"set_like.graded_monoid",
"set_like.nat_cast_mem_graded",
"subtype.ext",
"zero_mul"
] | Build a `gsemiring` instance for a collection of additive submonoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcomm_semiring [add_comm_monoid ι] [comm_semiring R] [set_like σ R]
[add_submonoid_class σ R] (A : ι → σ) [set_like.graded_monoid A] :
direct_sum.gcomm_semiring (λ i, A i) | { ..set_like.gcomm_monoid A,
..set_like.gsemiring A, } | instance | set_like.gcomm_semiring | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_comm_monoid",
"add_submonoid_class",
"comm_semiring",
"direct_sum.gcomm_semiring",
"set_like",
"set_like.gcomm_monoid",
"set_like.graded_monoid",
"set_like.gsemiring"
] | Build a `gcomm_semiring` instance for a collection of additive submonoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gring [add_monoid ι] [ring R] [set_like σ R] [add_subgroup_class σ R]
(A : ι → σ) [set_like.graded_monoid A] :
direct_sum.gring (λ i, A i) | { int_cast := λ z, ⟨z, set_like.int_cast_mem_graded _ _⟩,
int_cast_of_nat := λ n, subtype.ext $ int.cast_of_nat _,
int_cast_neg_succ_of_nat := λ n, subtype.ext $ int.cast_neg_succ_of_nat n,
..set_like.gsemiring A } | instance | set_like.gring | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"add_subgroup_class",
"direct_sum.gring",
"int.cast_neg_succ_of_nat",
"int.cast_of_nat",
"ring",
"set_like",
"set_like.graded_monoid",
"set_like.gsemiring",
"set_like.int_cast_mem_graded",
"subtype.ext"
] | Build a `gring` instance for a collection of additive subgroups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcomm_ring [add_comm_monoid ι] [comm_ring R] [set_like σ R]
[add_subgroup_class σ R] (A : ι → σ) [set_like.graded_monoid A] :
direct_sum.gcomm_ring (λ i, A i) | { ..set_like.gcomm_monoid A,
..set_like.gring A, } | instance | set_like.gcomm_ring | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_comm_monoid",
"add_subgroup_class",
"comm_ring",
"direct_sum.gcomm_ring",
"set_like",
"set_like.gcomm_monoid",
"set_like.graded_monoid",
"set_like.gring"
] | Build a `gcomm_semiring` instance for a collection of additive submonoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_ring_hom [add_monoid ι] [set_like.graded_monoid A] : (⨁ i, A i) →+* R | direct_sum.to_semiring (λ i, add_submonoid_class.subtype (A i)) rfl (λ _ _ _ _, rfl) | def | direct_sum.coe_ring_hom | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"direct_sum.to_semiring",
"set_like.graded_monoid"
] | The canonical ring isomorphism between `⨁ i, A i` and `R` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_ring_hom_of [add_monoid ι] [set_like.graded_monoid A] (i : ι)
(x : A i) : (coe_ring_hom A : _ →+* R) (of (λ i, A i) i x) = x | direct_sum.to_semiring_of _ _ _ _ _ | lemma | direct_sum.coe_ring_hom_of | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"direct_sum.to_semiring_of",
"set_like.graded_monoid"
] | The canonical ring isomorphism between `⨁ i, A i` and `R` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_apply [add_monoid ι] [set_like.graded_monoid A]
[Π (i : ι) (x : A i), decidable (x ≠ 0)] (r r' : ⨁ i, A i) (n : ι) :
((r * r') n : R) =
∑ ij in (r.support ×ˢ r'.support).filter (λ ij : ι × ι, ij.1 + ij.2 = n), r ij.1 * r' ij.2 | begin
rw [mul_eq_sum_support_ghas_mul, dfinsupp.finset_sum_apply, add_submonoid_class.coe_finset_sum],
simp_rw [coe_of_apply, ←finset.sum_filter, set_like.coe_ghas_mul],
end | lemma | direct_sum.coe_mul_apply | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"dfinsupp.finset_sum_apply",
"filter",
"set_like.coe_ghas_mul",
"set_like.graded_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_apply_eq_dfinsupp_sum [add_monoid ι] [set_like.graded_monoid A]
[Π (i : ι) (x : A i), decidable (x ≠ 0)] (r r' : ⨁ i, A i) (n : ι) :
((r * r') n : R) = r.sum (λ i ri, r'.sum (λ j rj, if i + j = n then ri * rj else 0)) | begin
simp only [mul_eq_dfinsupp_sum, dfinsupp.sum_apply],
iterate 2 { rw [dfinsupp.sum, add_submonoid_class.coe_finset_sum], congr, ext },
dsimp only, split_ifs,
{ subst h, rw of_eq_same, refl },
{ rw of_eq_of_ne _ _ _ _ h, refl },
end | lemma | direct_sum.coe_mul_apply_eq_dfinsupp_sum | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"dfinsupp.sum_apply",
"set_like.graded_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_mul_apply_aux [add_monoid ι] [set_like.graded_monoid A] {i : ι}
