statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
finprod_mem_of_eq_on_one (hf : s.eq_on f 1) : ∏ᶠ i ∈ s, f i = 1 | by { rw ← finprod_mem_one s, exact finprod_mem_congr rfl hf } | lemma | finprod_mem_of_eq_on_one | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_congr",
"finprod_mem_one"
] | If a function `f` equals `1` on a set `s`, then the product of `f i` over `i ∈ s` equals `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_ne_one_of_finprod_mem_ne_one (h : ∏ᶠ i ∈ s, f i ≠ 1) : ∃ x ∈ s, f x ≠ 1 | begin
by_contra' h',
exact h (finprod_mem_of_eq_on_one h')
end | lemma | exists_ne_one_of_finprod_mem_ne_one | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_of_eq_on_one"
] | If the product of `f i` over `i ∈ s` is not equal to `1`, then there is some `x ∈ s` such that
`f x ≠ 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_mul_distrib (hs : s.finite) :
∏ᶠ i ∈ s, f i * g i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ s, g i | finprod_mem_mul_distrib' (hs.inter_of_left _) (hs.inter_of_left _) | lemma | finprod_mem_mul_distrib | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_mul_distrib'"
] | Given a finite set `s`, the product of `f i * g i` over `i ∈ s` equals the product of `f i`
over `i ∈ s` times the product of `g i` over `i ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.map_finprod {f : α → M} (g : M →* N) (hf : (mul_support f).finite) :
g (∏ᶠ i, f i) = ∏ᶠ i, g (f i) | g.map_finprod_plift f $ hf.preimage $ equiv.plift.injective.inj_on _ | lemma | monoid_hom.map_finprod | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_pow (hf : (mul_support f).finite) (n : ℕ) :
(∏ᶠ i, f i) ^ n = ∏ᶠ i, f i ^ n | (pow_monoid_hom n).map_finprod hf | lemma | finprod_pow | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"pow_monoid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_finprod_mem' {f : α → M} (g : M →* N) (h₀ : (s ∩ mul_support f).finite) :
g (∏ᶠ j ∈ s, f j) = ∏ᶠ i ∈ s, g (f i) | begin
rw [g.map_finprod],
{ simp only [g.map_finprod_Prop] },
{ simpa only [finprod_eq_mul_indicator_apply, mul_support_mul_indicator] }
end | lemma | monoid_hom.map_finprod_mem' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_eq_mul_indicator_apply"
] | A more general version of `monoid_hom.map_finprod_mem` that requires `s ∩ mul_support f` rather
than `s` to be finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.map_finprod_mem (f : α → M) (g : M →* N) (hs : s.finite) :
g (∏ᶠ j ∈ s, f j) = ∏ᶠ i ∈ s, g (f i) | g.map_finprod_mem' (hs.inter_of_left _) | lemma | monoid_hom.map_finprod_mem | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [] | Given a monoid homomorphism `g : M →* N` and a function `f : α → M`, the value of `g` at the
product of `f i` over `i ∈ s` equals the product of `g (f i)` over `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.map_finprod_mem (g : M ≃* N) (f : α → M) {s : set α}
(hs : s.finite) : g (∏ᶠ i ∈ s, f i) = ∏ᶠ i ∈ s, g (f i) | g.to_monoid_hom.map_finprod_mem f hs | lemma | mul_equiv.map_finprod_mem | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_inv_distrib [division_comm_monoid G] (f : α → G) (hs : s.finite) :
∏ᶠ x ∈ s, (f x)⁻¹ = (∏ᶠ x ∈ s, f x)⁻¹ | ((mul_equiv.inv G).map_finprod_mem f hs).symm | lemma | finprod_mem_inv_distrib | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"division_comm_monoid",
"mul_equiv.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_div_distrib [division_comm_monoid G] (f g : α → G) (hs : s.finite) :
∏ᶠ i ∈ s, f i / g i = (∏ᶠ i ∈ s, f i) / ∏ᶠ i ∈ s, g i | by simp only [div_eq_mul_inv, finprod_mem_mul_distrib hs, finprod_mem_inv_distrib g hs] | lemma | finprod_mem_div_distrib | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"div_eq_mul_inv",
"division_comm_monoid",
"finprod_mem_inv_distrib",
"finprod_mem_mul_distrib"
] | Given a finite set `s`, the product of `f i / g i` over `i ∈ s` equals the product of `f i`
over `i ∈ s` divided by the product of `g i` over `i ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_empty : ∏ᶠ i ∈ (∅ : set α), f i = 1 | by simp | lemma | finprod_mem_empty | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [] | The product of any function over an empty set is `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_of_finprod_mem_ne_one (h : ∏ᶠ i ∈ s, f i ≠ 1) : s.nonempty | nonempty_iff_ne_empty.2 $ λ h', h $ h'.symm ▸ finprod_mem_empty | lemma | nonempty_of_finprod_mem_ne_one | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_empty"
] | A set `s` is nonempty if the product of some function over `s` is not equal to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_union_inter (hs : s.finite) (ht : t.finite) :
(∏ᶠ i ∈ s ∪ t, f i) * ∏ᶠ i ∈ s ∩ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i | begin
lift s to finset α using hs, lift t to finset α using ht,
classical,
rw [← finset.coe_union, ← finset.coe_inter],
simp only [finprod_mem_coe_finset, finset.prod_union_inter]
end | lemma | finprod_mem_union_inter | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_coe_finset",
"finset",
"finset.coe_inter",
"finset.coe_union",
"finset.prod_union_inter",
"lift"
] | Given finite sets `s` and `t`, the product of `f i` over `i ∈ s ∪ t` times the product of
`f i` over `i ∈ s ∩ t` equals the product of `f i` over `i ∈ s` times the product of `f i`
over `i ∈ t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_union_inter'
(hs : (s ∩ mul_support f).finite) (ht : (t ∩ mul_support f).finite) :
