P. 122 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12101-12200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	oddprmgt2 12101	An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
Theorem	oddprmge3 12102	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 12103	A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ)
Theorem	dvdsprm 12104	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 12105*	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 12106*	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 12107	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	isprm5lem 12108*	Lemma for isprm5 12109. The interesting direction (showing that one only needs to check prime divisors up to the square root of 𝑃). (Contributed by Jim Kingdon, 20-Oct-2024.)
⊢ (𝜑 → 𝑃 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))    &   ⊢ (𝜑 → 𝑋 ∈ (2...(𝑃 − 1)))    ⇒   ⊢ (𝜑 → ¬ 𝑋 ∥ 𝑃)
Theorem	isprm5 12109*	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	divgcdodd 12110	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))))
5.2.2  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 12113.
Theorem	coprm 12111	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 12112	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 12113	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 12114*	A number is prime iff it satisfies Euclid's lemma euclemma 12113. (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 12115	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 12116	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 12117	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))
Theorem	prmexpb 12118	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)) → ((𝑃↑𝑀) = (𝑄↑𝑁) ↔ (𝑃 = 𝑄 ∧ 𝑀 = 𝑁)))
Theorem	prmfac1 12119	The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (!‘𝑁)) → 𝑃 ≤ 𝑁)
Theorem	rpexp 12120	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) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))
Theorem	rpexp1i 12121	Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑀) gcd 𝐵) = 1))
Theorem	rpexp12i 12122	Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑀) gcd (𝐵↑𝑁)) = 1))
Theorem	prmndvdsfaclt 12123	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)))
Theorem	cncongrprm 12124	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 ∧ ¬ 𝑃 ∥ 𝐶)) → (((𝐴 · 𝐶) mod 𝑃) = ((𝐵 · 𝐶) mod 𝑃) ↔ (𝐴 mod 𝑃) = (𝐵 mod 𝑃)))
Theorem	isevengcd2 12125	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))
Theorem	isoddgcd1 12126	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))
Theorem	3lcm2e6 12127	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
5.2.3  Non-rationality of square root of 2
Theorem	sqrt2irrlem 12128	Lemma for sqrt2irr 12129. This is the core of the proof: - if 𝐴 / 𝐵 = √(2), then 𝐴 and 𝐵 are even, so 𝐴 / 2 and 𝐵 / 2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵))    ⇒   ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ))
Theorem	sqrt2irr 12129	The square root of 2 is not rational. That is, for any rational number, (√‘2) does not equal it. However, if we were to say "the square root of 2 is irrational" that would mean something stronger: "for any rational number, (√‘2) is apart from it" (the two statements are equivalent given excluded middle). See sqrt2irrap 12147 for the proof that the square root of two is irrational. The proof's core is proven in sqrt2irrlem 12128, which shows that if 𝐴 / 𝐵 = √(2), then 𝐴 and 𝐵 are even, so 𝐴 / 2 and 𝐵 / 2 are smaller representatives, which is absurd. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ (√‘2) ∉ ℚ
Theorem	sqrt2re 12130	The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
⊢ (√‘2) ∈ ℝ
Theorem	sqrt2irr0 12131	The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
⊢ (√‘2) ∈ (ℝ ∖ ℚ)
Theorem	pw2dvdslemn 12132*	Lemma for pw2dvds 12133. If a natural number has some power of two which does not divide it, there is a highest power of two which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
Theorem	pw2dvds 12133*	A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.)
⊢ (𝑁 ∈ ℕ → ∃𝑚 ∈ ℕ0 ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
Theorem	pw2dvdseulemle 12134	Lemma for pw2dvdseu 12135. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0)    &   ⊢ (𝜑 → (2↑𝐴) ∥ 𝑁)    &   ⊢ (𝜑 → ¬ (2↑(𝐵 + 1)) ∥ 𝑁)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	pw2dvdseu 12135*	A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.)
⊢ (𝑁 ∈ ℕ → ∃!𝑚 ∈ ℕ0 ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
Theorem	oddpwdclemxy 12136*	Lemma for oddpwdc 12141. Another way of stating that decomposing a natural number into a power of two and an odd number is unique. (Contributed by Jim Kingdon, 16-Nov-2021.)