(r : A i) (r' : ⨁ i, A i) {j n : ι} (H : ∀ (x : ι), i + x = n ↔ x = j) :
((of _ i r * r') n : R) = r * r' j | begin
classical,
rw coe_mul_apply_eq_dfinsupp_sum,
apply (dfinsupp.sum_single_index _).trans, swap,
{ simp_rw [zero_mem_class.coe_zero, zero_mul, if_t_t], exact dfinsupp.sum_zero },
simp_rw [dfinsupp.sum, H, finset.sum_ite_eq'],
split_ifs, refl,
rw [dfinsupp.not_mem_support_iff.mp h, zero_mem_class.coe_ze... | lemma | direct_sum.coe_of_mul_apply_aux | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"mul_zero",
"set_like.graded_monoid",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_of_apply_aux [add_monoid ι] [set_like.graded_monoid A]
(r : ⨁ i, A i) {i : ι} (r' : A i) {j n : ι} (H : ∀ (x : ι), x + i = n ↔ x = j) :
((r * of _ i r') n : R) = r j * r' | begin
classical,
rw [coe_mul_apply_eq_dfinsupp_sum, dfinsupp.sum_comm],
apply (dfinsupp.sum_single_index _).trans, swap,
{ simp_rw [zero_mem_class.coe_zero, mul_zero, if_t_t], exact dfinsupp.sum_zero },
simp_rw [dfinsupp.sum, H, finset.sum_ite_eq'],
split_ifs, refl,
rw [dfinsupp.not_mem_support_iff.mp h, ... | lemma | direct_sum.coe_mul_of_apply_aux | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"mul_zero",
"set_like.graded_monoid",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_mul_apply_add [add_left_cancel_monoid ι] [set_like.graded_monoid A]
{i : ι} (r : A i) (r' : ⨁ i, A i) (j : ι) :
((of _ i r * r') (i + j) : R) = r * r' j | coe_of_mul_apply_aux _ _ _ (λ x, ⟨λ h, add_left_cancel h, λ h, h ▸ rfl⟩) | lemma | direct_sum.coe_of_mul_apply_add | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_left_cancel_monoid",
"set_like.graded_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_of_apply_add [add_right_cancel_monoid ι] [set_like.graded_monoid A]
(r : ⨁ i, A i) {i : ι} (r' : A i) (j : ι) :
((r * of _ i r') (j + i) : R) = r j * r' | coe_mul_of_apply_aux _ _ _ (λ x, ⟨λ h, add_right_cancel h, λ h, h ▸ rfl⟩) | lemma | direct_sum.coe_mul_of_apply_add | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_right_cancel_monoid",
"set_like.graded_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_mul_apply_of_not_le
{i : ι} (r : A i) (r' : ⨁ i, A i) (n : ι)
(h : ¬ i ≤ n) : ((of _ i r * r') n : R) = 0 | begin
classical,
rw coe_mul_apply_eq_dfinsupp_sum,
apply (dfinsupp.sum_single_index _).trans, swap,
{ simp_rw [zero_mem_class.coe_zero, zero_mul, if_t_t], exact dfinsupp.sum_zero },
{ rw [dfinsupp.sum, finset.sum_ite_of_false _ _ (λ x _ H, _), finset.sum_const_zero],
exact h ((self_le_add_right i x).trans... | lemma | direct_sum.coe_of_mul_apply_of_not_le | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_of_apply_of_not_le
(r : ⨁ i, A i) {i : ι} (r' : A i) (n : ι)
(h : ¬ i ≤ n) : ((r * of _ i r') n : R) = 0 | begin
classical,
rw [coe_mul_apply_eq_dfinsupp_sum, dfinsupp.sum_comm],
apply (dfinsupp.sum_single_index _).trans, swap,
{ simp_rw [zero_mem_class.coe_zero, mul_zero, if_t_t], exact dfinsupp.sum_zero },
{ rw [dfinsupp.sum, finset.sum_ite_of_false _ _ (λ x _ H, _), finset.sum_const_zero],
exact h ((self_le... | lemma | direct_sum.coe_mul_of_apply_of_not_le | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_of_apply_of_le (r : ⨁ i, A i) {i : ι} (r' : A i) (n : ι)
(h : i ≤ n) : ((r * of _ i r') n : R) = r (n - i) * r' | coe_mul_of_apply_aux _ _ _ (λ x, (eq_tsub_iff_add_eq_of_le h).symm) | lemma | direct_sum.coe_mul_of_apply_of_le | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"eq_tsub_iff_add_eq_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_mul_apply_of_le {i : ι} (r : A i) (r' : ⨁ i, A i) (n : ι)
(h : i ≤ n) : ((of _ i r * r') n : R) = r * r' (n - i) | coe_of_mul_apply_aux _ _ _ (λ x, by rw [eq_tsub_iff_add_eq_of_le h, add_comm]) | lemma | direct_sum.coe_of_mul_apply_of_le | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"eq_tsub_iff_add_eq_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_of_apply (r : ⨁ i, A i) {i : ι} (r' : A i) (n : ι) [decidable (i ≤ n)] :
((r * of _ i r') n : R) = if i ≤ n then r (n - i) * r' else 0 | by { split_ifs, exacts [coe_mul_of_apply_of_le _ _ _ n h, coe_mul_of_apply_of_not_le _ _ _ n h] } | lemma | direct_sum.coe_mul_of_apply | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_mul_apply {i : ι} (r : A i) (r' : ⨁ i, A i) (n : ι) [decidable (i ≤ n)] :
