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

Theoremisprm7 15401* One need only check prime divisors of 𝑃 up to 𝑃 in order to ensure primality. This version of isprm5 15400 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 15402* 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 15403 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 15406.

Theoremcoprm 15404 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 15405 Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)
((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑃 gcd 𝑄) = 1 ↔ 𝑃𝑄))

Theoremeuclemma 15406 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 15407* A number is prime iff it satisfies Euclid's lemma euclemma 15406. (Contributed by Mario Carneiro, 6-Sep-2015.)
(𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ‘2) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (𝑃 ∥ (𝑥 · 𝑦) → (𝑃𝑥𝑃𝑦))))

Theoremprmdvdsexp 15408 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 15409 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 15410 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 15411 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 15412 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 15413 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 15414 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 15415 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 15416 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 15417 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 15418 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 15419 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 15420 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 15421 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 15422 Extend class notation to include canonical numerator function.
class numer

Syntaxcdenom 15423 Extend class notation to include canonical denominator function.
class denom

Definitiondf-numer 15424* 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 15425* 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 15426* 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 15427* 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 15428 Lemma for qnumcl 15429 and qdencl 15430. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))

Theoremqnumcl 15429 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ)

Theoremqdencl 15430 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ)

Theoremfnum 15431 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer:ℚ⟶ℤ

Theoremfden 15432 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom:ℚ⟶ℕ

Theoremqnumdenbi 15433 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 15434 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)

Theoremqeqnumdivden 15435 Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → 𝐴 = ((numer‘𝐴) / (denom‘𝐴)))

Theoremqmuldeneqnum 15436 Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (𝐴 · (denom‘𝐴)) = (numer‘𝐴))

Theoremdivnumden 15437 Calculate the reduced form of a quotient using gcd. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((numer‘(𝐴 / 𝐵)) = (𝐴 / (𝐴 gcd 𝐵)) ∧ (denom‘(𝐴 / 𝐵)) = (𝐵 / (𝐴 gcd 𝐵))))

Theoremdivdenle 15438 Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (denom‘(𝐴 / 𝐵)) ≤ 𝐵)

Theoremqnumgt0 15439 A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (0 < 𝐴 ↔ 0 < (numer‘𝐴)))

Theoremqgt0numnn 15440 A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)
((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (numer‘𝐴) ∈ ℕ)

Theoremnn0gcdsq 15441 Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))

Theoremzgcdsq 15442 nn0gcdsq 15441 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))

Theoremnumdensq 15443 Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → ((numer‘(𝐴↑2)) = ((numer‘𝐴)↑2) ∧ (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2)))

Theoremnumsq 15444 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (numer‘(𝐴↑2)) = ((numer‘𝐴)↑2))

Theoremdensq 15445 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2))

Theoremqden1elz 15446 A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → ((denom‘𝐴) = 1 ↔ 𝐴 ∈ ℤ))

Theoremzsqrtelqelz 15447 If an integer has a rational square root, that root is must be an integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)
((𝐴 ∈ ℤ ∧ (√‘𝐴) ∈ ℚ) → (√‘𝐴) ∈ ℤ)

Theoremnonsq 15448 Any integer strictly between two adjacent squares has an irrational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) ∧ ((𝐵↑2) < 𝐴𝐴 < ((𝐵 + 1)↑2))) → ¬ (√‘𝐴) ∈ ℚ)

6.2.4  Euler's theorem

Syntaxcodz 15449 Extend class notation with the order function on the class of integers mod N.
class od

Syntaxcphi 15450 Extend class notation with the Euler phi function.
class ϕ

Definitiondf-odz 15451* Define the order function on the class of integers mod N. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
od = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )))

Definitiondf-phi 15452* Define the Euler phi function (also called _ Euler totient function_), which counts the number of integers less than 𝑛 and coprime to it, see definition in [ApostolNT] p. 25. (Contributed by Mario Carneiro, 23-Feb-2014.)
ϕ = (𝑛 ∈ ℕ ↦ (#‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))

Theoremphival 15453* Value of the Euler ϕ function. (Contributed by Mario Carneiro, 23-Feb-2014.)
(𝑁 ∈ ℕ → (ϕ‘𝑁) = (#‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))

Theoremphicl2 15454 Bounds and closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 23-Feb-2014.)
(𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ (1...𝑁))

Theoremphicl 15455 Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 28-Feb-2014.)
(𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)

Theoremphibndlem 15456* Lemma for phibnd 15457. (Contributed by Mario Carneiro, 23-Feb-2014.)
(𝑁 ∈ (ℤ‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))