⊢ ((((𝑋 ∈ ℕ ∧ ¬ 2 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((2↑𝑌) · 𝑋)) → (𝑋 = (𝐴 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))))
Theorem	oddpwdclemdvds 12137*	Lemma for oddpwdc 12141. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ (𝐴 ∈ ℕ → (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))) ∥ 𝐴)
Theorem	oddpwdclemndvds 12138*	Lemma for oddpwdc 12141. A natural number is not divisible by one more than the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ (𝐴 ∈ ℕ → ¬ (2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴)
Theorem	oddpwdclemodd 12139*	Lemma for oddpwdc 12141. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.)
⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (𝐴 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))))
Theorem	oddpwdclemdc 12140*	Lemma for oddpwdc 12141. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.)
⊢ ((((𝑋 ∈ ℕ ∧ ¬ 2 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((2↑𝑌) · 𝑋)) ↔ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))))
Theorem	oddpwdc 12141*	The function 𝐹 that decomposes a number into its "odd" and "even" parts, which is to say the largest power of two and largest odd divisor of a number, is a bijection from pairs of a nonnegative integer and an odd number to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.)
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    ⇒   ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
Theorem	sqpweven 12142*	The greatest power of two dividing the square of an integer is an even power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    ⇒   ⊢ (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(◡𝐹‘(𝐴↑2))))
Theorem	2sqpwodd 12143*	The greatest power of two dividing twice the square of an integer is an odd power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    ⇒   ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))))
Theorem	sqne2sq 12144	The square of a natural number can never be equal to two times the square of a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴↑2) ≠ (2 · (𝐵↑2)))
Theorem	znege1 12145	The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≠ 𝐵) → 1 ≤ (abs‘(𝐴 − 𝐵)))
Theorem	sqrt2irraplemnn 12146	Lemma for sqrt2irrap 12147. The square root of 2 is apart from a positive rational expressed as a numerator and denominator. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (√‘2) # (𝐴 / 𝐵))
Theorem	sqrt2irrap 12147	The square root of 2 is irrational. That is, for any rational number, (√‘2) is apart from it. In the absence of excluded middle, we can distinguish between this and "the square root of 2 is not rational" which is sqrt2irr 12129. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ (𝑄 ∈ ℚ → (√‘2) # 𝑄)
5.2.4  Properties of the canonical representation of a rational
Syntax	cnumer 12148	Extend class notation to include canonical numerator function.
class numer
Syntax	cdenom 12149	Extend class notation to include canonical denominator function.
class denom
Definition	df-numer 12150*	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)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))))
Definition	df-denom 12151*	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)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))))
Theorem	qnumval 12152*	Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))))
Theorem	qdenval 12153*	Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))))
Theorem	qnumdencl 12154	Lemma for qnumcl 12155 and qdencl 12156. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))
Theorem	qnumcl 12155	The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ)
Theorem	qdencl 12156	The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ)
Theorem	fnum 12157	Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ numer:ℚ⟶ℤ
Theorem	fden 12158	Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ denom:ℚ⟶ℕ
Theorem	qnumdenbi 12159	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)))
Theorem	qnumdencoprm 12160	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)
Theorem	qeqnumdivden 12161	Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℚ → 𝐴 = ((numer‘𝐴) / (denom‘𝐴)))
Theorem	qmuldeneqnum 12162	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))
Theorem	divnumden 12163	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))))
Theorem	divdenle 12164	Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (denom‘(𝐴 / 𝐵)) ≤ 𝐵)
Theorem	qnumgt0 12165	A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (0 < 𝐴 ↔ 0 < (numer‘𝐴)))
Theorem	qgt0numnn 12166	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)
Theorem	nn0gcdsq 12167	Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
Theorem	zgcdsq 12168	nn0gcdsq 12167 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)))
Theorem	numdensq 12169	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)))
Theorem	numsq 12170	Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (numer‘(𝐴↑2)) = ((numer‘𝐴)↑2))
Theorem	densq 12171	Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2))
Theorem	qden1elz 12172	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))
Theorem	nn0sqrtelqelz 12173	If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.)