((of _ i r * r') n : R) = if i ≤ n then r * r' (n - i) else 0 | by { split_ifs, exacts [coe_of_mul_apply_of_le _ _ _ n h, coe_of_mul_apply_of_not_le _ _ _ n h] } | lemma | direct_sum.coe_of_mul_apply | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
galgebra [add_monoid ι]
[comm_semiring S] [semiring R] [algebra S R]
(A : ι → submodule S R) [set_like.graded_monoid A] :
direct_sum.galgebra S (λ i, A i) | { to_fun := ((algebra.linear_map S R).cod_restrict (A 0) $
set_like.algebra_map_mem_graded A).to_add_monoid_hom,
map_one := subtype.ext $ by exact (algebra_map S R).map_one,
map_mul := λ x y, sigma.subtype_ext (add_zero 0).symm $ (algebra_map S R).map_mul _ _,
commutes := λ r ⟨i, xi⟩,
sigma.subtype_ext ((... | instance | submodule.galgebra | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"algebra",
"algebra.commutes",
"algebra.linear_map",
"algebra.smul_def",
"algebra_map",
"comm_semiring",
"direct_sum.galgebra",
"map_mul",
"map_one",
"semiring",
"set_like.algebra_map_mem_graded",
"set_like.graded_monoid",
"sigma.subtype_ext",
"submodule",
"subtype.ext"
] | Build a `galgebra` instance for a collection of `submodule`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.coe_galgebra_to_fun [add_monoid ι]
[comm_semiring S] [semiring R] [algebra S R]
(A : ι → submodule S R) [set_like.graded_monoid A] (s : S) :
↑(@direct_sum.galgebra.to_fun _ S (λ i, A i) _ _ _ _ _ _ _ s) = (algebra_map S R s : R) | rfl | lemma | submodule.set_like.coe_galgebra_to_fun | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"algebra",
"algebra_map",
"comm_semiring",
"semiring",
"set_like.graded_monoid",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_power_graded_monoid
[comm_semiring S] [semiring R] [algebra S R] (p : submodule S R) :
set_like.graded_monoid (λ i : ℕ, p ^ i) | { one_mem := by { rw [←one_le, pow_zero], exact le_rfl },
mul_mem := λ i j p q hp hq, by { rw pow_add, exact submodule.mul_mem_mul hp hq } } | instance | submodule.nat_power_graded_monoid | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"algebra",
"comm_semiring",
"le_rfl",
"pow_add",
"pow_zero",
"semiring",
"set_like.graded_monoid",
"submodule",
"submodule.mul_mem_mul"
] | A direct sum of powers of a submodule of an algebra has a multiplicative structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.coe_alg_hom [add_monoid ι]
[comm_semiring S] [semiring R] [algebra S R]
(A : ι → submodule S R) [set_like.graded_monoid A] : (⨁ i, A i) →ₐ[S] R | direct_sum.to_algebra S _ (λ i, (A i).subtype) rfl (λ _ _ _ _, rfl) (λ _, rfl) | def | direct_sum.coe_alg_hom | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"algebra",
"comm_semiring",
"direct_sum.to_algebra",
"semiring",
"set_like.graded_monoid",
"submodule"
] | The canonical algebra isomorphism between `⨁ i, A i` and `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.supr_eq_to_submodule_range [add_monoid ι]
[comm_semiring S] [semiring R] [algebra S R] (A : ι → submodule S R) [set_like.graded_monoid A] :
(⨆ i, A i) = (direct_sum.coe_alg_hom A).range.to_submodule | (submodule.supr_eq_range_dfinsupp_lsum A).trans $ set_like.coe_injective rfl | lemma | submodule.supr_eq_to_submodule_range | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"algebra",
"comm_semiring",
"direct_sum.coe_alg_hom",
"semiring",
"set_like.coe_injective",
"set_like.graded_monoid",
"submodule",
"submodule.supr_eq_range_dfinsupp_lsum"
] | The supremum of submodules that form a graded monoid is a subalgebra, and equal to the range of
`direct_sum.coe_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.coe_alg_hom_of [add_monoid ι]
[comm_semiring S] [semiring R] [algebra S R]
(A : ι → submodule S R) [set_like.graded_monoid A] (i : ι) (x : A i) :
direct_sum.coe_alg_hom A (direct_sum.of (λ i, A i) i x) = x | direct_sum.to_semiring_of _ rfl (λ _ _ _ _, rfl) _ _ | lemma | direct_sum.coe_alg_hom_of | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_monoid",
"algebra",
"comm_semiring",
"direct_sum.coe_alg_hom",
"direct_sum.of",
"direct_sum.to_semiring_of",
"semiring",
"set_like.graded_monoid",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.is_homogeneous_zero_submodule [has_zero ι]
[semiring S] [add_comm_monoid R] [module S R]
(A : ι → submodule S R) : set_like.is_homogeneous A (0 : R) | ⟨0, submodule.zero_mem _⟩ | lemma | set_like.is_homogeneous_zero_submodule | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"add_comm_monoid",
"module",
"semiring",
