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
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|---|---|---|---|---|---|---|---|---|---|---|
algebra_map_eq {R A : Type*} [comm_semiring R] [comm_semiring A] [semiring α]
[algebra R A] [algebra A α] (S : subalgebra R A) :
algebra_map S α = (algebra_map A α).comp S.val | rfl | lemma | subalgebra.algebra_map_eq | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_algebra_map {R A : Type*} [comm_semiring R] [comm_semiring A]
[algebra R A] (S : subalgebra R A) :
(algebra_map S A).srange = S.to_subsemiring | by rw [algebra_map_eq, algebra.id.map_eq_id, ring_hom.id_comp, ← to_subsemiring_subtype,
subsemiring.srange_subtype] | lemma | subalgebra.srange_algebra_map | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"algebra.id.map_eq_id",
"algebra_map",
"comm_semiring",
"ring_hom.id_comp",
"subalgebra",
"subsemiring.srange_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_algebra_map {R A : Type*} [comm_ring R] [comm_ring A]
[algebra R A] (S : subalgebra R A) :
(algebra_map S A).range = S.to_subring | by rw [algebra_map_eq, algebra.id.map_eq_id, ring_hom.id_comp, ← to_subring_subtype,
subring.range_subtype] | lemma | subalgebra.range_algebra_map | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"algebra.id.map_eq_id",
"algebra_map",
"comm_ring",
"ring_hom.id_comp",
"subalgebra",
"subring.range_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_smul_divisors_top [no_zero_divisors A] (S : subalgebra R A) :
no_zero_smul_divisors S A | ⟨λ c x h,
have (c : A) = 0 ∨ x = 0,
from eq_zero_or_eq_zero_of_mul_eq_zero h,
this.imp_left (@subtype.ext_iff _ _ c 0).mpr⟩ | instance | subalgebra.no_zero_smul_divisors_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"no_zero_divisors",
"no_zero_smul_divisors",
"subalgebra",
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.algebra_map_mem_center (r : R) : algebra_map R A r ∈ set.center A | by simp [algebra.commutes, set.mem_center_iff] | lemma | set.algebra_map_mem_center | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra.commutes",
"algebra_map",
"set.center",
"set.mem_center_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center : subalgebra R A | { algebra_map_mem' := set.algebra_map_mem_center,
.. subsemiring.center A } | def | subalgebra.center | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.algebra_map_mem_center",
"subalgebra",
"subsemiring.center"
] | The center of an algebra is the set of elements which commute with every element. They form a
subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_center : (center R A : set A) = set.center A | rfl | lemma | subalgebra.coe_center | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.center"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_to_subsemiring :
(center R A).to_subsemiring = subsemiring.center A | rfl | lemma | subalgebra.center_to_subsemiring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subsemiring.center"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_to_subring (R A : Type*) [comm_ring R] [ring A] [algebra R A] :
(center R A).to_subring = subring.center A | rfl | lemma | subalgebra.center_to_subring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"comm_ring",
"ring",
"subring.center"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_eq_top (A : Type*) [comm_semiring A] [algebra R A] : center R A = ⊤ | set_like.coe_injective (set.center_eq_univ A) | lemma | subalgebra.center_eq_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"comm_semiring",
"set.center_eq_univ",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_center_iff {a : A} : a ∈ center R A ↔ ∀ (b : A), b*a = a*b | iff.rfl | lemma | subalgebra.mem_center_iff | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.algebra_map_mem_centralizer
{s : set A} (r : R) : algebra_map R A r ∈ s.centralizer | λ a h, (algebra.commutes _ _).symm | lemma | set.algebra_map_mem_centralizer | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra.commutes",
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer (s : set A) : subalgebra R A | { algebra_map_mem' := set.algebra_map_mem_centralizer,
..subsemiring.centralizer s, } | def | subalgebra.centralizer | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.algebra_map_mem_centralizer",
"subalgebra",
"subsemiring.centralizer"
] | The centralizer of a set as a subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_centralizer (s : set A) : (centralizer R s : set A) = s.centralizer | rfl | lemma | subalgebra.coe_centralizer | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_centralizer_iff {s : set A} {z : A} :
z ∈ centralizer R s ↔ ∀ g ∈ s, g * z = z * g | iff.rfl | lemma | subalgebra.mem_centralizer_iff | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_le_centralizer (s) : center R A ≤ centralizer R s | s.center_subset_centralizer | lemma | subalgebra.center_le_centralizer | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_le (s t : set A) (h : s ⊆ t) :
