P. 158 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	3lcm2e6woprm 15701	The least common multiple of three and two is six. In contrast to 3lcm2e6 15811, this proof does not use the property of 2 and 3 being prime, therefore it is much longer. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (3 lcm 2) = 6
Theorem	6lcm4e12 15702	The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.)
⊢ (6 lcm 4) = ;12
Theorem	absproddvds 15703*	The absolute value of the product of the elements of a finite subset of the integers is divisible by each element of this subset. (Contributed by AV, 21-Aug-2020.)
⊢ (𝜑 → 𝑍 ⊆ ℤ)    &   ⊢ (𝜑 → 𝑍 ∈ Fin)    &   ⊢ 𝑃 = (abs‘∏𝑧 ∈ 𝑍 𝑧)    ⇒   ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑃)
Theorem	absprodnn 15704*	The absolute value of the product of the elements of a finite subset of the integers not containing 0 is a poitive integer. (Contributed by AV, 21-Aug-2020.)
⊢ (𝜑 → 𝑍 ⊆ ℤ)    &   ⊢ (𝜑 → 𝑍 ∈ Fin)    &   ⊢ 𝑃 = (abs‘∏𝑧 ∈ 𝑍 𝑧)    &   ⊢ (𝜑 → 0 ∉ 𝑍)    ⇒   ⊢ (𝜑 → 𝑃 ∈ ℕ)
Theorem	fissn0dvds 15705*	For each finite subset of the integers not containing 0 there is a positive integer which is divisible by each element of this subset. (Contributed by AV, 21-Aug-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → ∃𝑛 ∈ ℕ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛)
Theorem	fissn0dvdsn0 15706*	For each finite subset of the integers not containing 0 there is a positive integer which is divisible by each element of this subset. (Contributed by AV, 21-Aug-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛} ≠ ∅)
Theorem	lcmfval 15707*	Value of the lcm function. (lcm‘𝑍) is the least common multiple of the integers contained in the finite subset of integers 𝑍. If at least one of the elements of 𝑍 is 0, the result is defined conventionally as 0. (Contributed by AV, 21-Apr-2020.) (Revised by AV, 16-Sep-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) = if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )))
Theorem	lcmf0val 15708	The value, by convention, of the least common multiple for a set containing 0 is 0. (Contributed by AV, 21-Apr-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → (lcm‘𝑍) = 0)
Theorem	lcmfn0val 15709*	The value of the lcm function for a set without 0. (Contributed by AV, 21-Aug-2020.) (Revised by AV, 16-Sep-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → (lcm‘𝑍) = inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ))
Theorem	lcmfnnval 15710*	The value of the lcm function for a subset of the positive integers. (Contributed by AV, 21-Aug-2020.) (Revised by AV, 16-Sep-2020.)
⊢ ((𝑍 ⊆ ℕ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) = inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ))
Theorem	lcmfcllem 15711*	Lemma for lcmfn0cl 15712 and dvdslcmf 15717. (Contributed by AV, 21-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → (lcm‘𝑍) ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛})
Theorem	lcmfn0cl 15712	Closure of the lcm function. (Contributed by AV, 21-Aug-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → (lcm‘𝑍) ∈ ℕ)
Theorem	lcmfpr 15713	The value of the lcm function for an unordered pair is the value of the lcm operator for both elements. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (lcm‘{𝑀, 𝑁}) = (𝑀 lcm 𝑁))
Theorem	lcmfcl 15714	Closure of the lcm function. (Contributed by AV, 21-Aug-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) ∈ ℕ0)
Theorem	lcmfnncl 15715	Closure of the lcm function. (Contributed by AV, 20-Apr-2020.)
⊢ ((𝑍 ⊆ ℕ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) ∈ ℕ)
Theorem	lcmfeq0b 15716	The least common multiple of a set of integers is 0 iff at least one of its element is 0. (Contributed by AV, 21-Aug-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((lcm‘𝑍) = 0 ↔ 0 ∈ 𝑍))
Theorem	dvdslcmf 15717*	The least common multiple of a set of integers is divisible by each of its elements. (Contributed by AV, 22-Aug-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ∀𝑥 ∈ 𝑍 𝑥 ∥ (lcm‘𝑍))
Theorem	lcmfledvds 15718*	A positive integer which is divisible by all elements of a set of integers bounds the least common multiple of the set. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍) → ((𝐾 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾) → (lcm‘𝑍) ≤ 𝐾))
Theorem	lcmf 15719*	Characterization of the least common multiple of a set of integers (without 0): A positiven integer is the least common multiple of a set of integers iff it divides each of the elements of the set and every integer which divides each of the elements of the set is greater than or equal to this integer. (Contributed by AV, 22-Aug-2020.)