"set_like.is_homogeneous",
"submodule",
"submodule.zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.is_homogeneous.smul [comm_semiring S] [semiring R] [algebra S R]
{A : ι → submodule S R} {s : S}
{r : R} (hr : set_like.is_homogeneous A r) : set_like.is_homogeneous A (s • r) | let ⟨i, hi⟩ := hr in ⟨i, submodule.smul_mem _ _ hi⟩ | lemma | set_like.is_homogeneous.smul | algebra.direct_sum | src/algebra/direct_sum/internal.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"algebra.direct_sum.algebra"
] | [
"algebra",
"comm_semiring",
"semiring",
"set_like.is_homogeneous",
"submodule",
"submodule.smul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (b : R) (v : ⨁ i, M i) (i : ι) :
(b • v) i = b • (v i) | dfinsupp.smul_apply _ _ _ | lemma | direct_sum.smul_apply | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lmk : Π s : finset ι, (Π i : (↑s : set ι), M i.val) →ₗ[R] (⨁ i, M i) | dfinsupp.lmk | def | direct_sum.lmk | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.lmk",
"finset"
] | Create the direct sum given a family `M` of `R` modules indexed over `ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lof : Π i : ι, M i →ₗ[R] (⨁ i, M i) | dfinsupp.lsingle | def | direct_sum.lof | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.lsingle"
] | Inclusion of each component into the direct sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lof_eq_of (i : ι) (b : M i) : lof R ι M i b = of M i b | rfl | lemma | direct_sum.lof_eq_of | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_eq_lof (i : ι) (b : M i) :
dfinsupp.single i b = lof R ι M i b | rfl | lemma | direct_sum.single_eq_lof | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_smul (s : finset ι) (c : R) (x) : mk M s (c • x) = c • mk M s x | (lmk R ι M s).map_smul c x | theorem | direct_sum.mk_smul | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"finset"
] | Scalar multiplication commutes with direct sums. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_smul (i : ι) (c : R) (x) : of M i (c • x) = c • of M i x | (lof R ι M i).map_smul c x | theorem | direct_sum.of_smul | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | Scalar multiplication commutes with the inclusion of each component into the direct sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
support_smul [Π (i : ι) (x : M i), decidable (x ≠ 0)]
(c : R) (v : ⨁ i, M i) : (c • v).support ⊆ v.support | dfinsupp.support_smul _ _ | lemma | direct_sum.support_smul | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.support_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_module : (⨁ i, M i) →ₗ[R] N | dfinsupp.lsum ℕ φ | def | direct_sum.to_module | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.lsum"
] | The linear map constructed using the universal property of the coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_module_eq_coe_to_add_monoid :
(to_module R ι N φ : (⨁ i, M i) → N) = to_add_monoid (λ i, (φ i).to_add_monoid_hom) | rfl | lemma | direct_sum.coe_to_module_eq_coe_to_add_monoid | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | Coproducts in the categories of modules and additive monoids commute with the forgetful functor
from modules to additive monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_module_lof (i) (x : M i) : to_module R ι N φ (lof R ι M i x) = φ i x | to_add_monoid_of (λ i, (φ i).to_add_monoid_hom) i x | lemma | direct_sum.to_module_lof | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | The map constructed using the universal property gives back the original maps when
restricted to each component. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_module.unique (f : ⨁ i, M i) : ψ f = to_module R ι N (λ i, ψ.comp $ lof R ι M i) f | to_add_monoid.unique ψ.to_add_monoid_hom f | theorem | direct_sum.to_module.unique | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | Every linear map from a direct sum agrees with the one obtained by applying
the universal property to each of its components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map_ext ⦃ψ ψ' : (⨁ i, M i) →ₗ[R] N⦄
(H : ∀ i, ψ.comp (lof R ι M i) = ψ'.comp (lof R ι M i)) : ψ = ψ' | dfinsupp.lhom_ext' H | theorem | direct_sum.linear_map_ext | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.lhom_ext'"
] | Two `linear_map`s out of a direct sum are equal if they agree on the generators.