(∏ᶠ i ∈ s ∪ t, f i) * ∏ᶠ i ∈ s ∩ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i | begin
rw [← finprod_mem_inter_mul_support f s, ← finprod_mem_inter_mul_support f t,
← finprod_mem_union_inter hs ht, ← union_inter_distrib_right, finprod_mem_inter_mul_support,
← finprod_mem_inter_mul_support f (s ∩ t)],
congr' 2,
rw [inter_left_comm, inter_assoc, inter_assoc, inter_self, inter_left_comm]... | lemma | finprod_mem_union_inter' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_mem_inter_mul_support",
"finprod_mem_union_inter"
] | A more general version of `finprod_mem_union_inter` that requires `s ∩ mul_support f` and
`t ∩ mul_support f` rather than `s` and `t` to be finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_union' (hst : disjoint s t) (hs : (s ∩ mul_support f).finite)
(ht : (t ∩ mul_support f).finite) :
∏ᶠ i ∈ s ∪ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i | by rw [← finprod_mem_union_inter' hs ht, disjoint_iff_inter_eq_empty.1 hst, finprod_mem_empty,
mul_one] | lemma | finprod_mem_union' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"disjoint",
"finite",
"finprod_mem_empty",
"finprod_mem_union_inter'",
"mul_one"
] | A more general version of `finprod_mem_union` that requires `s ∩ mul_support f` and
`t ∩ mul_support f` rather than `s` and `t` to be finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_union (hst : disjoint s t) (hs : s.finite) (ht : t.finite) :
∏ᶠ i ∈ s ∪ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i | finprod_mem_union' hst (hs.inter_of_left _) (ht.inter_of_left _) | lemma | finprod_mem_union | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"disjoint",
"finprod_mem_union'"
] | Given two finite disjoint sets `s` and `t`, the product of `f i` over `i ∈ s ∪ t` equals the
product of `f i` over `i ∈ s` times the product of `f i` over `i ∈ t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_union'' (hst : disjoint (s ∩ mul_support f) (t ∩ mul_support f))
(hs : (s ∩ mul_support f).finite) (ht : (t ∩ mul_support f).finite) :
∏ᶠ i ∈ s ∪ t, f i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i | by rw [← finprod_mem_inter_mul_support f s, ← finprod_mem_inter_mul_support f t,
← finprod_mem_union hst hs ht, ← union_inter_distrib_right, finprod_mem_inter_mul_support] | lemma | finprod_mem_union'' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"disjoint",
"finite",
"finprod_mem_inter_mul_support",
"finprod_mem_union"
] | A more general version of `finprod_mem_union'` that requires `s ∩ mul_support f` and
`t ∩ mul_support f` rather than `s` and `t` to be disjoint | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_singleton : ∏ᶠ i ∈ ({a} : set α), f i = f a | by rw [← finset.coe_singleton, finprod_mem_coe_finset, finset.prod_singleton] | lemma | finprod_mem_singleton | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_coe_finset",
"finset.coe_singleton",
"finset.prod_singleton"
] | The product of `f i` over `i ∈ {a}` equals `f a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_cond_eq_left : ∏ᶠ i = a, f i = f a | finprod_mem_singleton | lemma | finprod_cond_eq_left | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_cond_eq_right : ∏ᶠ i (hi : a = i), f i = f a | by simp [@eq_comm _ a] | lemma | finprod_cond_eq_right | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_insert' (f : α → M) (h : a ∉ s) (hs : (s ∩ mul_support f).finite) :
∏ᶠ i ∈ insert a s, f i = f a * ∏ᶠ i ∈ s, f i | begin
rw [insert_eq, finprod_mem_union' _ _ hs, finprod_mem_singleton],
{ rwa disjoint_singleton_left },
{ exact (finite_singleton a).inter_of_left _ }
end | lemma | finprod_mem_insert' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_mem_singleton",
"finprod_mem_union'"
] | A more general version of `finprod_mem_insert` that requires `s ∩ mul_support f` rather than `s`
to be finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_insert (f : α → M) (h : a ∉ s) (hs : s.finite) :
∏ᶠ i ∈ insert a s, f i = f a * ∏ᶠ i ∈ s, f i | finprod_mem_insert' f h $ hs.inter_of_left _ | lemma | finprod_mem_insert | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_insert'"
] | Given a finite set `s` and an element `a ∉ s`, the product of `f i` over `i ∈ insert a s` equals
`f a` times the product of `f i` over `i ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_insert_of_eq_one_if_not_mem (h : a ∉ s → f a = 1) :
∏ᶠ i ∈ insert a s, f i = ∏ᶠ i ∈ s, f i | begin
refine finprod_mem_inter_mul_support_eq' _ _ _ (λ x hx, ⟨_, or.inr⟩),
rintro (rfl|hxs),
exacts [not_imp_comm.1 h hx, hxs]
end | lemma | finprod_mem_insert_of_eq_one_if_not_mem | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_inter_mul_support_eq'"
] | If `f a = 1` when `a ∉ s`, then the product of `f i` over `i ∈ insert a s` equals the product of
`f i` over `i ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_insert_one (h : f a = 1) : ∏ᶠ i ∈ insert a s, f i = ∏ᶠ i ∈ s, f i | finprod_mem_insert_of_eq_one_if_not_mem (λ _, h) | lemma | finprod_mem_insert_one | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_insert_of_eq_one_if_not_mem"
] | If `f a = 1`, then the product of `f i` over `i ∈ insert a s` equals the product of `f i` over
`i ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_dvd {f : α → N} (a : α) (hf : (mul_support f).finite) : f a ∣ finprod f | begin
by_cases ha : a ∈ mul_support f,
{ rw finprod_eq_prod_of_mul_support_to_finset_subset f hf (set.subset.refl _),
