P. 130 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12901-13000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prmind 12901*	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 12902	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 12903	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 12904	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 12905	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 12906	A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃))
Theorem	2prm 12907	2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
⊢ 2 ∈ ℙ
Theorem	3prm 12908	3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 3 ∈ ℙ
Theorem	4nprm 12909	4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
⊢ ¬ 4 ∈ ℙ
Theorem	prmdc 12910	Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
⊢ (𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ)
Theorem	prmuz2 12911	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 12912	A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
Theorem	prmm2nn0 12913	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 12914	An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
Theorem	oddprmge3 12915	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 12916	A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ)
Theorem	dvdsprm 12917	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 12918*	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 12919*	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 12920	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 12921*	Lemma for isprm5 12922. 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 12922*	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 12923	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 12926.
Theorem	coprm 12924	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 12925	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 12926	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 12927*	A number is prime iff it satisfies Euclid's lemma euclemma 12926. (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 12928	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 12929	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 12930	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 12931	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 12932	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 12933	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 12934	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 12935	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 12936	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 12937	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 12938	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 12939	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 12940	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 12941	Lemma for sqrt2irr 12942. 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 12942	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 12960 for the proof that the square root of two is irrational. The proof's core is proven in sqrt2irrlem 12941, 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 12943	The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
⊢ (√‘2) ∈ ℝ
Theorem	sqrt2irr0 12944	The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
⊢ (√‘2) ∈ (ℝ ∖ ℚ)
Theorem	pw2dvdslemn 12945*	Lemma for pw2dvds 12946. 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 12946*	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 12947	Lemma for pw2dvdseu 12948. 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 12948*	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 12949*	Lemma for oddpwdc 12954. 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 12950*	Lemma for oddpwdc 12954. 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 12951*	Lemma for oddpwdc 12954. 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 12952*	Lemma for oddpwdc 12954. 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 12953*	Lemma for oddpwdc 12954. 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 12954*	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 12955*	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 12956*	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 12957	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 12958	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 12959	Lemma for sqrt2irrap 12960. 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 12960	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 12942. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ (𝑄 ∈ ℚ → (√‘2) # 𝑄)
5.2.4  Properties of the canonical representation of a rational
Syntax	cnumer 12961	Extend class notation to include canonical numerator function.
class numer
Syntax	cdenom 12962	Extend class notation to include canonical denominator function.
class denom
Definition	df-numer 12963*	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 12964*	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 12965*	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 12966*	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 12967	Lemma for qnumcl 12968 and qdencl 12969. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))
Theorem	qnumcl 12968	The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ)
Theorem	qdencl 12969	The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ)
Theorem	fnum 12970	Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ numer:ℚ⟶ℤ
Theorem	fden 12971	Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ denom:ℚ⟶ℕ
Theorem	qnumdenbi 12972	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 12973	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 12974	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 12975	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 12976	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 12977	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 12978	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 12979	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 12980	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 12981	nn0gcdsq 12980 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 12982	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 12983	Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (numer‘(𝐴↑2)) = ((numer‘𝐴)↑2))
Theorem	densq 12984	Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2))
Theorem	qden1elz 12985	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 12986	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 12987	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 12988	Extend class notation with the order function on the class of integers modulo N.
class odℤ
Syntax	cphi 12989	Extend class notation with the Euler phi function.
class ϕ
Definition	df-odz 12990*	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 12991*	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 12992*	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 12993*	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 12994	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 12995	Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 28-Feb-2014.)
⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)
Theorem	phibndlem 12996*	Lemma for phibnd 12997. (Contributed by Mario Carneiro, 23-Feb-2014.)
⊢ (𝑁 ∈ (ℤ≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))
Theorem	phibnd 12997	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 12998	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 12999	Value of the Euler ϕ function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
⊢ (ϕ‘1) = 1
Theorem	dfphi2 13000*	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}))