See note [partially-applied ext lemmas]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lset_to_set (S T : set ι) (H : S ⊆ T) :
(⨁ (i : S), M i) →ₗ[R] (⨁ (i : T), M i) | to_module R _ _ $ λ i, lof R T (λ (i : subtype T), M i) ⟨i, H i.prop⟩ | def | direct_sum.lset_to_set | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | The inclusion of a subset of the direct summands
into a larger subset of the direct summands, as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv_fun_on_fintype [fintype ι] :
(⨁ i, M i) ≃ₗ[R] (Π i, M i) | { to_fun := coe_fn,
map_add' := λ f g, by { ext, simp only [add_apply, pi.add_apply] },
map_smul' := λ c f, by { ext, simp only [dfinsupp.coe_smul, ring_hom.id_apply] },
.. dfinsupp.equiv_fun_on_fintype } | def | direct_sum.linear_equiv_fun_on_fintype | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.coe_smul",
"dfinsupp.equiv_fun_on_fintype",
"fintype",
"ring_hom.id_apply"
] | Given `fintype α`, `linear_equiv_fun_on_fintype R` is the natural `R`-linear equivalence
between `⨁ i, M i` and `Π i, M i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv_fun_on_fintype_lof [fintype ι] [decidable_eq ι] (i : ι) (m : M i) :
(linear_equiv_fun_on_fintype R ι M) (lof R ι M i m) = pi.single i m | begin
ext a,
change (dfinsupp.equiv_fun_on_fintype (lof R ι M i m)) a = _,
convert _root_.congr_fun (dfinsupp.equiv_fun_on_fintype_single i m) a,
end | lemma | direct_sum.linear_equiv_fun_on_fintype_lof | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.equiv_fun_on_fintype",
"dfinsupp.equiv_fun_on_fintype_single",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv_fun_on_fintype_symm_single [fintype ι] [decidable_eq ι]
(i : ι) (m : M i) :
(linear_equiv_fun_on_fintype R ι M).symm (pi.single i m) = lof R ι M i m | begin
ext a,
change (dfinsupp.equiv_fun_on_fintype.symm (pi.single i m)) a = _,
rw (dfinsupp.equiv_fun_on_fintype_symm_single i m),
refl
end | lemma | direct_sum.linear_equiv_fun_on_fintype_symm_single | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.equiv_fun_on_fintype_symm_single",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv_fun_on_fintype_symm_coe [fintype ι] (f : ⨁ i, M i) :
(linear_equiv_fun_on_fintype R ι M).symm f = f | by { ext, simp [linear_equiv_fun_on_fintype], } | lemma | direct_sum.linear_equiv_fun_on_fintype_symm_coe | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lid (M : Type v) (ι : Type* := punit) [add_comm_monoid M] [module R M]
[unique ι] :
(⨁ (_ : ι), M) ≃ₗ[R] M | { .. direct_sum.id M ι,
.. to_module R ι M (λ i, linear_map.id) } | def | direct_sum.lid | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"add_comm_monoid",
"direct_sum.id",
"linear_map.id",
"module",
"unique"
] | The natural linear equivalence between `⨁ _ : ι, M` and `M` when `unique ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
component (i : ι) : (⨁ i, M i) →ₗ[R] M i | dfinsupp.lapply i | def | direct_sum.component | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.lapply"
] | The projection map onto one component, as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_eq_component (f : ⨁ i, M i) (i : ι) :
f i = component R ι M i f | rfl | lemma | direct_sum.apply_eq_component | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : ⨁ i, M i}
(h : ∀ i, component R ι M i f = component R ι M i g) : f = g | dfinsupp.ext h | lemma | direct_sum.ext | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : ⨁ i, M i} : f = g ↔
∀ i, component R ι M i f = component R ι M i g | ⟨λ h _, by rw h, ext R⟩ | lemma | direct_sum.ext_iff | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lof_apply (i : ι) (b : M i) : ((lof R ι M i) b) i = b | dfinsupp.single_eq_same | lemma | direct_sum.lof_apply | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.single_eq_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
component.lof_self (i : ι) (b : M i) :
component R ι M i ((lof R ι M i) b) = b | lof_apply R i b | lemma | direct_sum.component.lof_self | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
component.of (i j : ι) (b : M j) :
component R ι M i ((lof R ι M j) b) =
if h : j = i then eq.rec_on h b else 0 | dfinsupp.single_apply | lemma | direct_sum.component.of | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.single_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lequiv_congr_left (h : ι ≃ κ) : (⨁ i, M i) ≃ₗ[R] ⨁ k, M (h.symm k) | { map_smul' := dfinsupp.comap_domain'_smul _ _,
..equiv_congr_left h } | def | direct_sum.lequiv_congr_left | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.comap_domain'_smul"
] | Reindexing terms of a direct sum is linear. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lequiv_congr_left_apply (h : ι ≃ κ) (f : ⨁ i, M i) (k : κ) :