centralizer R t ≤ centralizer R s | set.centralizer_subset h | lemma | subalgebra.centralizer_le | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.centralizer_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_eq_top_iff_subset {s : set A} : centralizer R s = ⊤ ↔ s ⊆ center R A | set_like.ext'_iff.trans set.centralizer_eq_top_iff_subset | lemma | subalgebra.centralizer_eq_top_iff_subset | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.centralizer_eq_top_iff_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_univ : centralizer R set.univ = center R A | set_like.ext' (set.centralizer_univ A) | lemma | subalgebra.centralizer_univ | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"set.centralizer_univ",
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_finset_sum_eq_one_of_pow_smul_mem {S : Type*} [comm_ring S]
[algebra R S] (S' : subalgebra R S) {ι : Type*} (ι' : finset ι) (s : ι → S) (l : ι → S)
(e : ∑ i in ι', l i * s i = 1)
(hs : ∀ i, s i ∈ S') (hl : ∀ i, l i ∈ S') (x : S)
(H : ∀ i, ∃ (n : ℕ), (s i ^ n : S) • x ∈ S') : x ∈ S' | begin
classical,
suffices : x ∈ (algebra.of_id S' S).range.to_submodule,
{ obtain ⟨x, rfl⟩ := this, exact x.2 },
choose n hn using H,
let s' : ι → S' := λ x, ⟨s x, hs x⟩,
have : ideal.span (s' '' ι')= ⊤,
{ rw [ideal.eq_top_iff_one, ideal.span, finsupp.mem_span_iff_total],
refine ⟨(finsupp.of_support_f... | lemma | subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"algebra.of_id",
"comm_ring",
"finset",
"finset.le_sup",
"finsupp.mem_span_iff_total",
"finsupp.of_support_finite",
"finsupp.total_apply",
"finsupp.total_map_domain",
"ideal.eq_top_iff_one",
"ideal.span",
"ideal.span_pow_eq_top",
"pow_add",
"pow_mem",
"set.to_finite",
"subal... | Suppose we are given `∑ i, lᵢ * sᵢ = 1` in `S`, and `S'` a subalgebra of `S` that contains
`lᵢ` and `sᵢ`. To check that an `x : S` falls in `S'`, we only need to show that
`r ^ n • x ∈ M'` for some `n` for each `r : s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_of_span_eq_top_of_smul_pow_mem {S : Type*} [comm_ring S] [algebra R S]
(S' : subalgebra R S) (s : set S) (l : s →₀ S) (hs : finsupp.total s S S coe l = 1)
(hs' : s ⊆ S') (hl : ∀ i, l i ∈ S') (x : S)
(H : ∀ r : s, ∃ (n : ℕ), (r ^ n : S) • x ∈ S') : x ∈ S' | mem_of_finset_sum_eq_one_of_pow_smul_mem S' l.support coe l hs (λ x, hs' x.2) hl x H | lemma | subalgebra.mem_of_span_eq_top_of_smul_pow_mem | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"algebra",
"comm_ring",
"finsupp.total",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra_of_subsemiring (S : subsemiring R) : subalgebra ℕ R | { algebra_map_mem' := λ i, coe_nat_mem S i,
.. S } | def | subalgebra_of_subsemiring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"coe_nat_mem",
"subalgebra",
"subsemiring"
] | A subsemiring is a `ℕ`-subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_subalgebra_of_subsemiring {x : R} {S : subsemiring R} :
x ∈ subalgebra_of_subsemiring S ↔ x ∈ S | iff.rfl | lemma | mem_subalgebra_of_subsemiring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra_of_subsemiring",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra_of_subring (S : subring R) : subalgebra ℤ R | { algebra_map_mem' := λ i, int.induction_on i (by simpa using S.zero_mem)
(λ i ih, by simpa using S.add_mem ih S.one_mem)
(λ i ih, show ((-i - 1 : ℤ) : R) ∈ S, by { rw [int.cast_sub, int.cast_one],
exact S.sub_mem ih S.one_mem }),
.. S } | def | subalgebra_of_subring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"ih",
"int.cast_one",
"int.cast_sub",
"int.induction_on",
"subalgebra",
"subring"
] | A subring is a `ℤ`-subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_subalgebra_of_subring {x : R} {S : subring R} :
x ∈ subalgebra_of_subring S ↔ x ∈ S | iff.rfl | lemma | mem_subalgebra_of_subring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/basic.lean | [
"algebra.algebra.basic",
"data.set.Union_lift",
"linear_algebra.finsupp",
"ring_theory.ideal.operations"
] | [
"subalgebra_of_subring",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_to_submodule_le (S T : subalgebra R A) :
S.to_submodule * T.to_submodule ≤ (S ⊔ T).to_submodule | begin
rw submodule.mul_le,
intros y hy z hz,
show y * z ∈ (S ⊔ T),
exact mul_mem (algebra.mem_sup_left hy) (algebra.mem_sup_right hz),
end | theorem | subalgebra.mul_to_submodule_le | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/pointwise.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"ring_theory.subring.pointwise",
"ring_theory.adjoin.basic"
] | [
"algebra.mem_sup_left",
"algebra.mem_sup_right",
"subalgebra",
"submodule.mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self (S : subalgebra R A) : S.to_submodule * S.to_submodule = S.to_submodule | begin
apply le_antisymm,
{ refine (mul_to_submodule_le _ _).trans_eq _,
rw sup_idem },
{ intros x hx1,
rw ← mul_one x,
exact submodule.mul_mem_mul hx1 (show (1 : A) ∈ S, from one_mem S) }