Theoremphibnd 15457 A slightly tighter bound on the value of the Euler ϕ function. (Contributed by Mario Carneiro, 23-Feb-2014.)
(𝑁 ∈ (ℤ‘2) → (ϕ‘𝑁) ≤ (𝑁 − 1))

Theoremphicld 15458 Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → (ϕ‘𝑁) ∈ ℕ)

Theoremphi1 15459 Value of the Euler ϕ function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
(ϕ‘1) = 1

Theoremdfphi2 15460* Alternate definition of the Euler ϕ function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.)
(𝑁 ∈ ℕ → (ϕ‘𝑁) = (#‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}))

Theoremhashdvds 15461* The number of numbers in a given residue class in a finite set of integers. (Contributed by Mario Carneiro, 12-Mar-2014.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ (ℤ‘(𝐴 − 1)))    &   (𝜑𝐶 ∈ ℤ)       (𝜑 → (#‘{𝑥 ∈ (𝐴...𝐵) ∣ 𝑁 ∥ (𝑥𝐶)}) = ((⌊‘((𝐵𝐶) / 𝑁)) − (⌊‘(((𝐴 − 1) − 𝐶) / 𝑁))))

Theoremphiprmpw 15462 Value of the Euler ϕ function at a prime power. Theorem 2.5(a) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.)
((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (ϕ‘(𝑃𝐾)) = ((𝑃↑(𝐾 − 1)) · (𝑃 − 1)))

Theoremphiprm 15463 Value of the Euler ϕ function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
(𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))

Theoremcrth 15464* The Chinese Remainder Theorem: the function that maps 𝑥 to its remainder classes mod 𝑀 and mod 𝑁 is 1-1 and onto when 𝑀 and 𝑁 are coprime. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-May-2016.)
𝑆 = (0..^(𝑀 · 𝑁))    &   𝑇 = ((0..^𝑀) × (0..^𝑁))    &   𝐹 = (𝑥𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩)    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1))       (𝜑𝐹:𝑆1-1-onto𝑇)

Theoremphimullem 15465* Lemma for phimul 15466. (Contributed by Mario Carneiro, 24-Feb-2014.)
𝑆 = (0..^(𝑀 · 𝑁))    &   𝑇 = ((0..^𝑀) × (0..^𝑁))    &   𝐹 = (𝑥𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩)    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1))    &   𝑈 = {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1}    &   𝑉 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   𝑊 = {𝑦𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}       (𝜑 → (ϕ‘(𝑀 · 𝑁)) = ((ϕ‘𝑀) · (ϕ‘𝑁)))

Theoremphimul 15466 The Euler ϕ function is a multiplicative function, meaning that it distributes over multiplication at relatively prime arguments. Theorem 2.5(c) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (ϕ‘(𝑀 · 𝑁)) = ((ϕ‘𝑀) · (ϕ‘𝑁)))

Theoremeulerthlem1 15467* Lemma for eulerth 15469. (Contributed by Mario Carneiro, 8-May-2015.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   𝑇 = (1...(ϕ‘𝑁))    &   (𝜑𝐹:𝑇1-1-onto𝑆)    &   𝐺 = (𝑥𝑇 ↦ ((𝐴 · (𝐹𝑥)) mod 𝑁))       (𝜑𝐺:𝑇𝑆)

Theoremeulerthlem2 15468* Lemma for eulerth 15469. (Contributed by Mario Carneiro, 28-Feb-2014.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   𝑇 = (1...(ϕ‘𝑁))    &   (𝜑𝐹:𝑇1-1-onto𝑆)    &   𝐺 = (𝑥𝑇 ↦ ((𝐴 · (𝐹𝑥)) mod 𝑁))       (𝜑 → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁))

Theoremeulerth 15469 Euler's theorem, a generalization of Fermat's little theorem. If 𝐴 and 𝑁 are coprime, then 𝐴↑ϕ(𝑁)≡1 (mod 𝑁). This is Metamath 100 proof #10. Also called Euler-Fermat theorem, see theorem 5.17 in [ApostolNT] p. 113. (Contributed by Mario Carneiro, 28-Feb-2014.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁))

Theoremfermltl 15470 Fermat's little theorem. When 𝑃 is prime, 𝐴𝑃𝐴 (mod 𝑃) for any 𝐴, see theorem 5.19 in [ApostolNT] p. 114. (Contributed by Mario Carneiro, 28-Feb-2014.)
((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝐴𝑃) mod 𝑃) = (𝐴 mod 𝑃))