lequiv_congr_left R h f k = f (h.symm k) | equiv_congr_left_apply _ _ _ | lemma | direct_sum.lequiv_congr_left_apply | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_lcurry : (⨁ (i : Σ i, _), δ i.1 i.2) →ₗ[R] ⨁ i j, δ i j | { map_smul' := λ r, by convert (@dfinsupp.sigma_curry_smul _ _ _ δ _ _ _ r),
..sigma_curry } | def | direct_sum.sigma_lcurry | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.sigma_curry_smul"
] | `curry` as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_lcurry_apply (f : ⨁ (i : Σ i, _), δ i.1 i.2) (i : ι) (j : α i) :
sigma_lcurry R f i j = f ⟨i, j⟩ | sigma_curry_apply f i j | lemma | direct_sum.sigma_lcurry_apply | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_luncurry [Π i, decidable_eq (α i)] [Π i j, decidable_eq (δ i j)] :
(⨁ i j, δ i j) →ₗ[R] ⨁ (i : Σ i, _), δ i.1 i.2 | { map_smul' := dfinsupp.sigma_uncurry_smul,
..sigma_uncurry } | def | direct_sum.sigma_luncurry | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.sigma_uncurry_smul"
] | `uncurry` as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_luncurry_apply [Π i, decidable_eq (α i)] [Π i j, decidable_eq (δ i j)]
(f : ⨁ i j, δ i j) (i : ι) (j : α i) :
sigma_luncurry R f ⟨i, j⟩ = f i j | sigma_uncurry_apply f i j | lemma | direct_sum.sigma_luncurry_apply | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_lcurry_equiv
[Π i, decidable_eq (α i)] [Π i j, decidable_eq (δ i j)] :
(⨁ (i : Σ i, _), δ i.1 i.2) ≃ₗ[R] ⨁ i j, δ i j | { ..sigma_curry_equiv, ..sigma_lcurry R } | def | direct_sum.sigma_lcurry_equiv | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | `curry_equiv` as a linear equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lequiv_prod_direct_sum : (⨁ i, α i) ≃ₗ[R] α none × ⨁ i, α (some i) | { map_smul' := dfinsupp.equiv_prod_dfinsupp_smul,
..add_equiv_prod_direct_sum } | def | direct_sum.lequiv_prod_direct_sum | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"dfinsupp.equiv_prod_dfinsupp_smul"
] | Linear isomorphism obtained by separating the term of index `none` of a direct sum over
`option ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_linear_map : (⨁ i, A i) →ₗ[R] M | to_module R ι M (λ i, (A i).subtype) | def | direct_sum.coe_linear_map | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [] | The canonical embedding from `⨁ i, A i` to `M` where `A` is a collection of `submodule R M`
indexed by `ι`. This is `direct_sum.coe_add_monoid_hom` as a `linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_linear_map_of (i : ι) (x : A i) :
direct_sum.coe_linear_map A (of (λ i, A i) i x) = x | to_add_monoid_of _ _ _ | lemma | direct_sum.coe_linear_map_of | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"direct_sum.coe_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_internal.submodule_supr_eq_top (h : is_internal A) : supr A = ⊤ | begin
rw [submodule.supr_eq_range_dfinsupp_lsum, linear_map.range_eq_top],
exact function.bijective.surjective h,
end | lemma | direct_sum.is_internal.submodule_supr_eq_top | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"function.bijective.surjective",
"linear_map.range_eq_top",
"submodule.supr_eq_range_dfinsupp_lsum",
"supr"
] | If a direct sum of submodules is internal then the submodules span the module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_internal.submodule_independent (h : is_internal A) :
complete_lattice.independent A | complete_lattice.independent_of_dfinsupp_lsum_injective _ h.injective | lemma | direct_sum.is_internal.submodule_independent | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"complete_lattice.independent",
"complete_lattice.independent_of_dfinsupp_lsum_injective"
] | If a direct sum of submodules is internal then the submodules are independent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_internal.collected_basis
(h : is_internal A) {α : ι → Type*} (v : Π i, basis (α i) R (A i)) :
basis (Σ i, α i) R M | { repr :=
(linear_equiv.of_bijective (direct_sum.coe_linear_map A) h).symm ≪≫ₗ
(dfinsupp.map_range.linear_equiv (λ i, (v i).repr)) ≪≫ₗ
(sigma_finsupp_lequiv_dfinsupp R).symm } | def | direct_sum.is_internal.collected_basis | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"basis",
"dfinsupp.map_range.linear_equiv",
"direct_sum.coe_linear_map",
"linear_equiv.of_bijective",
"sigma_finsupp_lequiv_dfinsupp"
] | Given an internal direct sum decomposition of a module `M`, and a basis for each of the
components of the direct sum, the disjoint union of these bases is a basis for `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_internal.collected_basis_coe
(h : is_internal A) {α : ι → Type*} (v : Π i, basis (α i) R (A i)) :