end | theorem | subalgebra.mul_self | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/pointwise.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"ring_theory.subring.pointwise",
"ring_theory.adjoin.basic"
] | [
"mul_one",
"mul_self",
"subalgebra",
"submodule.mul_mem_mul",
"sup_idem"
] | As submodules, subalgebras are idempotent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_to_submodule {R : Type*} {A : Type*} [comm_semiring R] [comm_semiring A] [algebra R A]
(S T : subalgebra R A) :
S.to_submodule * T.to_submodule = (S ⊔ T).to_submodule | begin
refine le_antisymm (mul_to_submodule_le _ _) _,
rintros x (hx : x ∈ algebra.adjoin R (S ∪ T : set A)),
refine algebra.adjoin_induction hx (λ x hx, _) (λ r, _) (λ _ _, submodule.add_mem _)
(λ x y hx hy, _),
{ cases hx with hxS hxT,
{ rw ← mul_one x,
exact submodule.mul_mem_mul hxS (show (1 : ... | theorem | subalgebra.mul_to_submodule | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/pointwise.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"ring_theory.subring.pointwise",
"ring_theory.adjoin.basic"
] | [
"algebra",
"algebra.adjoin",
"algebra.adjoin_induction",
"algebra_map",
"comm_semiring",
"mul_assoc",
"mul_comm",
"mul_one",
"mul_self",
"one_mul",
"subalgebra",
"submodule.add_mem",
"submodule.mul_mem_mul"
] | When `A` is commutative, `subalgebra.mul_to_submodule_le` is strict. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pointwise_mul_action : mul_action R' (subalgebra R A) | { smul := λ a S, S.map (mul_semiring_action.to_alg_hom _ _ a),
one_smul := λ S,
(congr_arg (λ f, S.map f) (alg_hom.ext $ by exact one_smul R')).trans S.map_id,
mul_smul := λ a₁ a₂ S,
(congr_arg (λ f, S.map f) (alg_hom.ext $ by exact mul_smul _ _)).trans (S.map_map _ _).symm } | def | subalgebra.pointwise_mul_action | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/pointwise.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"ring_theory.subring.pointwise",
"ring_theory.adjoin.basic"
] | [
"alg_hom.ext",
"mul_action",
"mul_semiring_action.to_alg_hom",
"one_smul",
"subalgebra"
] | The action on a subalgebra corresponding to applying the action to every element.
This is available as an instance in the `pointwise` locale. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pointwise_smul (m : R') (S : subalgebra R A) : ↑(m • S) = m • (S : set A) | rfl | lemma | subalgebra.coe_pointwise_smul | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/pointwise.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"ring_theory.subring.pointwise",
"ring_theory.adjoin.basic"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_to_subsemiring (m : R') (S : subalgebra R A) :
(m • S).to_subsemiring = m • S.to_subsemiring | rfl | lemma | subalgebra.pointwise_smul_to_subsemiring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/pointwise.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"ring_theory.subring.pointwise",
"ring_theory.adjoin.basic"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_to_submodule (m : R') (S : subalgebra R A) :
(m • S).to_submodule = m • S.to_submodule | rfl | lemma | subalgebra.pointwise_smul_to_submodule | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/pointwise.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"ring_theory.subring.pointwise",
"ring_theory.adjoin.basic"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_to_subring {R' R A : Type*} [semiring R'] [comm_ring R] [ring A]
[mul_semiring_action R' A] [algebra R A] [smul_comm_class R' R A] (m : R') (S : subalgebra R A) :
(m • S).to_subring = m • S.to_subring | rfl | lemma | subalgebra.pointwise_smul_to_subring | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/pointwise.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"ring_theory.subring.pointwise",
"ring_theory.adjoin.basic"
] | [
"algebra",
"comm_ring",
"mul_semiring_action",
"ring",
"semiring",
"smul_comm_class",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_pointwise_smul (m : R') (r : A) (S : subalgebra R A) : r ∈ S → m • r ∈ m • S | (set.smul_mem_smul_set : _ → _ ∈ m • (S : set A)) | lemma | subalgebra.smul_mem_pointwise_smul | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/pointwise.lean | [
"algebra.algebra.operations",
"algebra.algebra.subalgebra.basic",
"ring_theory.subring.pointwise",
"ring_theory.adjoin.basic"
] | [
"set.smul_mem_smul_set",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lmul_algebra_map (x : R) :
algebra.lmul R A (algebra_map R A x) = algebra.lsmul R A x | eq.symm $ linear_map.ext $ smul_def x | lemma | algebra.lmul_algebra_map | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"algebra.lmul",
"algebra.lsmul",
"algebra_map",
"linear_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra (S₀ : subalgebra R S) : is_scalar_tower S₀ S A | of_algebra_map_eq $ λ x, rfl | instance | is_scalar_tower.subalgebra | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"is_scalar_tower",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra' (S₀ : subalgebra R S) : is_scalar_tower R S₀ A | @is_scalar_tower.of_algebra_map_eq R S₀ A _ _ _ _ _ _ $ λ _,