⊢ ((𝐾 ∈ ℕ ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍)) → (𝐾 = (lcm‘𝑍) ↔ (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾 ∧ ∀𝑘 ∈ ℕ (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑘 → 𝐾 ≤ 𝑘))))
Theorem	lcmf0 15720	The least common multiple of the empty set is 1. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ (lcm‘∅) = 1
Theorem	lcmfsn 15721	The least common multiple of a singleton is its absolute value. (Contributed by AV, 22-Aug-2020.)
⊢ (𝑀 ∈ ℤ → (lcm‘{𝑀}) = (abs‘𝑀))
Theorem	lcmftp 15722	The least common multiple of a triple of integers is the least common multiple of the third integer and the least common multiple of the first two integers. Although there would be a shorter proof using lcmfunsn 15730, this explicit proof (not based on induction) should be kept. (Proof modification is discouraged.) (Contributed by AV, 23-Aug-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))
Theorem	lcmfunsnlem1 15723*	Lemma for lcmfdvds 15728 and lcmfunsnlem 15727 (Induction step part 1). (Contributed by AV, 25-Aug-2020.)
⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → ∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
Theorem	lcmfunsnlem2lem1 15724*	Lemma 1 for lcmfunsnlem2 15726. (Contributed by AV, 26-Aug-2020.)
⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))))) → ∀𝑘 ∈ ℕ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))
Theorem	lcmfunsnlem2lem2 15725*	Lemma 2 for lcmfunsnlem2 15726. (Contributed by AV, 26-Aug-2020.)
⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
Theorem	lcmfunsnlem2 15726*	Lemma for lcmfunsn 15730 and lcmfunsnlem 15727 (Induction step part 2). (Contributed by AV, 26-Aug-2020.)
⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
Theorem	lcmfunsnlem 15727*	Lemma for lcmfdvds 15728 and lcmfunsn 15730. These two theorems must be proven simultaneously by induction on the cardinality of a finite set 𝑌, because they depend on each other. This can be seen by the two parts lcmfunsnlem1 15723 and lcmfunsnlem2 15726 of the induction step, each of them using both induction hypotheses. (Contributed by AV, 26-Aug-2020.)
⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛)))
Theorem	lcmfdvds 15728*	The least common multiple of a set of integers divides any integer which is divisible by all elements of the set. (Contributed by AV, 26-Aug-2020.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾 → (lcm‘𝑍) ∥ 𝐾))
Theorem	lcmfdvdsb 15729*	Biconditional form of lcmfdvds 15728. (Contributed by AV, 26-Aug-2020.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝐾 ↔ (lcm‘𝑍) ∥ 𝐾))
Theorem	lcmfunsn 15730	The lcm function for a union of a set of integer and a singleton. (Contributed by AV, 26-Aug-2020.)
⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin ∧ 𝑁 ∈ ℤ) → (lcm‘(𝑌 ∪ {𝑁})) = ((lcm‘𝑌) lcm 𝑁))
Theorem	lcmfun 15731	The lcm function for a union of sets of integers. (Contributed by AV, 27-Aug-2020.)
⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))
Theorem	lcmfass 15732	Associative law for the lcm function. (Contributed by AV, 27-Aug-2020.)
⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘({(lcm‘𝑌)} ∪ 𝑍)) = (lcm‘(𝑌 ∪ {(lcm‘𝑍)})))
Theorem	lcmf2a3a4e12 15733	The least common multiple of 2 , 3 and 4 is 12. (Contributed by AV, 27-Aug-2020.)
⊢ (lcm‘{2, 3, 4}) = ;12
Theorem	lcmflefac 15734	The least common multiple of all positive integers less than or equal to an integer is less than or equal to the factorial of the integer. (Contributed by AV, 16-Aug-2020.) (Revised by AV, 27-Aug-2020.)