⊢ ((𝐴 ∈ ℕ0 ∧ (√‘𝐴) ∈ ℚ) → (√‘𝐴) ∈ ℤ)
Theorem	nonsq 12174	Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ((𝐵↑2) < 𝐴 ∧ 𝐴 < ((𝐵 + 1)↑2))) → ¬ (√‘𝐴) ∈ ℚ)
5.2.5  Euler's theorem
Syntax	codz 12175	Extend class notation with the order function on the class of integers modulo N.
class odℤ
Syntax	cphi 12176	Extend class notation with the Euler phi function.
class ϕ
Definition	df-odz 12177*	Define the order function on the class of integers modulo N. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
⊢ odℤ = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥↑𝑚) − 1)}, ℝ, < )))
Definition	df-phi 12178*	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}))
Theorem	phivalfi 12179*	Finiteness of an expression used to define the Euler ϕ function. (Contributed by Jim Kingon, 28-May-2022.)
⊢ (𝑁 ∈ ℕ → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∈ Fin)
Theorem	phival 12180*	Value of the Euler ϕ function. (Contributed by Mario Carneiro, 23-Feb-2014.)
⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
Theorem	phicl2 12181	Bounds and closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 23-Feb-2014.)
⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ (1...𝑁))
Theorem	phicl 12182	Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 28-Feb-2014.)
⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)
Theorem	phibndlem 12183*	Lemma for phibnd 12184. (Contributed by Mario Carneiro, 23-Feb-2014.)
⊢ (𝑁 ∈ (ℤ≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))
Theorem	phibnd 12184	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))
Theorem	phicld 12185	Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
Theorem	phi1 12186	Value of the Euler ϕ function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
⊢ (ϕ‘1) = 1
Theorem	dfphi2 12187*	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}))
Theorem	hashdvds 12188*	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.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘(𝐴 − 1)))    &   ⊢ (𝜑 → 𝐶 ∈ ℤ)    ⇒   ⊢ (𝜑 → (♯‘{𝑥 ∈ (𝐴...𝐵) ∣ 𝑁 ∥ (𝑥 − 𝐶)}) = ((⌊‘((𝐵 − 𝐶) / 𝑁)) − (⌊‘(((𝐴 − 1) − 𝐶) / 𝑁))))
Theorem	phiprmpw 12189	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‘(𝑃↑𝐾)) = ((𝑃↑(𝐾 − 1)) · (𝑃 − 1)))
Theorem	phiprm 12190	Value of the Euler ϕ function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
Theorem	crth 12191*	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 = (𝑥 ∈ 𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩)    &   ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1))    ⇒   ⊢ (𝜑 → 𝐹:𝑆–1-1-onto→𝑇)
Theorem	phimullem 12192*	Lemma for phimul 12193. (Contributed by Mario Carneiro, 24-Feb-2014.)
⊢ 𝑆 = (0..^(𝑀 · 𝑁))    &   ⊢ 𝑇 = ((0..^𝑀) × (0..^𝑁))    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ⟨(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 = {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}    ⇒   ⊢ (𝜑 → (ϕ‘(𝑀 · 𝑁)) = ((ϕ‘𝑀) · (ϕ‘𝑁)))
Theorem	phimul 12193	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)))
Theorem	eulerthlem1 12194*	Lemma for eulerth 12200. (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))    &   ⊢ (𝜑 → 𝐹:𝑇–1-1-onto→𝑆)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑇 ↦ ((𝐴 · (𝐹‘𝑥)) mod 𝑁))    ⇒   ⊢ (𝜑 → 𝐺:𝑇⟶𝑆)
Theorem	eulerthlemfi 12195*	Lemma for eulerth 12200. The set 𝑆 is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    ⇒   ⊢ (𝜑 → 𝑆 ∈ Fin)
Theorem	eulerthlemrprm 12196*	Lemma for eulerth 12200. 𝑁 and ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    ⇒   ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1)
Theorem	eulerthlema 12197*	Lemma for eulerth 12200. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    ⇒   ⊢ (𝜑 → (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) mod 𝑁) = (∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) mod 𝑁))
Theorem	eulerthlemh 12198*	Lemma for eulerth 12200. A permutation of (1...(ϕ‘𝑁)). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.)
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    &   ⊢ 𝐻 = (◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)))    ⇒   ⊢ (𝜑 → 𝐻:(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))
Theorem	eulerthlemth 12199*	Lemma for eulerth 12200. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    ⇒   ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁))
Theorem	eulerth 12200	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))