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Theorem List for Metamath Proof Explorer - 15101-15200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisprm 15101* 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 15102 A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
 
Theoremprmz 15103 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 15104 The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
ℙ ⊆ ℕ
 
Theoremprmex 15105 The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
ℙ ∈ V
 
Theorem1nprm 15106 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
¬ 1 ∈ ℙ
 
Theorem1idssfct 15107* 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 15108* Lemma for isprm2 15109. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛𝑃} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛𝑃} = {1, 𝑃}))
 
Theoremisprm2 15109* 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 15110* 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 15111* 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 15112* A variation on prmind 15113 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 15113* 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 15114 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 15115 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 15116 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 15117 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 15118 A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
(𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃))
 
Theorem2prm 15119 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
2 ∈ ℙ
 
Theorem3prm 15120 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
3 ∈ ℙ
 
Theorem4nprm 15121 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
¬ 4 ∈ ℙ
 
Theoremprmuz2 15122 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 15123 A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
(𝑃 ∈ ℙ → 1 < 𝑃)
 
Theoremprmm2nn0 15124 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 15125 An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
(𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
 
Theoremoddprmge3 15126 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 15127 Obsolete version of oddprmge3 15126 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 15128 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ)
 
Theoremdvdsprm 15129 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 15130* 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 15131* 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 15132 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 15133* 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 15134* One need only check prime divisors of 𝑃 up to 𝑃 in order to ensure primality. This version of isprm5 15133 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 15135* 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 15136 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 15139.

 
Theoremcoprm 15137 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 15138 Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)
((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑃 gcd 𝑄) = 1 ↔ 𝑃𝑄))
 
Theoremeuclemma 15139 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 15140* A number is prime iff it satisfies Euclid's lemma euclemma 15139. (Contributed by Mario Carneiro, 6-Sep-2015.)
(𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ‘2) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (𝑃 ∥ (𝑥 · 𝑦) → (𝑃𝑥𝑃𝑦))))
 
Theoremprmdvdsexp 15141 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 15142 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 15143 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 15144 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 15145 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 15146 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 15147 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 15148 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 15149 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 15150 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 15151 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 15152 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 15153 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 15154 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 15155 Extend class notation to include canonical numerator function.
class numer
 
Syntaxcdenom 15156 Extend class notation to include canonical denominator function.
class denom
 
Definitiondf-numer 15157* 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 15158* 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 15159* 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 15160* 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 15161 Lemma for qnumcl 15162 and qdencl 15163. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))
 
Theoremqnumcl 15162 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ)
 
Theoremqdencl 15163 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ)
 
Theoremfnum 15164 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer:ℚ⟶ℤ
 
Theoremfden 15165 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom:ℚ⟶ℕ
 
Theoremqnumdenbi 15166 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 15167 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 15168 Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → 𝐴 = ((numer‘𝐴) / (denom‘𝐴)))
 
Theoremqmuldeneqnum 15169 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 15170 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 15171 Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (denom‘(𝐴 / 𝐵)) ≤ 𝐵)
 
Theoremqnumgt0 15172 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 15173 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 15174 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 15175 nn0gcdsq 15174 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 15176 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 15177 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (numer‘(𝐴↑2)) = ((numer‘𝐴)↑2))
 
Theoremdensq 15178 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2))
 
Theoremqden1elz 15179 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 15180 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 15181 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 15182 Extend class notation with the order function on the class of integers mod N.
class od
 
Syntaxcodzold 15183 Extend class notation with the order function on the class of integers mod N (old version).
class od
 
Syntaxcphi 15184 Extend class notation with the Euler phi function.
class ϕ
 
Definitiondf-odz 15185* 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-odzOLD 15186* Define the order function on the class of integers mod N. (Contributed by Mario Carneiro, 23-Feb-2014.) Obsolete version of df-odz 15185 as of 26-Sep-2020. (New usage is discouraged.)
od = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ sup({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )))
 
Definitiondf-phi 15187* 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 15188* 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 15189 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 15190 Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 28-Feb-2014.)
(𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)
 
Theoremphibndlem 15191* Lemma for phibnd 15192. (Contributed by Mario Carneiro, 23-Feb-2014.)
(𝑁 ∈ (ℤ‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))
 
Theoremphibnd 15192 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 15193 Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → (ϕ‘𝑁) ∈ ℕ)
 
Theoremphi1 15194 Value of the Euler ϕ function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
(ϕ‘1) = 1
 
Theoremdfphi2 15195* 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 15196* 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 15197 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 15198 Value of the Euler ϕ function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
(𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
 
Theoremcrth 15199* 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 15200* Lemma for phimul 15201. (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)))
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