⇑(h.collected_basis v) = λ a : Σ i, (α i), ↑(v a.1 a.2) | begin
funext a,
simp only [is_internal.collected_basis, to_module, coe_linear_map,
add_equiv.to_fun_eq_coe, basis.coe_of_repr, basis.repr_symm_apply, dfinsupp.lsum_apply_apply,
dfinsupp.map_range.linear_equiv_apply, dfinsupp.map_range.linear_equiv_symm,
dfinsupp.map_range_single, finsupp.total_single, l... | lemma | direct_sum.is_internal.collected_basis_coe | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"basis",
"basis.coe_of_repr",
"basis.repr_symm_apply",
"dfinsupp.map_range.linear_equiv_symm",
"dfinsupp.map_range_single",
"dfinsupp.sum_add_hom_single",
"finsupp.total_single",
"linear_equiv.of_bijective_apply",
"linear_equiv.symm_symm",
"linear_equiv.symm_trans_apply",
"one_smul",
"sigma_fi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_internal.collected_basis_mem
(h : is_internal A) {α : ι → Type*} (v : Π i, basis (α i) R (A i)) (a : Σ i, α i) :
h.collected_basis v a ∈ A a.1 | by simp | lemma | direct_sum.is_internal.collected_basis_mem | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_internal.is_compl {A : ι → submodule R M} {i j : ι} (hij : i ≠ j)
(h : (set.univ : set ι) = {i, j}) (hi : is_internal A) : is_compl (A i) (A j) | ⟨hi.submodule_independent.pairwise_disjoint hij,
codisjoint_iff.mpr $ eq.symm $ hi.submodule_supr_eq_top.symm.trans $
by rw [←Sup_pair, supr, ←set.image_univ, h, set.image_insert_eq, set.image_singleton]⟩ | lemma | direct_sum.is_internal.is_compl | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"is_compl",
"set.image_insert_eq",
"set.image_singleton",
"submodule",
"supr"
] | When indexed by only two distinct elements, `direct_sum.is_internal` implies
the two submodules are complementary. Over a `ring R`, this is true as an iff, as
`direct_sum.is_internal_iff_is_compl`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_internal_submodule_of_independent_of_supr_eq_top {A : ι → submodule R M}
(hi : complete_lattice.independent A) (hs : supr A = ⊤) : is_internal A | ⟨hi.dfinsupp_lsum_injective, linear_map.range_eq_top.1 $
(submodule.supr_eq_range_dfinsupp_lsum _).symm.trans hs⟩ | lemma | direct_sum.is_internal_submodule_of_independent_of_supr_eq_top | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"complete_lattice.independent",
"submodule",
"submodule.supr_eq_range_dfinsupp_lsum",
"supr"
] | Note that this is not generally true for `[semiring R]`; see
`complete_lattice.independent.dfinsupp_lsum_injective` for details. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_internal_submodule_iff_independent_and_supr_eq_top (A : ι → submodule R M) :
is_internal A ↔ complete_lattice.independent A ∧ supr A = ⊤ | ⟨λ i, ⟨i.submodule_independent, i.submodule_supr_eq_top⟩,
and.rec is_internal_submodule_of_independent_of_supr_eq_top⟩ | lemma | direct_sum.is_internal_submodule_iff_independent_and_supr_eq_top | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"complete_lattice.independent",
"submodule",
"supr"
] | `iff` version of `direct_sum.is_internal_submodule_of_independent_of_supr_eq_top`,
`direct_sum.is_internal.independent`, and `direct_sum.is_internal.supr_eq_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_internal_submodule_iff_is_compl (A : ι → submodule R M) {i j : ι} (hij : i ≠ j)
(h : (set.univ : set ι) = {i, j}) :
is_internal A ↔ is_compl (A i) (A j) | begin
have : ∀ k, k = i ∨ k = j := λ k, by simpa using set.ext_iff.mp h k,
rw [is_internal_submodule_iff_independent_and_supr_eq_top,
supr, ←set.image_univ, h, set.image_insert_eq, set.image_singleton, Sup_pair,
complete_lattice.independent_pair hij this],
exact ⟨λ ⟨hd, ht⟩, ⟨hd, codisjoint_iff.mpr ht⟩, λ... | lemma | direct_sum.is_internal_submodule_iff_is_compl | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"Sup_pair",
"complete_lattice.independent_pair",
"is_compl",
"set.image_insert_eq",
"set.image_singleton",
"submodule",
"supr"
] | If a collection of submodules has just two indices, `i` and `j`, then
`direct_sum.is_internal` is equivalent to `is_compl`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_internal.add_submonoid_independent {M : Type*} [add_comm_monoid M]
{A : ι → add_submonoid M} (h : is_internal A) :
complete_lattice.independent A | complete_lattice.independent_of_dfinsupp_sum_add_hom_injective _ h.injective | lemma | direct_sum.is_internal.add_submonoid_independent | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"add_comm_monoid",
"add_submonoid",
"complete_lattice.independent",
"complete_lattice.independent_of_dfinsupp_sum_add_hom_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_internal.add_subgroup_independent {M : Type*} [add_comm_group M]