exact finset.dvd_prod_of_mem f ((finite.mem_to_finset hf).mpr ha) },
{ rw nmem_mul_support.mp ha,
exact one_dvd (finprod f) }
end | lemma | finprod_mem_dvd | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod",
"finprod_eq_prod_of_mul_support_to_finset_subset",
"finset.dvd_prod_of_mem",
"one_dvd",
"set.subset.refl"
] | If the multiplicative support of `f` is finite, then for every `x` in the domain of `f`, `f x`
divides `finprod f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_pair (h : a ≠ b) : ∏ᶠ i ∈ ({a, b} : set α), f i = f a * f b | by { rw [finprod_mem_insert, finprod_mem_singleton], exacts [h, finite_singleton b] } | lemma | finprod_mem_pair | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_insert",
"finprod_mem_singleton"
] | The product of `f i` over `i ∈ {a, b}`, `a ≠ b`, is equal to `f a * f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_image' {s : set β} {g : β → α} (hg : (s ∩ mul_support (f ∘ g)).inj_on g) :
∏ᶠ i ∈ g '' s, f i = ∏ᶠ j ∈ s, f (g j) | begin
classical,
by_cases hs : (s ∩ mul_support (f ∘ g)).finite,
{ have hg : ∀ (x ∈ hs.to_finset) (y ∈ hs.to_finset), g x = g y → x = y,
by simpa only [hs.mem_to_finset],
rw [finprod_mem_eq_prod _ hs, ← finset.prod_image hg],
refine finprod_mem_eq_prod_of_inter_mul_support_eq f _,
rw [finset.coe... | lemma | finprod_mem_image' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_mem_eq_one_of_infinite",
"finprod_mem_eq_prod",
"finprod_mem_eq_prod_of_inter_mul_support_eq",
"finset.coe_image",
"finset.prod_image"
] | The product of `f y` over `y ∈ g '' s` equals the product of `f (g i)` over `s`
provided that `g` is injective on `s ∩ mul_support (f ∘ g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_image {s : set β} {g : β → α} (hg : s.inj_on g) :
∏ᶠ i ∈ g '' s, f i = ∏ᶠ j ∈ s, f (g j) | finprod_mem_image' $ hg.mono $ inter_subset_left _ _ | lemma | finprod_mem_image | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_image'"
] | The product of `f y` over `y ∈ g '' s` equals the product of `f (g i)` over `s` provided that
`g` is injective on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_range' {g : β → α} (hg : (mul_support (f ∘ g)).inj_on g) :
∏ᶠ i ∈ range g, f i = ∏ᶠ j, f (g j) | begin
rw [← image_univ, finprod_mem_image', finprod_mem_univ],
rwa univ_inter
end | lemma | finprod_mem_range' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_image'",
"finprod_mem_univ"
] | The product of `f y` over `y ∈ set.range g` equals the product of `f (g i)` over all `i`
provided that `g` is injective on `mul_support (f ∘ g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_range {g : β → α} (hg : injective g) : ∏ᶠ i ∈ range g, f i = ∏ᶠ j, f (g j) | finprod_mem_range' (hg.inj_on _) | lemma | finprod_mem_range | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_range'"
] | The product of `f y` over `y ∈ set.range g` equals the product of `f (g i)` over all `i`
provided that `g` is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_eq_of_bij_on {s : set α} {t : set β} {f : α → M} {g : β → M} (e : α → β)
(he₀ : s.bij_on e t) (he₁ : ∀ x ∈ s, f x = g (e x)) :
∏ᶠ i ∈ s, f i = ∏ᶠ j ∈ t, g j | begin
rw [← set.bij_on.image_eq he₀, finprod_mem_image he₀.2.1],
exact finprod_mem_congr rfl he₁
end | lemma | finprod_mem_eq_of_bij_on | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_congr",
"finprod_mem_image",
"set.bij_on.image_eq"
] | See also `finset.prod_bij`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_eq_of_bijective {f : α → M} {g : β → M} (e : α → β) (he₀ : bijective e)
(he₁ : ∀ x, f x = g (e x)) :
∏ᶠ i, f i = ∏ᶠ j, g j | begin
rw [← finprod_mem_univ f, ← finprod_mem_univ g],
exact finprod_mem_eq_of_bij_on _ (bijective_iff_bij_on_univ.mp he₀) (λ x _, he₁ x),
end | lemma | finprod_eq_of_bijective | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_eq_of_bij_on",
"finprod_mem_univ"
] | See `finprod_comp`, `fintype.prod_bijective` and `finset.prod_bij`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_comp {g : β → M} (e : α → β) (he₀ : function.bijective e) :
∏ᶠ i, g (e i) = ∏ᶠ j, g j | finprod_eq_of_bijective e he₀ (λ x, rfl) | lemma | finprod_comp | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_of_bijective"
] | See also `finprod_eq_of_bijective`, `fintype.prod_bijective` and `finset.prod_bij`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_comp_equiv (e : α ≃ β) {f : β → M} : ∏ᶠ i, f (e i) = ∏ᶠ i', f i' | finprod_comp e e.bijective | lemma | finprod_comp_equiv | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_set_coe_eq_finprod_mem (s : set α) : ∏ᶠ j : s, f j = ∏ᶠ i ∈ s, f i | begin
rw [← finprod_mem_range, subtype.range_coe],
exact subtype.coe_injective
end | lemma | finprod_set_coe_eq_finprod_mem | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_range",
"subtype.coe_injective",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_subtype_eq_finprod_cond (p : α → Prop) :
∏ᶠ j : subtype p, f j = ∏ᶠ i (hi : p i), f i | finprod_set_coe_eq_finprod_mem {i | p i} | lemma | finprod_subtype_eq_finprod_cond | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_set_coe_eq_finprod_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_inter_mul_diff' (t : set α) (h : (s ∩ mul_support f).finite) :
(∏ᶠ i ∈ s ∩ t, f i) * ∏ᶠ i ∈ s \ t, f i = ∏ᶠ i ∈ s, f i | begin
rw [← finprod_mem_union', inter_union_diff],
rw disjoint_iff_inf_le,
exacts [λ x hx, hx.2.2 hx.1.2, h.subset (λ x hx, ⟨hx.1.1, hx.2⟩),
h.subset (λ x hx, ⟨hx.1.1, hx.2⟩)],