Theoremprmdiv 15471 Show an explicit expression for the modular inverse of 𝐴 mod 𝑃. (Contributed by Mario Carneiro, 24-Jan-2015.)
𝑅 = ((𝐴↑(𝑃 − 2)) mod 𝑃)       ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃𝐴) → (𝑅 ∈ (1...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝐴 · 𝑅) − 1)))

Theoremprmdiveq 15472 The modular inverse of 𝐴 mod 𝑃 is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)
𝑅 = ((𝐴↑(𝑃 − 2)) mod 𝑃)       ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃𝐴) → ((𝑆 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝐴 · 𝑆) − 1)) ↔ 𝑆 = 𝑅))

Theoremprmdivdiv 15473 The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)
𝑅 = ((𝐴↑(𝑃 − 2)) mod 𝑃)       ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝐴 = ((𝑅↑(𝑃 − 2)) mod 𝑃))

Theoremhashgcdlem 15474* A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐴 = {𝑦 ∈ (0..^(𝑀 / 𝑁)) ∣ (𝑦 gcd (𝑀 / 𝑁)) = 1}    &   𝐵 = {𝑧 ∈ (0..^𝑀) ∣ (𝑧 gcd 𝑀) = 𝑁}    &   𝐹 = (𝑥𝐴 ↦ (𝑥 · 𝑁))       ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁𝑀) → 𝐹:𝐴1-1-onto𝐵)

Theoremhashgcdeq 15475* Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (#‘{𝑥 ∈ (0..^𝑀) ∣ (𝑥 gcd 𝑀) = 𝑁}) = if(𝑁𝑀, (ϕ‘(𝑀 / 𝑁)), 0))

Theoremphisum 15476* The divisor sum identity of the totient function. Theorem 2.2 in [ApostolNT] p. 26. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (ϕ‘𝑑) = 𝑁)

Theoremodzval 15477* Value of the order function. This is a function of functions; the inner argument selects the base (i.e. mod 𝑁 for some 𝑁, often prime) and the outer argument selects the integer or equivalence class (if you want to think about it that way) from the integers mod 𝑁. In order to ensure the supremum is well-defined, we only define the expression when 𝐴 and 𝑁 are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((od𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴𝑛) − 1)}, ℝ, < ))

Theoremodzcllem 15478 - Lemma for odzcl 15479, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (((od𝑁)‘𝐴) ∈ ℕ ∧ 𝑁 ∥ ((𝐴↑((od𝑁)‘𝐴)) − 1)))

Theoremodzcl 15479 The order of a group element is an integer. (Contributed by Mario Carneiro, 28-Feb-2014.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((od𝑁)‘𝐴) ∈ ℕ)

Theoremodzid 15480 Any element raised to the power of its order is 1. (Contributed by Mario Carneiro, 28-Feb-2014.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∥ ((𝐴↑((od𝑁)‘𝐴)) − 1))

Theoremodzdvds 15481 The only powers of 𝐴 that are congruent to 1 are the multiples of the order of 𝐴. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)
(((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∥ ((𝐴𝐾) − 1) ↔ ((od𝑁)‘𝐴) ∥ 𝐾))

Theoremodzphi 15482 The order of any group element is a divisor of the Euler ϕ function. (Contributed by Mario Carneiro, 28-Feb-2014.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((od𝑁)‘𝐴) ∥ (ϕ‘𝑁))

6.2.5  Arithmetic modulo a prime number

Theoremmodprm1div 15483 A prime number divides an integer minus 1 iff the integer modulo the prime number is 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝐴 mod 𝑃) = 1 ↔ 𝑃 ∥ (𝐴 − 1)))

Theoremm1dvdsndvds 15484 If an integer minus 1 is divisible by a prime number, the integer itself is not divisible by this prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 ∥ (𝐴 − 1) → ¬ 𝑃𝐴))

Theoremmodprminv 15485 Show an explicit expression for the modular inverse of 𝐴 mod 𝑃. This is an application of prmdiv 15471. (Contributed by Alexander van der Vekens, 15-May-2018.)
𝑅 = ((𝐴↑(𝑃 − 2)) mod 𝑃)       ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃𝐴) → (𝑅 ∈ (1...(𝑃 − 1)) ∧ ((𝐴 · 𝑅) mod 𝑃) = 1))