⊢ (𝑁 ∈ ℕ → (lcm‘(1...𝑁)) ≤ (!‘𝑁))
6.1.12  Coprimality and Euclid's lemma According to Wikipedia "Coprime integers", see https://en.wikipedia.org/wiki/Coprime_integers (16-Aug-2020) "[...] two integers a and b are said to be relatively prime, mutually prime, or coprime [...] if the only positive integer (factor) that divides both of them is 1. Consequently, any prime number that divides one does not divide the other. This is equivalent to their greatest common divisor (gcd) being 1.". In the following, we use this equivalent characterization to say that 𝐴 ∈ ℤ and 𝐵 ∈ ℤ are coprime (or relatively prime) if (𝐴 gcd 𝐵) = 1. The equivalence of the definitions is shown by coprmgcdb 15735. The negation, i.e. two integers are not coprime, can be expressed either by (𝐴 gcd 𝐵) ≠ 1, see ncoprmgcdne1b 15736, or equivalently by 1 < (𝐴 gcd 𝐵), see ncoprmgcdgt1b 15737. A proof of Euclid's lemma based on coprimality is provided in coprmdvds 15739 (see euclemma 15796 for a version of Euclid's lemma for primes).
Theorem	coprmgcdb 15735*	Two positive integers are coprime, i.e. the only positive integer that divides both of them is 1, iff their greatest common divisor is 1. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1) ↔ (𝐴 gcd 𝐵) = 1))
Theorem	ncoprmgcdne1b 15736*	Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) ↔ (𝐴 gcd 𝐵) ≠ 1))
Theorem	ncoprmgcdgt1b 15737*	Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is greater than 1. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) ↔ 1 < (𝐴 gcd 𝐵)))
Theorem	coprmdvds1 15738	If two positive integers are coprime, i.e. their greatest common divisor is 1, the only positive integer that divides both of them is 1. (Contributed by AV, 4-Aug-2021.)
⊢ ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐼 = 1))
Theorem	coprmdvds 15739	Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. Generalization of euclemma 15796. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 · 𝑁) ∧ (𝐾 gcd 𝑀) = 1) → 𝐾 ∥ 𝑁))
Theorem	coprmdvds2 15740	If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → ((𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾) → (𝑀 · 𝑁) ∥ 𝐾))
Theorem	mulgcddvds 15741	One half of rpmulgcd2 15742, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 gcd (𝑀 · 𝑁)) ∥ ((𝐾 gcd 𝑀) · (𝐾 gcd 𝑁)))
Theorem	rpmulgcd2 15742	If 𝑀 is relatively prime to 𝑁, then the GCD of 𝐾 with 𝑀 · 𝑁 is the product of the GCDs with 𝑀 and 𝑁 respectively. (Contributed by Mario Carneiro, 2-Jul-2015.)
⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → (𝐾 gcd (𝑀 · 𝑁)) = ((𝐾 gcd 𝑀) · (𝐾 gcd 𝑁)))
Theorem	qredeq 15743	Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) ∧ (𝑀 / 𝑁) = (𝑃 / 𝑄)) → (𝑀 = 𝑃 ∧ 𝑁 = 𝑄))
Theorem	qredeu 15744*	Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)
⊢ (𝐴 ∈ ℚ → ∃!𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))
Theorem	rpmul 15745	If 𝐾 is relatively prime to 𝑀 and to 𝑁, it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝐾 gcd 𝑀) = 1 ∧ (𝐾 gcd 𝑁) = 1) → (𝐾 gcd (𝑀 · 𝑁)) = 1))
Theorem	rpdvds 15746	If 𝐾 is relatively prime to 𝑁 then it is also relatively prime to any divisor 𝑀 of 𝑁. (Contributed by Mario Carneiro, 19-Jun-2015.)
⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝐾 gcd 𝑁) = 1 ∧ 𝑀 ∥ 𝑁)) → (𝐾 gcd 𝑀) = 1)
Theorem	coprmprod 15747*	The product of the elements of a sequence of pairwise coprime positive integers is coprime to a positive integer which is coprime to all integers of the sequence. (Contributed by AV, 18-Aug-2020.)
⊢ (((𝑀 ∈ Fin ∧ 𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1) → (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1))
Theorem	coprmproddvdslem 15748*	Lemma for coprmproddvds 15749: Induction step. (Contributed by AV, 19-Aug-2020.)
⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)))
Theorem	coprmproddvds 15749*	If a positive integer is divisible by each element of a set of pairwise coprime positive integers, then it is divisible by their product. (Contributed by AV, 19-Aug-2020.)