end | lemma | finprod_mem_inter_mul_diff' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"disjoint_iff_inf_le",
"finite",
"finprod_mem_union'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_inter_mul_diff (t : set α) (h : s.finite) :
(∏ᶠ i ∈ s ∩ t, f i) * ∏ᶠ i ∈ s \ t, f i = ∏ᶠ i ∈ s, f i | finprod_mem_inter_mul_diff' _ $ h.inter_of_left _ | lemma | finprod_mem_inter_mul_diff | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_inter_mul_diff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_mul_diff' (hst : s ⊆ t) (ht : (t ∩ mul_support f).finite) :
(∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t \ s, f i = ∏ᶠ i ∈ t, f i | by rw [← finprod_mem_inter_mul_diff' _ ht, inter_eq_self_of_subset_right hst] | lemma | finprod_mem_mul_diff' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_mem_inter_mul_diff'"
] | A more general version of `finprod_mem_mul_diff` that requires `t ∩ mul_support f` rather than
`t` to be finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_mul_diff (hst : s ⊆ t) (ht : t.finite) :
(∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t \ s, f i = ∏ᶠ i ∈ t, f i | finprod_mem_mul_diff' hst (ht.inter_of_left _) | lemma | finprod_mem_mul_diff | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_mul_diff'"
] | Given a finite set `t` and a subset `s` of `t`, the product of `f i` over `i ∈ s`
times the product of `f i` over `t \ s` equals the product of `f i` over `i ∈ t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_Union [finite ι] {t : ι → set α} (h : pairwise (disjoint on t))
(ht : ∀ i, (t i).finite) :
∏ᶠ a ∈ (⋃ i : ι, t i), f a = ∏ᶠ i, ∏ᶠ a ∈ t i, f a | begin
casesI nonempty_fintype ι,
lift t to ι → finset α using ht,
classical,
rw [← bUnion_univ, ← finset.coe_univ, ← finset.coe_bUnion,
finprod_mem_coe_finset, finset.prod_bUnion],
{ simp only [finprod_mem_coe_finset, finprod_eq_prod_of_fintype] },
{ exact λ x _ y _ hxy, finset.disjoint_coe.1 (h hxy) }
... | lemma | finprod_mem_Union | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"disjoint",
"finite",
"finprod_eq_prod_of_fintype",
"finprod_mem_coe_finset",
"finset",
"finset.coe_bUnion",
"finset.coe_univ",
"finset.prod_bUnion",
"lift",
"nonempty_fintype",
"pairwise"
] | Given a family of pairwise disjoint finite sets `t i` indexed by a finite type, the product of
`f a` over the union `⋃ i, t i` is equal to the product over all indexes `i` of the products of
`f a` over `a ∈ t i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_bUnion {I : set ι} {t : ι → set α} (h : I.pairwise_disjoint t) (hI : I.finite)
(ht : ∀ i ∈ I, (t i).finite) :
∏ᶠ a ∈ ⋃ x ∈ I, t x, f a = ∏ᶠ i ∈ I, ∏ᶠ j ∈ t i, f j | begin
haveI := hI.fintype,
rw [bUnion_eq_Union, finprod_mem_Union, ← finprod_set_coe_eq_finprod_mem],
exacts [λ x y hxy, h x.2 y.2 (subtype.coe_injective.ne hxy), λ b, ht b b.2]
end | lemma | finprod_mem_bUnion | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_mem_Union",
"finprod_set_coe_eq_finprod_mem"
] | Given a family of sets `t : ι → set α`, a finite set `I` in the index type such that all sets
`t i`, `i ∈ I`, are finite, if all `t i`, `i ∈ I`, are pairwise disjoint, then the product of `f a`
over `a ∈ ⋃ i ∈ I, t i` is equal to the product over `i ∈ I` of the products of `f a` over
`a ∈ t i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_sUnion {t : set (set α)} (h : t.pairwise_disjoint id) (ht₀ : t.finite)
(ht₁ : ∀ x ∈ t, set.finite x) :
∏ᶠ a ∈ ⋃₀ t, f a = ∏ᶠ s ∈ t, ∏ᶠ a ∈ s, f a | by { rw set.sUnion_eq_bUnion, exact finprod_mem_bUnion h ht₀ ht₁ } | lemma | finprod_mem_sUnion | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_bUnion",
"set.finite",
"set.sUnion_eq_bUnion"
] | If `t` is a finite set of pairwise disjoint finite sets, then the product of `f a`
over `a ∈ ⋃₀ t` is the product over `s ∈ t` of the products of `f a` over `a ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_finprod_cond_ne (a : α) (hf : (mul_support f).finite) :
f a * (∏ᶠ i ≠ a, f i) = ∏ᶠ i, f i | begin
classical,
rw [finprod_eq_prod _ hf],
have h : ∀ x : α, f x ≠ 1 → (x ≠ a ↔ x ∈ hf.to_finset \ {a}),
{ intros x hx,
rw [finset.mem_sdiff, finset.mem_singleton, finite.mem_to_finset, mem_mul_support],
exact ⟨λ h, and.intro hx h, λ h, h.2⟩,},
rw [finprod_cond_eq_prod_of_cond_iff f h, finset.sdiff_s... | lemma | mul_finprod_cond_ne | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_cond_eq_prod_of_cond_iff",
"finprod_eq_prod",
"finset.mem_sdiff",
"finset.mem_singleton",
"finset.mul_prod_erase",
"finset.prod_erase",
"finset.sdiff_singleton_eq_erase",
"not_not",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_comm {s : set α} {t : set β} (f : α → β → M) (hs : s.finite) (ht : t.finite) :
∏ᶠ i ∈ s, ∏ᶠ j ∈ t, f i j = ∏ᶠ j ∈ t, ∏ᶠ i ∈ s, f i j | begin
lift s to finset α using hs, lift t to finset β using ht,
simp only [finprod_mem_coe_finset],
exact finset.prod_comm
end | lemma | finprod_mem_comm | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_coe_finset",
"finset",
"finset.prod_comm",
"lift"
] | If `s : set α` and `t : set β` are finite sets, then taking the product over `s` commutes with
taking the product over `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_induction (p : M → Prop) (hp₀ : p 1)
(hp₁ : ∀ x y, p x → p y → p (x * y)) (hp₂ : ∀ x ∈ s, p $ f x) :
p (∏ᶠ i ∈ s, f i) | finprod_induction _ hp₀ hp₁ $ λ x, finprod_induction _ hp₀ hp₁ $ hp₂ x | lemma | finprod_mem_induction | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_induction"
