File: Mathlib/Algebra/BigOperators/Associated.lean

full_name: _aux_Mathlib_Algebra_BigOperators_Associated___unexpand_Associated_1
code: local infixl:50 " ~ᵤ " => Associated
goal: Lean.PrettyPrinter.Unexpander

full_name: «_aux_Mathlib_Algebra_BigOperators_Associated___macroRules_term_~ᵤ__1»
code: local infixl:50 " ~ᵤ " => Associated
goal: Lean.Macro

full_name: Prime.exists_mem_multiset_dvd
code: theorem exists_mem_multiset_dvd (hp : Prime p) {s : Multiset α} : p ∣ s.prod → ∃ a ∈ s, p ∣ a 
goal: p ∣ s.prod → ∃ a ∈ s, p ∣ a

full_name: Prime.exists_mem_multiset_map_dvd
code: theorem exists_mem_multiset_map_dvd (hp : Prime p) {s : Multiset β} {f : β → α} :
    p ∣ (s.map f).prod → ∃ a ∈ s, p ∣ f a 
goal: p ∣ (Multiset.map f s).prod → ∃ a ∈ s, p ∣ f a

full_name: Prime.exists_mem_finset_dvd
code: theorem exists_mem_finset_dvd (hp : Prime p) {s : Finset β} {f : β → α} :
    p ∣ s.prod f → ∃ i ∈ s, p ∣ f i 
goal: p ∣ s.prod f → ∃ i ∈ s, p ∣ f i

full_name: Prod.associated_iff
code: theorem Prod.associated_iff {M N : Type*} [Monoid M] [Monoid N] {x z : M × N} :
    x ~ᵤ z ↔ x.1 ~ᵤ z.1 ∧ x.2 ~ᵤ z.2 
goal: x ~ᵤ z ↔ x.1 ~ᵤ z.1 ∧ x.2 ~ᵤ z.2

full_name: Associated.prod
code: theorem Associated.prod {M : Type*} [CommMonoid M] {ι : Type*} (s : Finset ι) (f : ι → M)
    (g : ι → M) (h : ∀ i, i ∈ s → (f i) ~ᵤ (g i)) : (∏ i ∈ s, f i) ~ᵤ (∏ i ∈ s, g i) 
goal: ∏ i ∈ s, f i ~ᵤ ∏ i ∈ s, g i

full_name: exists_associated_mem_of_dvd_prod
code: theorem exists_associated_mem_of_dvd_prod [CancelCommMonoidWithZero α] {p : α} (hp : Prime p)
    {s : Multiset α} : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q 
goal: (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q

full_name: Multiset.prod_primes_dvd
code: theorem Multiset.prod_primes_dvd [CancelCommMonoidWithZero α]
    [∀ a : α, DecidablePred (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 
goal: s.prod ∣ n

full_name: Finset.prod_primes_dvd
code: theorem Finset.prod_primes_dvd [CancelCommMonoidWithZero α] [Unique αˣ] {s : Finset α} (n : α)
    (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) : (∏ p ∈ s, p) ∣ n 
goal: ∏ p ∈ s, p ∣ n

full_name: Associates.prod_mk
code: theorem prod_mk {p : Multiset α} : (p.map Associates.mk).prod = Associates.mk p.prod 
goal: (Multiset.map Associates.mk p).prod = Associates.mk p.prod

full_name: Associates.finset_prod_mk
code: theorem finset_prod_mk {p : Finset β} {f : β → α} :
    (∏ i ∈ p, Associates.mk (f i)) = Associates.mk (∏ i ∈ p, f i) 
goal: ∏ i ∈ p, Associates.mk (f i) = Associates.mk (∏ i ∈ p, f i)

full_name: Associates.rel_associated_iff_map_eq_map
code: theorem rel_associated_iff_map_eq_map {p q : Multiset α} :
    Multiset.Rel Associated p q ↔ p.map Associates.mk = q.map Associates.mk 
goal: Multiset.Rel Associated p q ↔ Multiset.map Associates.mk p = Multiset.map Associates.mk q

full_name: Associates.prod_eq_one_iff
code: theorem prod_eq_one_iff {p : Multiset (Associates α)} :
    p.prod = 1 ↔ ∀ a ∈ p, (a : Associates α) = 1 
goal: p.prod = 1 ↔ ∀ a ∈ p, a = 1

full_name: Associates.prod_le_prod
code: theorem prod_le_prod {p q : Multiset (Associates α)} (h : p ≤ q) : p.prod ≤ q.prod 
goal: p.prod ≤ q.prod

full_name: Associates.exists_mem_multiset_le_of_prime
code: theorem exists_mem_multiset_le_of_prime {s : Multiset (Associates α)} {p : Associates α}
    (hp : Prime p) : p ≤ s.prod → ∃ a ∈ s, p ≤ a 
goal: p ≤ s.prod → ∃ a ∈ s, p ≤ a

full_name: Multiset.prod_ne_zero_of_prime
code: theorem prod_ne_zero_of_prime [CancelCommMonoidWithZero α] [Nontrivial α] (s : Multiset α)
    (h : ∀ x ∈ s, Prime x) : s.prod ≠ 0 
goal: s.prod ≠ 0

full_name: Prime.dvd_finset_prod_iff
code: theorem 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 
goal: p ∣ S.prod g ↔ ∃ a ∈ S, p ∣ g a

full_name: Prime.not_dvd_finset_prod
code: theorem Prime.not_dvd_finset_prod {S : Finset α} {p : M} (pp : Prime p) {g : α → M}
    (hS : ∀ a ∈ S, ¬p ∣ g a) : ¬p ∣ S.prod g 
goal: ¬p ∣ S.prod g

full_name: Prime.dvd_finsupp_prod_iff
code: theorem 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) 
goal: p ∣ f.prod g ↔ ∃ a ∈ f.support, p ∣ g a (f a)

full_name: Prime.not_dvd_finsupp_prod
code: theorem Prime.not_dvd_finsupp_prod {f : α →₀ M} {g : α → M → ℕ} {p : ℕ} (pp : Prime p)
    (hS : ∀ a ∈ f.support, ¬p ∣ g a (f a)) : ¬p ∣ f.prod g 
goal: ¬p ∣ f.prod g