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Theorem List for Metamath Proof Explorer - 15401-15500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlcmfunsnlem 15401* Lemma for lcmfdvds 15402 and lcmfunsn 15404. 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 15397 and lcmfunsnlem2 15400 of the induction step, each of them using both induction hypotheses. (Contributed by AV, 26-Aug-2020.)
((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛)))

Theoremlcmfdvds 15402* 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) → (∀𝑚𝑍 𝑚𝐾 → (lcm𝑍) ∥ 𝐾))

Theoremlcmfdvdsb 15403* Biconditional form of lcmfdvds 15402. (Contributed by AV, 26-Aug-2020.)
((𝐾 ∈ ℤ ∧ 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (∀𝑚𝑍 𝑚𝐾 ↔ (lcm𝑍) ∥ 𝐾))

Theoremlcmfunsn 15404 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})) = ((lcm𝑌) lcm 𝑁))

Theoremlcmfun 15405 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‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍)))

Theoremlcmfass 15406 Associative law for the lcm function. (Contributed by AV, 27-Aug-2020.)
(((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘({(lcm𝑌)} ∪ 𝑍)) = (lcm‘(𝑌 ∪ {(lcm𝑍)})))

Theoremlcmf2a3a4e12 15407 The least common multiple of 2 , 3 and 4 is 12. (Contributed by AV, 27-Aug-2020.)
(lcm‘{2, 3, 4}) = 12

Theoremlcmflefac 15408 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 15409. The negation, i.e. two integers are not coprime, can be expressed either by (𝐴 gcd 𝐵) ≠ 1, see ncoprmgcdne1b 15410, or equivalently by 1 < (𝐴 gcd 𝐵), see ncoprmgcdgt1b 15411.

A proof of Euclid's lemma based on coprimality is provided in coprmdvds 15413 (see euclemma 15472 for a version of Euclid's lemma for primes).

Theoremcoprmgcdb 15409* 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 ((𝑖𝐴𝑖𝐵) → 𝑖 = 1) ↔ (𝐴 gcd 𝐵) = 1))

Theoremncoprmgcdne1b 15410* 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)(𝑖𝐴𝑖𝐵) ↔ (𝐴 gcd 𝐵) ≠ 1))

Theoremncoprmgcdgt1b 15411* 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)(𝑖𝐴𝑖𝐵) ↔ 1 < (𝐴 gcd 𝐵)))

Theoremcoprmdvds1 15412 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 ∧ 𝐼𝐹𝐼𝐺) → 𝐼 = 1))

Theoremcoprmdvds 15413 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 15472. (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) → 𝐾𝑁))

TheoremcoprmdvdsOLD 15414 If an integer divides the product of two integers and is coprime to one of them, then it divides the other. (Contributed by Paul Chapman, 22-Jun-2011.) Obsolete version of coprmdvds 15413 as of 10-Jul-2021. (New usage is discouraged.) (Proof modification is discouraged.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 · 𝑁) ∧ (𝐾 gcd 𝑀) = 1) → 𝐾𝑁))

Theoremcoprmdvds2 15415 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) → ((𝑀𝐾𝑁𝐾) → (𝑀 · 𝑁) ∥ 𝐾))

Theoremmulgcddvds 15416 One half of rpmulgcd2 15417, 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)))

Theoremrpmulgcd2 15417 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)))

Theoremqredeq 15418 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 = 𝑃𝑁 = 𝑄))

Theoremqredeu 15419* Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)
(𝐴 ∈ ℚ → ∃!𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))

Theoremrpmul 15420 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))

Theoremrpdvds 15421 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 ∧ 𝑀𝑁)) → (𝐾 gcd 𝑀) = 1)

Theoremcoprmprod 15422* 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 ∧ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1) → (∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))

Theoremcoprmproddvdslem 15423* Lemma for coprmproddvds 15424: Induction step. (Contributed by AV, 19-Aug-2020.)
((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ∧ ∀𝑚𝑦 (𝐹𝑚) ∥ 𝐾)) → ∏𝑚𝑦 (𝐹𝑚) ∥ 𝐾) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) ∥ 𝐾)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) ∥ 𝐾)))

Theoremcoprmproddvds 15424* 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) ∧ (∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ∧ ∀𝑚𝑀 (𝐹𝑚) ∥ 𝐾)) → ∏𝑚𝑀 (𝐹𝑚) ∥ 𝐾)

6.1.13  Cancellability of congruences

Theoremcongr 15425* 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 𝑎𝑏 (mod 𝑚) if 𝑚 divides the difference 𝑎𝑏", 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) = (𝐴𝐵)))

Theoremdivgcdcoprm0 15426 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)

Theoremdivgcdcoprmex 15427* 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))

Theoremcncongr1 15428 One direction of the bicondition in cncongr 15430. 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)))

Theoremcncongr2 15429 The other direction of the bicondition in cncongr 15430. (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)))

Theoremcncongr 15430 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)))

Theoremcncongrcoprm 15431 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 4350.