(is_scalar_tower.algebra_map_apply R S A _ : _) | instance | is_scalar_tower.subalgebra' | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply",
"is_scalar_tower.of_algebra_map_eq",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars (U : subalgebra S A) : subalgebra R A | { algebra_map_mem' := λ x, by { rw algebra_map_apply R S A, exact U.algebra_map_mem _ },
.. U } | def | subalgebra.restrict_scalars | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"algebra_map_apply",
"restrict_scalars",
"subalgebra"
] | Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret
`U` as an `R`-subalgebra of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_scalars {U : subalgebra S A} :
(restrict_scalars R U : set A) = (U : set A) | rfl | lemma | subalgebra.coe_restrict_scalars | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"restrict_scalars",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_top : restrict_scalars R (⊤ : subalgebra S A) = ⊤ | set_like.coe_injective rfl | lemma | subalgebra.restrict_scalars_top | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"restrict_scalars",
"set_like.coe_injective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_to_submodule {U : subalgebra S A} :
(U.restrict_scalars R).to_submodule = U.to_submodule.restrict_scalars R | set_like.coe_injective rfl | lemma | subalgebra.restrict_scalars_to_submodule | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"set_like.coe_injective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_restrict_scalars {U : subalgebra S A} {x : A} :
x ∈ restrict_scalars R U ↔ x ∈ U | iff.rfl | lemma | subalgebra.mem_restrict_scalars | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"restrict_scalars",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_injective :
function.injective (restrict_scalars R : subalgebra S A → subalgebra R A) | λ U V H, ext $ λ x, by rw [← mem_restrict_scalars R, H, mem_restrict_scalars] | lemma | subalgebra.restrict_scalars_injective | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"restrict_scalars",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_restrict_scalars (U : subalgebra S A) (f : U →ₐ[S] B) : U.restrict_scalars R →ₐ[R] B | f.restrict_scalars R | def | subalgebra.of_restrict_scalars | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"subalgebra"
] | Produces an `R`-algebra map from `U.restrict_scalars R` given an `S`-algebra map from `U`.
This is a special case of `alg_hom.restrict_scalars` that can be helpful in elaboration. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_range_to_alg_hom (t : set A) :
(algebra.adjoin (to_alg_hom R S A).range t).restrict_scalars R =
(algebra.adjoin S t).restrict_scalars R | subalgebra.ext $ λ z,
show z ∈ subsemiring.closure (set.range (algebra_map (to_alg_hom R S A).range A) ∪ t : set A) ↔
z ∈ subsemiring.closure (set.range (algebra_map S A) ∪ t : set A),
from suffices set.range (algebra_map (to_alg_hom R S A).range A) = set.range (algebra_map S A),
by rw this,
by { ext z, exact ⟨λ ⟨⟨... | theorem | is_scalar_tower.adjoin_range_to_alg_hom | algebra.algebra.subalgebra | src/algebra/algebra/subalgebra/tower.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower"
] | [
"algebra.adjoin",
"algebra_map",
"restrict_scalars",
"set.range",
"subalgebra.ext",
"subsemiring.closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_multiset_dvd {s : multiset α} :
p ∣ s.prod → ∃ a ∈ s, p ∣ a | multiset.induction_on s (λ h, (hp.not_dvd_one h).elim) $
λ a s ih h,
have p ∣ a * s.prod, by simpa using h,
match hp.dvd_or_dvd this with
| or.inl h := ⟨a, multiset.mem_cons_self a s, h⟩
| or.inr h := let ⟨a, has, h⟩ := ih h in ⟨a, multiset.mem_cons_of_mem has, h⟩
end | lemma | prime.exists_mem_multiset_dvd | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"ih",
"multiset",
"multiset.induction_on",
"multiset.mem_cons_of_mem",
"multiset.mem_cons_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_multiset_map_dvd {s : multiset β} {f : β → α} :
p ∣ (s.map f).prod → ∃ a ∈ s, p ∣ f a | λ h, by simpa only [exists_prop, multiset.mem_map, exists_exists_and_eq_and]
using hp.exists_mem_multiset_dvd h | lemma | prime.exists_mem_multiset_map_dvd | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"exists_exists_and_eq_and",
"exists_prop",
"multiset",
"multiset.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_finset_dvd {s : finset β} {f : β → α} :
p ∣ s.prod f → ∃ i ∈ s, p ∣ f i | hp.exists_mem_multiset_map_dvd | lemma | prime.exists_mem_finset_dvd | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_associated_mem_of_dvd_prod [cancel_comm_monoid_with_zero α] {p : α}
(hp : prime p) {s : multiset α} : (∀ r ∈ s, prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q | multiset.induction_on s (by simp [mt is_unit_iff_dvd_one.2 hp.not_unit])
(λ a s ih hs hps, begin
rw [multiset.prod_cons] at hps,
cases hp.dvd_or_dvd hps with h h,
{ have hap := hs a (multiset.mem_cons.2 (or.inl rfl)),