Theoremmodprminveq 15486 The modular inverse of 𝐴 mod 𝑃 is unique. (Contributed by Alexander van der Vekens, 17-May-2018.)
𝑅 = ((𝐴↑(𝑃 − 2)) mod 𝑃)       ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃𝐴) → ((𝑆 ∈ (0...(𝑃 − 1)) ∧ ((𝐴 · 𝑆) mod 𝑃) = 1) ↔ 𝑆 = 𝑅))

Theoremvfermltl 15487 Variant of Fermat's little theorem if 𝐴 is not a multiple of 𝑃, see theorem 5.18 in [ApostolNT] p. 113. (Contributed by AV, 21-Aug-2020.) (Proof shortened by AV, 5-Sep-2020.)
((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃𝐴) → ((𝐴↑(𝑃 − 1)) mod 𝑃) = 1)

TheoremvfermltlALT 15488 Alternate proof of vfermltl 15487, not using Euler's theorem. (Contributed by AV, 21-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃𝐴) → ((𝐴↑(𝑃 − 1)) mod 𝑃) = 1)

Theorempowm2modprm 15489 If an integer minus 1 is divisible by a prime number, then the integer to the power of the prime number minus 2 is 1 modulo the prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 ∥ (𝐴 − 1) → ((𝐴↑(𝑃 − 2)) mod 𝑃) = 1))

Theoremreumodprminv 15490* For any prime number and for any positive integer less than this prime number, there is a unique modular inverse of this positive integer. (Contributed by Alexander van der Vekens, 12-May-2018.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1..^𝑃)) → ∃!𝑖 ∈ (1...(𝑃 − 1))((𝑁 · 𝑖) mod 𝑃) = 1)

Theoremmodprm0 15491* For two positive integers less than a given prime number there is always a nonnegative integer (less than the given prime number) so that the sum of one of the two positive integers and the other of the positive integers multiplied by the nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 17-May-2018.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1..^𝑃) ∧ 𝐼 ∈ (1..^𝑃)) → ∃𝑗 ∈ (0..^𝑃)((𝐼 + (𝑗 · 𝑁)) mod 𝑃) = 0)

Theoremnnnn0modprm0 15492* For a positive integer and a nonnegative integer both less than a given prime number there is always a second nonnegative integer (less than the given prime number) so that the sum of this second nonnegative integer multiplied with the positive integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 8-Nov-2018.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1..^𝑃) ∧ 𝐼 ∈ (0..^𝑃)) → ∃𝑗 ∈ (0..^𝑃)((𝐼 + (𝑗 · 𝑁)) mod 𝑃) = 0)

Theoremmodprmn0modprm0 15493* For an integer not being 0 modulo a given prime number and a nonnegative integer less than the prime number, there is always a second nonnegative integer (less than the given prime number) so that the sum of this second nonnegative integer multiplied with the integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 10-Nov-2018.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ (𝑁 mod 𝑃) ≠ 0) → (𝐼 ∈ (0..^𝑃) → ∃𝑗 ∈ (0..^𝑃)((𝐼 + (𝑗 · 𝑁)) mod 𝑃) = 0))

6.2.6  Pythagorean Triples

Theoremcoprimeprodsq 15494 If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of gcd and square. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(((𝐴 ∈ ℕ0𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐶↑2) = (𝐴 · 𝐵) → 𝐴 = ((𝐴 gcd 𝐶)↑2)))

Theoremcoprimeprodsq2 15495 If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of gcd and square. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐶↑2) = (𝐴 · 𝐵) → 𝐵 = ((𝐵 gcd 𝐶)↑2)))

Theoremoddprm 15496 A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)
(𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℕ)

Theoremnnoddn2prm 15497 A prime not equal to 2 is an odd positive integer. (Contributed by AV, 28-Jun-2021.)
(𝑁 ∈ (ℙ ∖ {2}) → (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁))

Theoremoddn2prm 15498 A prime not equal to 2 is odd. (Contributed by AV, 28-Jun-2021.)
(𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑁)

Theoremnnoddn2prmb 15499 A number is a prime number not equal to 2 iff it is an odd prime number. Conversion theorem for two representations of odd primes. (Contributed by AV, 14-Jul-2021.)
(𝑁 ∈ (ℙ ∖ {2}) ↔ (𝑁 ∈ ℙ ∧ ¬ 2 ∥ 𝑁))

Theoremprm23lt5 15500 A prime less than 5 is either 2 or 3. (Contributed by AV, 5-Jul-2021.)
((𝑃 ∈ ℙ ∧ 𝑃 < 5) → (𝑃 = 2 ∨ 𝑃 = 3))

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