{A : ι → add_subgroup M} (h : is_internal A) :
complete_lattice.independent A | complete_lattice.independent_of_dfinsupp_sum_add_hom_injective' _ h.injective | lemma | direct_sum.is_internal.add_subgroup_independent | algebra.direct_sum | src/algebra/direct_sum/module.lean | [
"algebra.direct_sum.basic",
"linear_algebra.dfinsupp"
] | [
"add_comm_group",
"add_subgroup",
"complete_lattice.independent",
"complete_lattice.independent_of_dfinsupp_sum_add_hom_injective'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gnon_unital_non_assoc_semiring [has_add ι] [Π i, add_comm_monoid (A i)] extends
graded_monoid.ghas_mul A | (mul_zero : ∀ {i j} (a : A i), mul a (0 : A j) = 0)
(zero_mul : ∀ {i j} (b : A j), mul (0 : A i) b = 0)
(mul_add : ∀ {i j} (a : A i) (b c : A j), mul a (b + c) = mul a b + mul a c)
(add_mul : ∀ {i j} (a b : A i) (c : A j), mul (a + b) c = mul a c + mul b c) | class | direct_sum.gnon_unital_non_assoc_semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_comm_monoid",
"graded_monoid.ghas_mul",
"mul_zero",
"zero_mul"
] | A graded version of `non_unital_non_assoc_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gsemiring [add_monoid ι] [Π i, add_comm_monoid (A i)] extends
gnon_unital_non_assoc_semiring A, graded_monoid.gmonoid A | (nat_cast : ℕ → A 0)
(nat_cast_zero : nat_cast 0 = 0)
(nat_cast_succ : ∀ n : ℕ, nat_cast (n + 1) = nat_cast n + graded_monoid.ghas_one.one) | class | direct_sum.gsemiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_comm_monoid",
"add_monoid",
"graded_monoid.gmonoid"
] | A graded version of `semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcomm_semiring [add_comm_monoid ι] [Π i, add_comm_monoid (A i)] extends
gsemiring A, graded_monoid.gcomm_monoid A | class | direct_sum.gcomm_semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_comm_monoid",
"graded_monoid.gcomm_monoid"
] | A graded version of `comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gring [add_monoid ι] [Π i, add_comm_group (A i)] extends gsemiring A | (int_cast : ℤ → A 0)
(int_cast_of_nat : ∀ n : ℕ, int_cast n = nat_cast n)
(int_cast_neg_succ_of_nat : ∀ n : ℕ, int_cast (-(n+1 : ℕ)) = -nat_cast (n+1 : ℕ)) | class | direct_sum.gring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_comm_group",
"add_monoid"
] | A graded version of `ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcomm_ring [add_comm_monoid ι] [Π i, add_comm_group (A i)] extends
gring A, gcomm_semiring A | class | direct_sum.gcomm_ring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_comm_group",
"add_comm_monoid"
] | A graded version of `comm_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_eq_of_graded_monoid_eq {A : ι → Type*} [Π (i : ι), add_comm_monoid (A i)]
{i j : ι} {a : A i} {b : A j} (h : graded_monoid.mk i a = graded_monoid.mk j b) :
direct_sum.of A i a = direct_sum.of A j b | dfinsupp.single_eq_of_sigma_eq h | lemma | direct_sum.of_eq_of_graded_monoid_eq | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_comm_monoid",
"dfinsupp.single_eq_of_sigma_eq",
"direct_sum.of",
"graded_monoid.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gmul_hom {i j} : A i →+ A j →+ A (i + j) | { to_fun := λ a,
{ to_fun := λ b, graded_monoid.ghas_mul.mul a b,
map_zero' := gnon_unital_non_assoc_semiring.mul_zero _,
map_add' := gnon_unital_non_assoc_semiring.mul_add _ },
map_zero' := add_monoid_hom.ext $ λ a, gnon_unital_non_assoc_semiring.zero_mul a,
map_add' := λ a₁ a₂, add_monoid_hom.ext $ λ b,... | def | direct_sum.gmul_hom | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | The piecewise multiplication from the `has_mul` instance, as a bundled homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom : (⨁ i, A i) →+ (⨁ i, A i) →+ ⨁ i, A i | direct_sum.to_add_monoid $ λ i,
add_monoid_hom.flip $ direct_sum.to_add_monoid $ λ j, add_monoid_hom.flip $
(direct_sum.of A _).comp_hom.comp $ gmul_hom A | def | direct_sum.mul_hom | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"direct_sum.of",
"direct_sum.to_add_monoid",
"mul_hom"
] | The multiplication from the `has_mul` instance, as a bundled homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom_of_of {i j} (a : A i) (b : A j) :
mul_hom A (of _ i a) (of _ j b) = of _ (i + j) (graded_monoid.ghas_mul.mul a b) | begin
unfold mul_hom,
rw [to_add_monoid_of, flip_apply, to_add_monoid_of, flip_apply, coe_comp, function.comp_app,
comp_hom_apply_apply, coe_comp, function.comp_app, gmul_hom_apply_apply],
end | lemma | direct_sum.mul_hom_of_of | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"mul_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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