exact ⟨a, multiset.mem_cons_self a _, hp.associated_of_dvd hap h⟩ },
{ rcases ... | lemma | exists_associated_mem_of_dvd_prod | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"cancel_comm_monoid_with_zero",
"ih",
"multiset",
"multiset.induction_on",
"multiset.mem_cons_self",
"multiset.prod_cons",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset.prod_primes_dvd
[cancel_comm_monoid_with_zero α] [Π a : α, decidable_pred (associated a)]
{s : multiset α} (n : α) (h : ∀ a ∈ s, prime a) (div : ∀ a ∈ s, a ∣ n)
(uniq : ∀ a, s.countp (associated a) ≤ 1) :
s.prod ∣ n | begin
induction s using multiset.induction_on with a s induct n primes divs generalizing n,
{ simp only [multiset.prod_zero, one_dvd] },
{ rw multiset.prod_cons,
obtain ⟨k, rfl⟩ : a ∣ n := div a (multiset.mem_cons_self a s),
apply mul_dvd_mul_left a,
refine induct
(λ a ha, h a (multiset.mem_cons... | lemma | multiset.prod_primes_dvd | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"associated",
"associated.refl",
"cancel_comm_monoid_with_zero",
"mul_dvd_mul_left",
"multiset",
"multiset.countp_cons_of_pos",
"multiset.countp_le_of_le",
"multiset.countp_pos",
"multiset.induction_on",
"multiset.le_cons_self",
"multiset.mem_cons_of_mem",
"multiset.mem_cons_self",
"multiset... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.prod_primes_dvd
[cancel_comm_monoid_with_zero α] [unique αˣ]
{s : finset α} (n : α) (h : ∀ a ∈ s, prime a) (div : ∀ a ∈ s, a ∣ n) :
(∏ p in s, p) ∣ n | begin
classical,
exact multiset.prod_primes_dvd n
(by simpa only [multiset.map_id', finset.mem_def] using h)
(by simpa only [multiset.map_id', finset.mem_def] using div)
(by simp only [multiset.map_id', associated_eq_eq, multiset.countp_eq_card_filter,
←multiset.count_eq_card_filter_eq, ←multise... | lemma | finset.prod_primes_dvd | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"associated_eq_eq",
"cancel_comm_monoid_with_zero",
"finset",
"finset.mem_def",
"multiset.countp_eq_card_filter",
"multiset.map_id'",
"multiset.prod_primes_dvd",
"prime",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mk {p : multiset α} : (p.map associates.mk).prod = associates.mk p.prod | multiset.induction_on p (by simp) $ λ a s ih, by simp [ih, associates.mk_mul_mk] | theorem | associates.prod_mk | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"associates.mk",
"associates.mk_mul_mk",
"ih",
"multiset",
"multiset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_prod_mk {p : finset β} {f : β → α} :
∏ i in p, associates.mk (f i) = associates.mk (∏ i in p, f i) | by rw [finset.prod_eq_multiset_prod, ← multiset.map_map, prod_mk, ← finset.prod_eq_multiset_prod] | theorem | associates.finset_prod_mk | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"associates.mk",
"finset",
"finset.prod_eq_multiset_prod",
"multiset.map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_associated_iff_map_eq_map {p q : multiset α} :
multiset.rel associated p q ↔ p.map associates.mk = q.map associates.mk | by { rw [← multiset.rel_eq, multiset.rel_map], simp only [mk_eq_mk_iff_associated] } | theorem | associates.rel_associated_iff_map_eq_map | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"associated",
"associates.mk",
"multiset",
"multiset.rel",
"multiset.rel_eq",
"multiset.rel_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_one_iff {p : multiset (associates α)} :
p.prod = 1 ↔ (∀a ∈ p, (a:associates α) = 1) | multiset.induction_on p
(by simp)
(by simp [mul_eq_one_iff, or_imp_distrib, forall_and_distrib] {contextual := tt}) | theorem | associates.prod_eq_one_iff | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"associates",
"forall_and_distrib",
"mul_eq_one_iff",
"multiset",
"multiset.induction_on",
"or_imp_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod {p q : multiset (associates α)} (h : p ≤ q) : p.prod ≤ q.prod | begin
haveI := classical.dec_eq (associates α),
haveI := classical.dec_eq α,
suffices : p.prod ≤ (p + (q - p)).prod, { rwa [add_tsub_cancel_of_le h] at this },
suffices : p.prod * 1 ≤ p.prod * (q - p).prod, { simpa },
exact mul_mono (le_refl p.prod) one_le
end | theorem | associates.prod_le_prod | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"add_tsub_cancel_of_le",
"associates",
"classical.dec_eq",
"multiset",
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_multiset_le_of_prime {s : multiset (associates α)} {p : associates α}
(hp : prime p) :
p ≤ s.prod → ∃a∈s, p ≤ a | multiset.induction_on s (assume ⟨d, eq⟩, (hp.ne_one (mul_eq_one_iff.1 eq.symm).1).elim) $
assume a s ih h,
have p ≤ a * s.prod, by simpa using h,
match prime.le_or_le hp this with
| or.inl h := ⟨a, multiset.mem_cons_self a s, h⟩
| or.inr h := let ⟨a, has, h⟩ := ih h in ⟨a, multiset.mem_cons_of_mem has, h⟩
end | lemma | associates.exists_mem_multiset_le_of_prime | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"associates",
"ih",
"multiset",
"multiset.induction_on",
"multiset.mem_cons_of_mem",