] | To prove a property of a finite product, it suffices to prove that the property is
multiplicative and holds on factors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_cond_nonneg {R : Type*} [ordered_comm_semiring R] {p : α → Prop} {f : α → R}
(hf : ∀ x, p x → 0 ≤ f x) :
0 ≤ ∏ᶠ x (h : p x), f x | finprod_nonneg $ λ x, finprod_nonneg $ hf x | lemma | finprod_cond_nonneg | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_nonneg",
"ordered_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_le_finprod {M : Type*} [ordered_comm_monoid M] (i : α) {f : α → M}
(hf : (mul_support f).finite) (h : ∀ j, 1 ≤ f j) :
f i ≤ ∏ᶠ j, f j | by classical;
calc f i ≤ ∏ j in insert i hf.to_finset, f j :
finset.single_le_prod' (λ j hj, h j) (finset.mem_insert_self _ _)
... = ∏ᶠ j, f j :
(finprod_eq_prod_of_mul_support_to_finset_subset _ hf (finset.subset_insert _ _)).symm | lemma | single_le_finprod | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_eq_prod_of_mul_support_to_finset_subset",
"finset.mem_insert_self",
"finset.single_le_prod'",
"finset.subset_insert",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_zero {M₀ : Type*} [comm_monoid_with_zero M₀] (f : α → M₀) (x : α)
(hx : f x = 0) (hf : (mul_support f).finite) :
∏ᶠ x, f x = 0 | begin
nontriviality,
rw [finprod_eq_prod f hf],
refine finset.prod_eq_zero (hf.mem_to_finset.2 _) hx,
simp [hx]
end | lemma | finprod_eq_zero | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"comm_monoid_with_zero",
"finite",
"finprod_eq_prod",
"finset.prod_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_prod_comm (s : finset β) (f : α → β → M)
(h : ∀ b ∈ s, (mul_support (λ a, f a b)).finite) :
∏ᶠ a : α, ∏ b in s, f a b = ∏ b in s, ∏ᶠ a : α, f a b | begin
have hU : mul_support (λ a, ∏ b in s, f a b) ⊆
(s.finite_to_set.bUnion (λ b hb, h b (finset.mem_coe.1 hb))).to_finset,
{ rw finite.coe_to_finset,
intros x hx,
simp only [exists_prop, mem_Union, ne.def, mem_mul_support, finset.mem_coe],
contrapose! hx,
rw [mem_mul_support, not_not, finset.p... | lemma | finprod_prod_comm | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"exists_prop",
"finite",
"finprod_eq_prod_of_mul_support_subset",
"finset",
"finset.mem_coe",
"finset.prod_comm",
"finset.prod_congr",
"finset.prod_const_one",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_finprod_comm (s : finset α) (f : α → β → M)
(h : ∀ a ∈ s, (mul_support (f a)).finite) :
∏ a in s, ∏ᶠ b : β, f a b = ∏ᶠ b : β, ∏ a in s, f a b | (finprod_prod_comm s (λ b a, f a b) h).symm | lemma | prod_finprod_comm | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_prod_comm",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_finsum {R : Type*} [semiring R] (f : α → R) (r : R) (h : (support f).finite) :
r * ∑ᶠ a : α, f a = ∑ᶠ a : α, r * f a | (add_monoid_hom.mul_left r).map_finsum h | lemma | mul_finsum | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"add_monoid_hom.mul_left",
"finite",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finsum_mul {R : Type*} [semiring R] (f : α → R) (r : R) (h : (support f).finite) :
(∑ᶠ a : α, f a) * r = ∑ᶠ a : α, f a * r | (add_monoid_hom.mul_right r).map_finsum h | lemma | finsum_mul | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"add_monoid_hom.mul_right",
"finite",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.mul_support_of_fiberwise_prod_subset_image [decidable_eq β]
(s : finset α) (f : α → M) (g : α → β) :
mul_support (λ b, (s.filter (λ a, g a = b)).prod f) ⊆ s.image g | begin
simp only [finset.coe_image, set.mem_image, finset.mem_coe, function.support_subset_iff],
intros b h,
suffices : (s.filter (λ (a : α), g a = b)).nonempty,
{ simpa only [s.fiber_nonempty_iff_mem_image g b, finset.mem_image, exists_prop], },
exact finset.nonempty_of_prod_ne_one h,
end | lemma | finset.mul_support_of_fiberwise_prod_subset_image | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"exists_prop",
"finset",
"finset.coe_image",
"finset.mem_coe",
"finset.mem_image",
"finset.nonempty_of_prod_ne_one",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_finset_product' [decidable_eq α] [decidable_eq β]
(s : finset (α × β)) (f : α × β → M) :
∏ᶠ ab (h : ab ∈ s), f ab =
∏ᶠ a b (h : b ∈ (s.filter (λ ab, prod.fst ab = a)).image prod.snd), f (a, b) | begin
have : ∀ a, ∏ (i : β) in (s.filter (λ ab, prod.fst ab = a)).image prod.snd, f (a, i) =
(finset.filter (λ ab, prod.fst ab = a) s).prod f,
{ refine (λ a, finset.prod_bij (λ b _, (a, b)) _ _ _ _); -- `finish` closes these goals
try { simp, done },
suffices : ∀ a' b, (a', b) ∈ s → a' = a → (a, b) ∈ s ... | lemma | finprod_mem_finset_product' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_prod_of_mul_support_subset",
"finprod_mem_finset_eq_prod",
"finset",
"finset.filter",
"finset.mem_image",
"finset.prod_bij",
"finset.prod_fiberwise_of_maps_to"
] | Note that `b ∈ (s.filter (λ ab, prod.fst ab = a)).image prod.snd` iff `(a, b) ∈ s` so we can
simplify the right hand side of this lemma. However the form stated here is more useful for
iterating this lemma, e.g., if we have `f : α × β × γ → M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_finset_product (s : finset (α × β)) (f : α × β → M) :
∏ᶠ ab (h : ab ∈ s), f ab = ∏ᶠ a b (h : (a, b) ∈ s), f (a, b) | by { classical, rw finprod_mem_finset_product', simp, } | lemma | finprod_mem_finset_product | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_finset_product'",
"finset"
] | See also `finprod_mem_finset_product'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_finset_product₃ {γ : Type*}