Syntaxcprime 15432 Extend the definition of a class to include the set of prime numbers.
class

Definitiondf-prm 15433* Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)
ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2𝑜}

Theoremisprm 15434* 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 ∣ 𝑛𝑃} ≈ 2𝑜))

Theoremprmnn 15435 A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑃 ∈ ℙ → 𝑃 ∈ ℕ)

Theoremprmz 15436 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)

Theoremprmssnn 15437 The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
ℙ ⊆ ℕ

Theoremprmex 15438 The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
ℙ ∈ V

Theorem1nprm 15439 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
¬ 1 ∈ ℙ

Theorem1idssfct 15440* The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℕ → {1, 𝑁} ⊆ {𝑛 ∈ ℕ ∣ 𝑛𝑁})

Theoremisprm2lem 15441* Lemma for isprm2 15442. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛𝑃} = {1, 𝑃}))

Theoremisprm2 15442* 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 (𝑧𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃))))

Theoremisprm3 15443* 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)) ¬ 𝑧𝑃))

Theoremisprm4 15444* 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)(𝑧𝑃𝑧 = 𝑃)))

Theoremprmind2 15445* A variation on prmind 15446 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝑥 = 1 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = 𝑧 → (𝜑𝜃))    &   (𝑥 = (𝑦 · 𝑧) → (𝜑𝜏))    &   (𝑥 = 𝐴 → (𝜑𝜂))    &   𝜓    &   ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑)    &   ((𝑦 ∈ (ℤ‘2) ∧ 𝑧 ∈ (ℤ‘2)) → ((𝜒𝜃) → 𝜏))       (𝐴 ∈ ℕ → 𝜂)

Theoremprmind 15446* Perform induction over the multiplicative structure of . 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 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = 𝑧 → (𝜑𝜃))    &   (𝑥 = (𝑦 · 𝑧) → (𝜑𝜏))    &   (𝑥 = 𝐴 → (𝜑𝜂))    &   𝜓    &   (𝑥 ∈ ℙ → 𝜑)    &   ((𝑦 ∈ (ℤ‘2) ∧ 𝑧 ∈ (ℤ‘2)) → ((𝜒𝜃) → 𝜏))       (𝐴 ∈ ℕ → 𝜂)

Theoremdvdsprime 15447 If 𝑀 divides a prime, then 𝑀 is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)
((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ) → (𝑀𝑃 ↔ (𝑀 = 𝑃𝑀 = 1)))

Theoremnprm 15448 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)

Theoremnprmi 15449 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

Theoremdvdsnprmd 15450 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)    &   (𝜑𝐴 < 𝑁)    &   (𝜑𝐴𝑁)       (𝜑 → ¬ 𝑁 ∈ ℙ)

Theoremprm2orodd 15451 A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
(𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃))

Theorem2prm 15452 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
2 ∈ ℙ

Theorem3prm 15453 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
3 ∈ ℙ

Theorem4nprm 15454 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
¬ 4 ∈ ℙ

Theoremprmuz2 15455 A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
(𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))

Theoremprmgt1 15456 A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
(𝑃 ∈ ℙ → 1 < 𝑃)

Theoremprmm2nn0 15457 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)

Theoremoddprmgt2 15458 An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
(𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)

Theoremoddprmge3 15459 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))

Theoremprmn2uzge3OLD 15460 Obsolete version of oddprmge3 15459 as of 20-Aug-2021. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Proof shortened by AV, 20-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → 𝑃 ∈ (ℤ‘3))

Theoremsqnprm 15461 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ)

Theoremdvdsprm 15462 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) → (𝑁𝑃𝑁 = 𝑃))

Theoremexprmfct 15463* 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 𝑝𝑁)

Theoremprmdvdsfz 15464* 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 (𝑝𝑁𝑝𝐼))

Theoremnprmdvds1 15465 No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
(𝑃 ∈ ℙ → ¬ 𝑃 ∥ 1)

Theoremisprm5 15466* 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 → ¬ 𝑧𝑃)))

Theoremisprm7 15467* One need only check prime divisors of 𝑃 up to 𝑃 in order to ensure primality. This version of isprm5 15466 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) ¬ 𝑧𝑃))

Theoremmaxprmfct 15468* 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 ∣ 𝑧𝑁}       (𝑁 ∈ (ℤ‘2) → ((𝑆 ⊆ ℤ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦𝑆 𝑦𝑥) ∧ sup(𝑆, ℝ, < ) ∈ 𝑆))

Theoremdivgcdodd 15469 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 15472.