"multiset.mem_cons_self",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_ne_zero_of_prime [cancel_comm_monoid_with_zero α] [nontrivial α]
(s : multiset α) (h : ∀ x ∈ s, prime x) : s.prod ≠ 0 | multiset.prod_ne_zero (λ h0, prime.ne_zero (h 0 h0) rfl) | lemma | multiset.prod_ne_zero_of_prime | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"cancel_comm_monoid_with_zero",
"multiset",
"multiset.prod_ne_zero",
"nontrivial",
"prime",
"prime.ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime.dvd_finset_prod_iff {S : finset α} {p : M} (pp : prime p) (g : α → M) :
p ∣ S.prod g ↔ ∃ a ∈ S, p ∣ g a | ⟨pp.exists_mem_finset_dvd, λ ⟨a, ha1, ha2⟩, dvd_trans ha2 (dvd_prod_of_mem g ha1)⟩ | lemma | prime.dvd_finset_prod_iff | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"dvd_trans",
"finset",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime.dvd_finsupp_prod_iff {f: α →₀ M} {g : α → M → ℕ} {p : ℕ} (pp : prime p) :
p ∣ f.prod g ↔ ∃ a ∈ f.support, p ∣ g a (f a) | prime.dvd_finset_prod_iff pp _ | lemma | prime.dvd_finsupp_prod_iff | algebra.big_operators | src/algebra/big_operators/associated.lean | [
"algebra.associated",
"algebra.big_operators.finsupp"
] | [
"prime",
"prime.dvd_finset_prod_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod [comm_monoid β] (s : finset α) (f : α → β) : β | (s.1.map f).prod | def | finset.prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset"
] | `∏ x in s, f x` is the product of `f x`
as `x` ranges over the elements of the finite set `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mk [comm_monoid β] (s : multiset α) (hs : s.nodup) (f : α → β) :
(⟨s, hs⟩ : finset α).prod f = (s.map f).prod | rfl | lemma | finset.prod_mk | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_val [comm_monoid α] (s : finset α) : s.1.prod = s.prod id | by rw [finset.prod, multiset.map_id] | lemma | finset.prod_val | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset",
"finset.prod",
"multiset.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_multiset_prod [comm_monoid β] (s : finset α) (f : α → β) :
∏ x in s, f x = (s.1.map f).prod | rfl | lemma | finset.prod_eq_multiset_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_val [comm_monoid β] (s : finset α) (f : α → β) :
(s.1.map f).prod = ∏ a in s, f a | rfl | lemma | finset.prod_map_val | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_fold [comm_monoid β] (s : finset α) (f : α → β) :
∏ x in s, f x = s.fold (*) 1 f | rfl | theorem | finset.prod_eq_fold | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_multiset_singleton (s : finset α) :
s.sum (λ x, {x}) = s.val | by simp only [sum_eq_multiset_sum, multiset.sum_map_singleton] | lemma | finset.sum_multiset_singleton | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset",
"multiset.sum_map_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod [comm_monoid β] [comm_monoid γ] {G : Type*} [monoid_hom_class G β γ] (g : G)
(f : α → β) (s : finset α) :
g (∏ x in s, f x) = ∏ x in s, g (f x) | by simp only [finset.prod_eq_multiset_prod, map_multiset_prod, multiset.map_map] | lemma | map_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset",
"finset.prod_eq_multiset_prod",
"map_multiset_prod",
"monoid_hom_class",
"multiset.map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_prod [comm_monoid β] [comm_monoid γ] (g : β →* γ) (f : α → β)
(s : finset α) : g (∏ x in s, f x) = ∏ x in s, g (f x) | map_prod g f s | lemma | monoid_hom.map_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset",
"map_prod"
] | Deprecated: use `_root_.map_prod` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.map_prod [comm_monoid β] [comm_monoid γ] (g : β ≃* γ) (f : α → β)
(s : finset α) : g (∏ x in s, f x) = ∏ x in s, g (f x) | map_prod g f s | lemma | mul_equiv.map_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset",
"map_prod"
] | Deprecated: use `_root_.map_prod` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_list_prod [semiring β] [semiring γ] (f : β →+* γ) (l : list β) :
f l.prod = (l.map f).prod | map_list_prod f l | lemma | ring_hom.map_list_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"map_list_prod",
"semiring"
] | Deprecated: use `_root_.map_list_prod` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_list_sum [non_assoc_semiring β] [non_assoc_semiring γ]
(f : β →+* γ) (l : list β) :
f l.sum = (l.map f).sum | map_list_sum f l | lemma | ring_hom.map_list_sum | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"non_assoc_semiring"
] | Deprecated: use `_root_.map_list_sum` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.unop_map_list_prod [semiring β] [semiring γ] (f : β →+* γᵐᵒᵖ)
(l : list β) : mul_opposite.unop (f l.prod) = (l.map (mul_opposite.unop ∘ f)).reverse.prod | unop_map_list_prod f l | lemma | ring_hom.unop_map_list_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"mul_opposite.unop",
"semiring",
"unop_map_list_prod"
] | A morphism into the opposite ring acts on the product by acting on the reversed elements.