(s : finset (α × β × γ)) (f : α × β × γ → M) :
∏ᶠ abc (h : abc ∈ s), f abc = ∏ᶠ a b c (h : (a, b, c) ∈ s), f (a, b, c) | by { classical, rw finprod_mem_finset_product', simp_rw finprod_mem_finset_product', simp, } | lemma | finprod_mem_finset_product₃ | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_finset_product'",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_curry (f : α × β → M) (hf : (mul_support f).finite) :
∏ᶠ ab, f ab = ∏ᶠ a b, f (a, b) | begin
have h₁ : ∀ a, ∏ᶠ (h : a ∈ hf.to_finset), f a = f a, { simp, },
have h₂ : ∏ᶠ a, f a = ∏ᶠ a (h : a ∈ hf.to_finset), f a, { simp, },
simp_rw [h₂, finprod_mem_finset_product, h₁],
end | lemma | finprod_curry | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_mem_finset_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_curry₃ {γ : Type*} (f : α × β × γ → M) (h : (mul_support f).finite) :
∏ᶠ abc, f abc = ∏ᶠ a b c, f (a, b, c) | by { rw finprod_curry f h, congr, ext a, rw finprod_curry, simp [h], } | lemma | finprod_curry₃ | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_curry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_dmem {s : set α} [decidable_pred (∈ s)] (f : (Π (a : α), a ∈ s → M)) :
∏ᶠ (a : α) (h : a ∈ s), f a h = ∏ᶠ (a : α) (h : a ∈ s), if h' : a ∈ s then f a h' else 1 | finprod_congr (λ a, finprod_congr (λ ha, (dif_pos ha).symm)) | lemma | finprod_dmem | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_emb_domain' {f : α → β} (hf : injective f) [decidable_pred (∈ set.range f)]
(g : α → M) :
∏ᶠ (b : β), (if h : b ∈ set.range f then g (classical.some h) else 1) = ∏ᶠ (a : α), g a | begin
simp_rw [← finprod_eq_dif],
rw [finprod_dmem, finprod_mem_range hf, finprod_congr (λ a, _)],
rw [dif_pos (set.mem_range_self a), hf (classical.some_spec (set.mem_range_self a))]
end | lemma | finprod_emb_domain' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_congr",
"finprod_dmem",
"finprod_eq_dif",
"finprod_mem_range",
"set.mem_range_self",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_emb_domain (f : α ↪ β) [decidable_pred (∈ set.range f)] (g : α → M) :
∏ᶠ (b : β), (if h : b ∈ set.range f then g (classical.some h) else 1) = ∏ᶠ (a : α), g a | finprod_emb_domain' f.injective g | lemma | finprod_emb_domain | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_emb_domain'",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod [has_zero M] [comm_monoid N] (f : α →₀ M) (g : α → M → N) : N | ∏ a in f.support, g a (f a) | def | finsupp.prod | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"comm_monoid"
] | `prod f g` is the product of `g a (f a)` over the support of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_of_support_subset (f : α →₀ M) {s : finset α}
(hs : f.support ⊆ s) (g : α → M → N) (h : ∀ i ∈ s, g i 0 = 1) :
f.prod g = ∏ x in s, g x (f x) | finset.prod_subset hs $ λ x hxs hx, h x hxs ▸ congr_arg (g x) $ not_mem_support_iff.1 hx | lemma | finsupp.prod_of_support_subset | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"finset",
"finset.prod_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_fintype [fintype α] (f : α →₀ M) (g : α → M → N) (h : ∀ i, g i 0 = 1) :
f.prod g = ∏ i, g i (f i) | f.prod_of_support_subset (subset_univ _) g (λ x _, h x) | lemma | finsupp.prod_fintype | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_single_index {a : α} {b : M} {h : α → M → N} (h_zero : h a 0 = 1) :
(single a b).prod h = h a b | calc (single a b).prod h = ∏ x in {a}, h x (single a b x) :
prod_of_support_subset _ support_single_subset h $
λ x hx, (mem_singleton.1 hx).symm ▸ h_zero
... = h a b : by simp | lemma | finsupp.prod_single_index | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_range_index {f : M → M'} {hf : f 0 = 0} {g : α →₀ M} {h : α → M' → N}
(h0 : ∀a, h a 0 = 1) : (map_range f hf g).prod h = g.prod (λa b, h a (f b)) | finset.prod_subset support_map_range $ λ _ _ H,
by rw [not_mem_support_iff.1 H, h0] | lemma | finsupp.prod_map_range_index | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"finset.prod_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_zero_index {h : α → M → N} : (0 : α →₀ M).prod h = 1 | rfl | lemma | finsupp.prod_zero_index | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comm (f : α →₀ M) (g : β →₀ M') (h : α → M → β → M' → N) :
f.prod (λ x v, g.prod (λ x' v', h x v x' v')) = g.prod (λ x' v', f.prod (λ x v, h x v x' v')) | finset.prod_comm | lemma | finsupp.prod_comm | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"finset.prod_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_ite_eq [decidable_eq α] (f : α →₀ M) (a : α) (b : α → M → N) :
f.prod (λ x v, ite (a = x) (b x v) 1) = ite (a ∈ f.support) (b a (f a)) 1 | by { dsimp [finsupp.prod], rw f.support.prod_ite_eq, } | lemma | finsupp.prod_ite_eq | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"finsupp.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_ite_self_eq
[decidable_eq α] {N : Type*} [add_comm_monoid N] (f : α →₀ N) (a : α) :
f.sum (λ x v, ite (a = x) v 0) = f a | begin
classical,
convert f.sum_ite_eq a (λ x, id),
simp [ite_eq_right_iff.2 eq.symm]
end | lemma | finsupp.sum_ite_self_eq | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_ite_eq' [decidable_eq α] (f : α →₀ M) (a : α) (b : α → M → N) :
f.prod (λ x v, ite (x = a) (b x v) 1) = ite (a ∈ f.support) (b a (f a)) 1 | by { dsimp [finsupp.prod], rw f.support.prod_ite_eq', } | lemma | finsupp.prod_ite_eq' | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"finsupp.prod"