⊢ (((𝑀 ⊆ ℕ ∧ 𝑀 ∈ Fin) ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾)
6.1.13  Cancellability of congruences
Theorem	congr 15750*	Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer 𝐴 is congruent to an integer 𝐵 modulo 𝑀 if their difference is a multiple of 𝑀. See also the definition in [ApostolNT] p. 104: "... 𝑎 is congruent to 𝑏 modulo 𝑚, and we write 𝑎≡𝑏 (mod 𝑚) if 𝑚 divides the difference 𝑎 − 𝑏", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence, 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℕ) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = (𝐴 − 𝐵)))
Theorem	divgcdcoprm0 15751	Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
Theorem	divgcdcoprmex 15752*	Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1))
Theorem	cncongr1 15753	One direction of the bicondition in cncongr 15755. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) → (𝐴 mod 𝑀) = (𝐵 mod 𝑀)))
Theorem	cncongr2 15754	The other direction of the bicondition in cncongr 15755. (Contributed by AV, 11-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) → ((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁)))
Theorem	cncongr 15755	Cancellability of Congruences (see ProofWiki "Cancellability of Congruences, https://proofwiki.org/wiki/Cancellability_of_Congruences, 10-Jul-2021): Two products with a common factor are congruent modulo a positive integer iff the other factors are congruent modulo the integer divided by the greates common divisor of the integer and the common factor. See also Theorem 5.4 "Cancellation law" in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) ↔ (𝐴 mod 𝑀) = (𝐵 mod 𝑀)))
Theorem	cncongrcoprm 15756	Corollary 1 of Cancellability of Congruences: Two products with a common factor are congruent modulo an integer being coprime to the common factor iff the other factors are congruent modulo the integer. (Contributed by AV, 13-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ (𝐶 gcd 𝑁) = 1)) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) ↔ (𝐴 mod 𝑁) = (𝐵 mod 𝑁)))
6.2  Elementary prime number theory
6.2.1  Elementary properties Remark: to represent odd prime numbers, i.e., all prime numbers except 2, the idiom 𝑃 ∈ (ℙ ∖ {2}) is used. It is a little bit shorter than (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2). Both representations can be converted into each other by eldifsn 4536.
Syntax	cprime 15757	Extend the definition of a class to include the set of prime numbers.
class ℙ
Definition	df-prm 15758*	Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o}
Theorem	isprm 15759*	The predicate "is a prime number". A prime number is a positive integer with exactly two positive divisors. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o))
Theorem	prmnn 15760	A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
Theorem	prmz 15761	A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.)
⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
Theorem	prmssnn 15762	The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
⊢ ℙ ⊆ ℕ
Theorem	prmex 15763	The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
⊢ ℙ ∈ V
Theorem	1nprm 15764	1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
⊢ ¬ 1 ∈ ℙ
Theorem	1idssfct 15765*	The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑁 ∈ ℕ → {1, 𝑁} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁})
Theorem	isprm2lem 15766*	Lemma for isprm2 15767. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃}))
Theorem	isprm2 15767*	The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only positive divisors are 1 and itself. Definition in [ApostolNT] p. 16. (Contributed by Paul Chapman, 26-Oct-2012.)
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ ℕ (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃))))
Theorem	isprm3 15768*	The predicate "is a prime number". A prime number is an integer greater than or equal to 2 with no divisors strictly between 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.)
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ (2...(𝑃 − 1)) ¬ 𝑧 ∥ 𝑃))
Theorem	isprm4 15769*	The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only divisor greater than or equal to 2 is itself. (Contributed by Paul Chapman, 26-Oct-2012.)
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ (ℤ≥‘2)(𝑧 ∥ 𝑃 → 𝑧 = 𝑃)))
Theorem	prmind2 15770*	A variation on prmind 15771 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂))    &   ⊢ 𝜓    &   ⊢ ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑)    &   ⊢ ((𝑦 ∈ (ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜂)
Theorem	prmind 15771*	Perform induction over the multiplicative structure of ℕ. If a property 𝜑(𝑥) holds for the primes and 1 and is preserved under multiplication, then it holds for every positive integer. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂))    &   ⊢ 𝜓    &   ⊢ (𝑥 ∈ ℙ → 𝜑)    &   ⊢ ((𝑦 ∈ (ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜂)
Theorem	dvdsprime 15772	If 𝑀 divides a prime, then 𝑀 is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ) → (𝑀 ∥ 𝑃 ↔ (𝑀 = 𝑃 ∨ 𝑀 = 1)))
Theorem	nprm 15773	A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → ¬ (𝐴 · 𝐵) ∈ ℙ)
Theorem	nprmi 15774	An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    &   ⊢ 1 < 𝐴    &   ⊢ 1 < 𝐵    &   ⊢ (𝐴 · 𝐵) = 𝑁    ⇒   ⊢ ¬ 𝑁 ∈ ℙ
Theorem	dvdsnprmd 15775	If a number is divisible by an integer greater than 1 and less then the number, the number is not prime. (Contributed by AV, 24-Jul-2021.)