Deprecated: use `_root_.unop_map_list_prod` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_multiset_prod [comm_semiring β] [comm_semiring γ] (f : β →+* γ)
(s : multiset β) :
f s.prod = (s.map f).prod | map_multiset_prod f s | lemma | ring_hom.map_multiset_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_semiring",
"map_multiset_prod",
"multiset"
] | Deprecated: use `_root_.map_multiset_prod` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_multiset_sum [non_assoc_semiring β] [non_assoc_semiring γ]
(f : β →+* γ) (s : multiset β) :
f s.sum = (s.map f).sum | map_multiset_sum f s | lemma | ring_hom.map_multiset_sum | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"multiset",
"non_assoc_semiring"
] | Deprecated: use `_root_.map_multiset_sum` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_prod [comm_semiring β] [comm_semiring γ] (g : β →+* γ) (f : α → β)
(s : finset α) :
g (∏ x in s, f x) = ∏ x in s, g (f x) | map_prod g f s | lemma | ring_hom.map_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_semiring",
"finset",
"map_prod"
] | Deprecated: use `_root_.map_prod` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_sum [non_assoc_semiring β] [non_assoc_semiring γ]
(g : β →+* γ) (f : α → β) (s : finset α) :
g (∑ x in s, f x) = ∑ x in s, g (f x) | map_sum g f s | lemma | ring_hom.map_sum | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset",
"non_assoc_semiring"
] | Deprecated: use `_root_.map_sum` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.coe_finset_prod [mul_one_class β] [comm_monoid γ] (f : α → β →* γ) (s : finset α) :
⇑(∏ x in s, f x) = ∏ x in s, f x | (monoid_hom.coe_fn β γ).map_prod _ _ | lemma | monoid_hom.coe_finset_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset",
"map_prod",
"monoid_hom.coe_fn",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.finset_prod_apply [mul_one_class β] [comm_monoid γ] (f : α → β →* γ)
(s : finset α) (b : β) : (∏ x in s, f x) b = ∏ x in s, f x b | (monoid_hom.eval b).map_prod _ _ | lemma | monoid_hom.finset_prod_apply | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"comm_monoid",
"finset",
"map_prod",
"monoid_hom.eval",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_empty : ∏ x in ∅, f x = 1 | rfl | lemma | finset.prod_empty | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_of_empty [is_empty α] (s : finset α) : ∏ i in s, f i = 1 | by rw [eq_empty_of_is_empty s, prod_empty] | lemma | finset.prod_of_empty | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset",
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_cons (h : a ∉ s) : (∏ x in (cons a s h), f x) = f a * ∏ x in s, f x | fold_cons h | lemma | finset.prod_cons | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_insert [decidable_eq α] : a ∉ s → (∏ x in (insert a s), f x) = f a * ∏ x in s, f x | fold_insert | lemma | finset.prod_insert | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_insert_of_eq_one_if_not_mem [decidable_eq α] (h : a ∉ s → f a = 1) :
∏ x in insert a s, f x = ∏ x in s, f x | begin
by_cases hm : a ∈ s,
{ simp_rw insert_eq_of_mem hm },
{ rw [prod_insert hm, h hm, one_mul] },
end | lemma | finset.prod_insert_of_eq_one_if_not_mem | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"one_mul"
] | The product of `f` over `insert a s` is the same as
the product over `s`, as long as `a` is in `s` or `f a = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_insert_one [decidable_eq α] (h : f a = 1) :
∏ x in insert a s, f x = ∏ x in s, f x | prod_insert_of_eq_one_if_not_mem (λ _, h) | lemma | finset.prod_insert_one | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [] | The product of `f` over `insert a s` is the same as the product over `s`, as long as `f a = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_singleton : (∏ x in (singleton a), f x) = f a | eq.trans fold_singleton $ mul_one _ | lemma | finset.prod_singleton | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_pair [decidable_eq α] {a b : α} (h : a ≠ b) :
(∏ x in ({a, b} : finset α), f x) = f a * f b | by rw [prod_insert (not_mem_singleton.2 h), prod_singleton] | lemma | finset.prod_pair | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_const_one : (∏ x in s, (1 : β)) = 1 | by simp only [finset.prod, multiset.map_const, multiset.prod_replicate, one_pow] | lemma | finset.prod_const_one | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset.prod",