] | A restatement of `prod_ite_eq` with the equality test reversed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_ite_self_eq'
[decidable_eq α] {N : Type*} [add_comm_monoid N] (f : α →₀ N) (a : α) :
f.sum (λ x v, ite (x = a) v 0) = f a | begin
classical,
convert f.sum_ite_eq' a (λ x, id),
simp [ite_eq_right_iff.2 eq.symm]
end | lemma | finsupp.sum_ite_self_eq' | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_pow [fintype α] (f : α →₀ ℕ) (g : α → N) :
f.prod (λ a b, g a ^ b) = ∏ a, g a ^ (f a) | f.prod_fintype _ $ λ a, pow_zero _ | lemma | finsupp.prod_pow | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"fintype",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
on_finset_prod {s : finset α} {f : α → M} {g : α → M → N}
(hf : ∀a, f a ≠ 0 → a ∈ s) (hg : ∀ a, g a 0 = 1) :
(on_finset s f hf).prod g = ∏ a in s, g a (f a) | finset.prod_subset support_on_finset_subset $ by simp [*] { contextual := tt } | lemma | finsupp.on_finset_prod | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"finset",
"finset.prod_subset"
] | If `g` maps a second argument of 0 to 1, then multiplying it over the
result of `on_finset` is the same as multiplying it over the original
`finset`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_prod_erase (f : α →₀ M) (y : α) (g : α → M → N) (hyf : y ∈ f.support) :
g y (f y) * (erase y f).prod g = f.prod g | begin
classical,
rw [finsupp.prod, finsupp.prod, ←finset.mul_prod_erase _ _ hyf, finsupp.support_erase,
finset.prod_congr rfl],
intros h hx,
rw finsupp.erase_ne (ne_of_mem_erase hx),
end | lemma | finsupp.mul_prod_erase | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"finset.prod_congr",
"finsupp.erase_ne",
"finsupp.prod",
"finsupp.support_erase"
] | Taking a product over `f : α →₀ M` is the same as multiplying the value on a single element
`y ∈ f.support` by the product over `erase y f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_prod_erase' (f : α →₀ M) (y : α) (g : α → M → N) (hg : ∀ (i : α), g i 0 = 1) :
g y (f y) * (erase y f).prod g = f.prod g | begin
classical,
by_cases hyf : y ∈ f.support,
{ exact finsupp.mul_prod_erase f y g hyf },
{ rw [not_mem_support_iff.mp hyf, hg y, erase_of_not_mem_support hyf, one_mul] },
end | lemma | finsupp.mul_prod_erase' | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"finsupp.mul_prod_erase",
"one_mul"
] | Generalization of `finsupp.mul_prod_erase`: if `g` maps a second argument of 0 to 1,
then its product over `f : α →₀ M` is the same as multiplying the value on any element
`y : α` by the product over `erase y f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.submonoid_class.finsupp_prod_mem {S : Type*} [set_like S N] [submonoid_class S N]
(s : S) (f : α →₀ M) (g : α → M → N) (h : ∀ c, f c ≠ 0 → g c (f c) ∈ s) : f.prod g ∈ s | prod_mem $ λ i hi, h _ (finsupp.mem_support_iff.mp hi) | lemma | submonoid_class.finsupp_prod_mem | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"prod_mem",
"set_like",
"submonoid_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_congr {f : α →₀ M} {g1 g2 : α → M → N}
(h : ∀ x ∈ f.support, g1 x (f x) = g2 x (f x)) : f.prod g1 = f.prod g2 | finset.prod_congr rfl h | lemma | finsupp.prod_congr | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"finset.prod_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_finsupp_prod [has_zero M] [comm_monoid N] [comm_monoid P] {H : Type*}
[monoid_hom_class H N P] (h : H) (f : α →₀ M) (g : α → M → N) :
h (f.prod g) = f.prod (λ a b, h (g a b)) | map_prod h _ _ | lemma | map_finsupp_prod | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"comm_monoid",
"map_prod",
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv.map_finsupp_prod [has_zero M] [comm_monoid N] [comm_monoid P]
(h : N ≃* P) (f : α →₀ M) (g : α → M → N) : h (f.prod g) = f.prod (λ a b, h (g a b)) | map_finsupp_prod h f g | lemma | mul_equiv.map_finsupp_prod | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"comm_monoid",
"map_finsupp_prod"
] | Deprecated, use `_root_.map_finsupp_prod` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.map_finsupp_prod [has_zero M] [comm_monoid N] [comm_monoid P]
(h : N →* P) (f : α →₀ M) (g : α → M → N) : h (f.prod g) = f.prod (λ a b, h (g a b)) | map_finsupp_prod h f g | lemma | monoid_hom.map_finsupp_prod | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"comm_monoid",
"map_finsupp_prod"
] | Deprecated, use `_root_.map_finsupp_prod` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_finsupp_sum [has_zero M] [semiring R] [semiring S]
(h : R →+* S) (f : α →₀ M) (g : α → M → R) : h (f.sum g) = f.sum (λ a b, h (g a b)) | map_finsupp_sum h f g | lemma | ring_hom.map_finsupp_sum | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"semiring"
] | Deprecated, use `_root_.map_finsupp_sum` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_finsupp_prod [has_zero M] [comm_semiring R] [comm_semiring S]
(h : R →+* S) (f : α →₀ M) (g : α → M → R) : h (f.prod g) = f.prod (λ a b, h (g a b)) | map_finsupp_prod h f g | lemma | ring_hom.map_finsupp_prod | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"comm_semiring",
"map_finsupp_prod"
] | Deprecated, use `_root_.map_finsupp_prod` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.coe_finsupp_prod [has_zero β] [monoid N] [comm_monoid P]
(f : α →₀ β) (g : α → β → N →* P) :