⊢ (𝜑 → 1 < 𝐴)    &   ⊢ (𝜑 → 𝐴 < 𝑁)    &   ⊢ (𝜑 → 𝐴 ∥ 𝑁)    ⇒   ⊢ (𝜑 → ¬ 𝑁 ∈ ℙ)
Theorem	prm2orodd 15776	A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃))
Theorem	2prm 15777	2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
⊢ 2 ∈ ℙ
Theorem	3prm 15778	3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 3 ∈ ℙ
Theorem	4nprm 15779	4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
⊢ ¬ 4 ∈ ℙ
Theorem	prmuz2 15780	A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
Theorem	prmgt1 15781	A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
Theorem	prmm2nn0 15782	Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
⊢ (𝑃 ∈ ℙ → (𝑃 − 2) ∈ ℕ0)
Theorem	oddprmgt2 15783	An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
Theorem	oddprmge3 15784	An odd prime is greater than or equal to 3. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 20-Aug-2021.)
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘3))
Theorem	sqnprm 15785	A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ)
Theorem	dvdsprm 15786	An integer greater than or equal to 2 divides a prime number iff it is equal to it. (Contributed by Paul Chapman, 26-Oct-2012.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑃 ∈ ℙ) → (𝑁 ∥ 𝑃 ↔ 𝑁 = 𝑃))
Theorem	exprmfct 15787*	Every integer greater than or equal to 2 has a prime factor. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 20-Jun-2015.)
⊢ (𝑁 ∈ (ℤ≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
Theorem	prmdvdsfz 15788*	Each integer greater than 1 and less then or equal to a fixed number is divisible by a prime less then or equal to this fixed number. (Contributed by AV, 15-Aug-2020.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∃𝑝 ∈ ℙ (𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼))
Theorem	nprmdvds1 15789	No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
⊢ (𝑃 ∈ ℙ → ¬ 𝑃 ∥ 1)
Theorem	isprm5 15790*	One need only check prime divisors of 𝑃 up to √𝑃 in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
Theorem	isprm7 15791*	One need only check prime divisors of 𝑃 up to √𝑃 in order to ensure primality. This version of isprm5 15790 combines the primality and bound on 𝑧 into a finite interval of prime numbers. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ ((2...(⌊‘(√‘𝑃))) ∩ ℙ) ¬ 𝑧 ∥ 𝑃))
Theorem	maxprmfct 15792*	The set of prime factors of an integer greater than or equal to 2 satisfies the conditions to have a supremum, and that supremum is a member of the set. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ 𝑆 = {𝑧 ∈ ℙ ∣ 𝑧 ∥ 𝑁}    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘2) → ((𝑆 ⊆ ℤ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) ∧ sup(𝑆, ℝ, < ) ∈ 𝑆))
Theorem	divgcdodd 15793	Either 𝐴 / (𝐴 gcd 𝐵) is odd or 𝐵 / (𝐴 gcd 𝐵) is odd. (Contributed by Scott Fenton, 19-Apr-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)) ∨ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))))
6.2.2  Coprimality and Euclid's lemma (cont.) This section is about coprimality with respect to primes, and a special version of Euclid's lemma for primes is provided, see euclemma 15796.
Theorem	coprm 15794	A prime number either divides an integer or is coprime to it, but not both. Theorem 1.8 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
Theorem	prmrp 15795	Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑃 gcd 𝑄) = 1 ↔ 𝑃 ≠ 𝑄))
Theorem	euclemma 15796	Euclid's lemma. A prime number divides the product of two integers iff it divides at least one of them. Theorem 1.9 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ (𝑀 · 𝑁) ↔ (𝑃 ∥ 𝑀 ∨ 𝑃 ∥ 𝑁)))
Theorem	isprm6 15797*	A number is prime iff it satisfies Euclid's lemma euclemma 15796. (Contributed by Mario Carneiro, 6-Sep-2015.)
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦))))
Theorem	prmdvdsexp 15798	A prime divides a positive power of an integer iff it divides the integer. (Contributed by Mario Carneiro, 24-Feb-2014.) (Revised by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ (𝐴↑𝑁) ↔ 𝑃 ∥ 𝐴))
Theorem	prmdvdsexpb 15799	A prime divides a positive power of another iff they are equal. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 24-Feb-2014.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ (𝑄↑𝑁) ↔ 𝑃 = 𝑄))
Theorem	prmdvdsexpr 15800	If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑃 ∥ (𝑄↑𝑁) → 𝑃 = 𝑄))