"multiset.map_const",
"multiset.prod_replicate",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_image [decidable_eq α] {s : finset γ} {g : γ → α} :
(∀ x ∈ s, ∀ y ∈ s, g x = g y → x = y) → (∏ x in (s.image g), f x) = ∏ x in s, f (g x) | fold_image | lemma | finset.prod_image | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map (s : finset α) (e : α ↪ γ) (f : γ → β) :
(∏ x in (s.map e), f x) = ∏ x in s, f (e x) | by rw [finset.prod, finset.map_val, multiset.map_map]; refl | lemma | finset.prod_map | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset",
"finset.map_val",
"finset.prod",
"multiset.map_map",
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_congr (h : s₁ = s₂) : (∀ x ∈ s₂, f x = g x) → s₁.prod f = s₂.prod g | by rw [h]; exact fold_congr | lemma | finset.prod_congr | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_disj_union (h) : ∏ x in s₁.disj_union s₂ h, f x = (∏ x in s₁, f x) * ∏ x in s₂, f x | by { refine eq.trans _ (fold_disj_union h), rw one_mul, refl } | lemma | finset.prod_disj_union | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_disj_Union (s : finset ι) (t : ι → finset α) (h) :
∏ x in s.disj_Union t h, f x = ∏ i in s, ∏ x in t i, f x | begin
refine eq.trans _ (fold_disj_Union h),
dsimp [finset.prod, multiset.prod, multiset.fold, finset.disj_Union, finset.fold],
congr',
exact prod_const_one.symm,
end | lemma | finset.prod_disj_Union | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset",
"finset.disj_Union",
"finset.fold",
"finset.prod",
"multiset.fold",
"multiset.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_union_inter [decidable_eq α] :
(∏ x in (s₁ ∪ s₂), f x) * (∏ x in (s₁ ∩ s₂), f x) = (∏ x in s₁, f x) * (∏ x in s₂, f x) | fold_union_inter | lemma | finset.prod_union_inter | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_union [decidable_eq α] (h : disjoint s₁ s₂) :
(∏ x in (s₁ ∪ s₂), f x) = (∏ x in s₁, f x) * (∏ x in s₂, f x) | by rw [←prod_union_inter, (disjoint_iff_inter_eq_empty.mp h)]; exact (mul_one _).symm | lemma | finset.prod_union | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"disjoint",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_filter_mul_prod_filter_not (s : finset α) (p : α → Prop) [decidable_pred p]
[decidable_pred (λ x, ¬p x)] (f : α → β) :
(∏ x in s.filter p, f x) * (∏ x in s.filter (λ x, ¬p x), f x) = ∏ x in s, f x | begin
haveI := classical.dec_eq α,
rw [← prod_union (disjoint_filter_filter_neg _ _ p), filter_union_filter_neg_eq]
end | lemma | finset.prod_filter_mul_prod_filter_not | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"classical.dec_eq",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_to_list (s : finset α) (f : α → β) : (s.to_list.map f).prod = s.prod f | by rw [finset.prod, ← multiset.coe_prod, ← multiset.coe_map, finset.coe_to_list] | lemma | finset.prod_to_list | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset",
"finset.coe_to_list",
"finset.prod",
"multiset.coe_map",
"multiset.coe_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.equiv.perm.prod_comp (σ : equiv.perm α) (s : finset α) (f : α → β)
(hs : {a | σ a ≠ a} ⊆ s) :
(∏ x in s, f (σ x)) = ∏ x in s, f x | by { convert (prod_map _ σ.to_embedding _).symm, exact (map_perm hs).symm } | lemma | equiv.perm.prod_comp | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"equiv.perm",
"finset",
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.equiv.perm.prod_comp' (σ : equiv.perm α) (s : finset α) (f : α → α → β)
(hs : {a | σ a ≠ a} ⊆ s) :
(∏ x in s, f (σ x) x) = ∏ x in s, f x (σ.symm x) | by { convert σ.prod_comp s (λ x, f x (σ.symm x)) hs, ext, rw equiv.symm_apply_apply } | lemma | equiv.perm.prod_comp' | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"equiv.perm",
"equiv.symm_apply_apply",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compl.prod_mul_prod {s t : finset α} (h : is_compl s t) (f : α → β) :
(∏ i in s, f i) * (∏ i in t, f i) = ∏ i, f i | (finset.prod_disj_union h.disjoint).symm.trans $
by { classical, rw [finset.disj_union_eq_union, ← finset.sup_eq_union, h.sup_eq_top]; refl } | lemma | is_compl.prod_mul_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset",
"finset.disj_union_eq_union",
"finset.prod_disj_union",
"finset.sup_eq_union",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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