⇑(f.prod g) = f.prod (λ i fi, g i fi) | monoid_hom.coe_finset_prod _ _ | lemma | monoid_hom.coe_finsupp_prod | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"comm_monoid",
"monoid",
"monoid_hom.coe_finset_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.finsupp_prod_apply [has_zero β] [monoid N] [comm_monoid P]
(f : α →₀ β) (g : α → β → N →* P) (x : N) :
f.prod g x = f.prod (λ i fi, g i fi x) | monoid_hom.finset_prod_apply _ _ _ | lemma | monoid_hom.finsupp_prod_apply | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"comm_monoid",
"monoid",
"monoid_hom.finset_prod_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_multiset_sum [add_comm_monoid M] (s : multiset M) (a : α) :
single a s.sum = (s.map (single a)).sum | multiset.induction_on s (single_zero _) $ λ a s ih,
by rw [multiset.sum_cons, single_add, ih, multiset.map_cons, multiset.sum_cons] | lemma | finsupp.single_multiset_sum | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid",
"ih",
"multiset",
"multiset.induction_on",
"multiset.map_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_finset_sum [add_comm_monoid M] (s : finset ι) (f : ι → M) (a : α) :
single a (∑ b in s, f b) = ∑ b in s, single a (f b) | begin
transitivity,
apply single_multiset_sum,
rw [multiset.map_map],
refl
end | lemma | finsupp.single_finset_sum | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid",
"finset",
"multiset.map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_sum [has_zero M] [add_comm_monoid N] (s : ι →₀ M) (f : ι → M → N) (a : α) :
single a (s.sum f) = s.sum (λd c, single a (f d c)) | single_finset_sum _ _ _ | lemma | finsupp.single_sum | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_neg_index [add_group G] [comm_monoid M] {g : α →₀ G} {h : α → G → M}
(h0 : ∀a, h a 0 = 1) :
(-g).prod h = g.prod (λa b, h a (- b)) | prod_map_range_index h0 | lemma | finsupp.prod_neg_index | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_group",
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_sum_apply [add_comm_monoid N] (S : finset ι) (f : ι → α →₀ N) (a : α) :
(∑ i in S, f i) a = ∑ i in S, f i a | (apply_add_hom a : (α →₀ N) →+ _).map_sum _ _ | lemma | finsupp.finset_sum_apply | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_apply [has_zero M] [add_comm_monoid N]
{f : α →₀ M} {g : α → M → β →₀ N} {a₂ : β} :
(f.sum g) a₂ = f.sum (λa₁ b, g a₁ b a₂) | finset_sum_apply _ _ _ | lemma | finsupp.sum_apply | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_finset_sum [add_comm_monoid N] (S : finset ι) (f : ι → α →₀ N) :
⇑(∑ i in S, f i) = ∑ i in S, f i | (coe_fn_add_hom : (α →₀ N) →+ _).map_sum _ _ | lemma | finsupp.coe_finset_sum | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sum [has_zero M] [add_comm_monoid N] (f : α →₀ M) (g : α → M → β →₀ N) :
⇑(f.sum g) = f.sum (λ a₁ b, g a₁ b) | coe_finset_sum _ _ | lemma | finsupp.coe_sum | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_sum [decidable_eq β] [has_zero M] [add_comm_monoid N]
{f : α →₀ M} {g : α → M → (β →₀ N)} :
(f.sum g).support ⊆ f.support.bUnion (λa, (g a (f a)).support) | have ∀ c, f.sum (λ a b, g a b c) ≠ 0 → (∃ a, f a ≠ 0 ∧ ¬ (g a (f a)) c = 0),
from assume a₁ h,
let ⟨a, ha, ne⟩ := finset.exists_ne_zero_of_sum_ne_zero h in
⟨a, mem_support_iff.mp ha, ne⟩,
by simpa only [finset.subset_iff, mem_support_iff, finset.mem_bUnion, sum_apply, exists_prop] | lemma | finsupp.support_sum | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid",
"exists_prop",
"finset.mem_bUnion",
"finset.subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_finset_sum [decidable_eq β] [add_comm_monoid M] {s : finset α} {f : α → (β →₀ M)} :
(finset.sum s f).support ⊆ s.bUnion (λ x, (f x).support) | begin
rw ←finset.sup_eq_bUnion,
induction s using finset.cons_induction_on with a s ha ih,
{ refl },
{ rw [finset.sum_cons, finset.sup_cons],
exact support_add.trans (finset.union_subset_union (finset.subset.refl _) ih), },
end | lemma | finsupp.support_finset_sum | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid",
"finset",
"finset.cons_induction_on",
"finset.subset.refl",
"finset.sup_cons",
"finset.union_subset_union",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_zero [has_zero M] [add_comm_monoid N] {f : α →₀ M} :
f.sum (λa b, (0 : N)) = 0 | finset.sum_const_zero | lemma | finsupp.sum_zero | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mul [has_zero M] [comm_monoid N] {f : α →₀ M} {h₁ h₂ : α → M → N} :
f.prod (λa b, h₁ a b * h₂ a b) = f.prod h₁ * f.prod h₂ | finset.prod_mul_distrib | lemma | finsupp.prod_mul | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"comm_monoid",
"finset.prod_mul_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_inv [has_zero M] [comm_group G] {f : α →₀ M}
{h : α → M → G} : f.prod (λa b, (h a b)⁻¹) = (f.prod h)⁻¹ | (map_prod ((monoid_hom.id G)⁻¹) _ _).symm | lemma | finsupp.prod_inv | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"comm_group",
"map_prod",
"monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_sub [has_zero M] [add_comm_group G] {f : α →₀ M}
{h₁ h₂ : α → M → G} :
f.sum (λa b, h₁ a b - h₂ a b) = f.sum h₁ - f.sum h₂ | finset.sum_sub_distrib | lemma | finsupp.sum_sub | algebra.big_operators | src/algebra/big_operators/finsupp.lean | [
"data.finsupp.indicator",
"algebra.big_operators.pi",
"algebra.big_operators.ring",
"algebra.big_operators.order",
"group_theory.submonoid.membership"
] | [
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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