Theoremcoprm 15470 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) → (¬ 𝑃𝑁 ↔ (𝑃 gcd 𝑁) = 1))

Theoremprmrp 15471 Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)
((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑃 gcd 𝑄) = 1 ↔ 𝑃𝑄))

Theoremeuclemma 15472 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) ↔ (𝑃𝑀𝑃𝑁)))

Theoremisprm6 15473* A number is prime iff it satisfies Euclid's lemma euclemma 15472. (Contributed by Mario Carneiro, 6-Sep-2015.)
(𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ‘2) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (𝑃 ∥ (𝑥 · 𝑦) → (𝑃𝑥𝑃𝑦))))

Theoremprmdvdsexp 15474 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 ∥ (𝐴𝑁) ↔ 𝑃𝐴))

Theoremprmdvdsexpb 15475 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 ∥ (𝑄𝑁) ↔ 𝑃 = 𝑄))

Theoremprmdvdsexpr 15476 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 ∥ (𝑄𝑁) → 𝑃 = 𝑄))

Theoremprmexpb 15477 Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.)
(((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((𝑃𝑀) = (𝑄𝑁) ↔ (𝑃 = 𝑄𝑀 = 𝑁)))

Theoremprmfac1 15478 The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)
((𝑁 ∈ ℕ0𝑃 ∈ ℙ ∧ 𝑃 ∥ (!‘𝑁)) → 𝑃𝑁)

Theoremrpexp 15479 If two numbers 𝐴 and 𝐵 are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))

Theoremrpexp1i 15480 Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑀) gcd 𝐵) = 1))

Theoremrpexp12i 15481 Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑀) gcd (𝐵𝑁)) = 1))

Theoremprmndvdsfaclt 15482 A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑁 < 𝑃 → ¬ 𝑃 ∥ (!‘𝑁)))

Theoremncoprmlnprm 15483 If two positive integers are not coprime, the larger of them is not a prime number. (Contributed by AV, 9-Aug-2020.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → (1 < (𝐴 gcd 𝐵) → 𝐵 ∉ ℙ))

Theoremcncongrprm 15484 Corollary 2 of Cancellability of Congruences: Two products with a common factor are congruent modulo a prime number not dividing the common factor iff the other factors are congruent modulo the prime number. (Contributed by AV, 13-Jul-2021.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑃 ∈ ℙ ∧ ¬ 𝑃𝐶)) → (((𝐴 · 𝐶) mod 𝑃) = ((𝐵 · 𝐶) mod 𝑃) ↔ (𝐴 mod 𝑃) = (𝐵 mod 𝑃)))

Theoremisevengcd2 15485 The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.)
(𝑍 ∈ ℤ → (2 ∥ 𝑍 ↔ (2 gcd 𝑍) = 2))

Theoremisoddgcd1 15486 The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.)
(𝑍 ∈ ℤ → (¬ 2 ∥ 𝑍 ↔ (2 gcd 𝑍) = 1))

Theorem3lcm2e6 15487 The least common multiple of three and two is six. The operands are unequal primes and thus coprime, so the result is (the absolute value of) their product. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 27-Aug-2020.)
(3 lcm 2) = 6

6.2.3  Properties of the canonical representation of a rational

Syntaxcnumer 15488 Extend class notation to include canonical numerator function.
class numer

Syntaxcdenom 15489 Extend class notation to include canonical denominator function.
class denom

Definitiondf-numer 15490* The canonical numerator of a rational is the numerator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer = (𝑦 ∈ ℚ ↦ (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥))))))

Definitiondf-denom 15491* The canonical denominator of a rational is the denominator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom = (𝑦 ∈ ℚ ↦ (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥))))))

Theoremqnumval 15492* Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))))

Theoremqdenval 15493* Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝐴 = ((1st𝑥) / (2nd𝑥))))))

Theoremqnumdencl 15494 Lemma for qnumcl 15495 and qdencl 15496. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))

Theoremqnumcl 15495 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ)

Theoremqdencl 15496 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ)

Theoremfnum 15497 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer:ℚ⟶ℤ

Theoremfden 15498 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom:ℚ⟶ℕ

Theoremqnumdenbi 15499 Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶)) ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))

Theoremqnumdencoprm 15500 The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → ((numer‘𝐴) gcd (denom‘𝐴)) = 1)

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