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Theorem List for Intuitionistic Logic Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprmgt1 11801 A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
(𝑃 ∈ ℙ → 1 < 𝑃)

Theoremprmm2nn0 11802 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 11803 An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
(𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)

Theoremoddprmge3 11804 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))

Theoremsqnprm 11805 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ)

Theoremdvdsprm 11806 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 11807* 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 11808* 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 11809 No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
(𝑃 ∈ ℙ → ¬ 𝑃 ∥ 1)

Theoremdivgcdodd 11810 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 11813.

Theoremcoprm 11811 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 11812 Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)
((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑃 gcd 𝑄) = 1 ↔ 𝑃𝑄))

Theoremeuclemma 11813 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 11814* A number is prime iff it satisfies Euclid's lemma euclemma 11813. (Contributed by Mario Carneiro, 6-Sep-2015.)
(𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ‘2) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (𝑃 ∥ (𝑥 · 𝑦) → (𝑃𝑥𝑃𝑦))))

Theoremprmdvdsexp 11815 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 11816 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 11817 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 11818 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 11819 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 11820 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 11821 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 11822 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 11823 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)))

Theoremcncongrprm 11824 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 11825 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 11826 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 11827 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

Theoremsqrt2irrlem 11828 Lemma for sqrt2irr 11829. 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.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → (√‘2) = (𝐴 / 𝐵))       (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ))

Theoremsqrt2irr 11829 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 11847 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 11828, 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

Theoremsqrt2re 11830 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
(√‘2) ∈ ℝ

Theoremsqrt2irr0 11831 The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
(√‘2) ∈ (ℝ ∖ ℚ)

Theorempw2dvdslemn 11832* Lemma for pw2dvds 11833. 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))

Theorempw2dvds 11833* 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))

Theorempw2dvdseulemle 11834 Lemma for pw2dvdseu 11835. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)    &   (𝜑 → (2↑𝐴) ∥ 𝑁)    &   (𝜑 → ¬ (2↑(𝐵 + 1)) ∥ 𝑁)       (𝜑𝐴𝐵)

Theorempw2dvdseu 11835* 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))

Theoremoddpwdclemxy 11836* Lemma for oddpwdc 11841. 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))))

Theoremoddpwdclemdvds 11837* Lemma for oddpwdc 11841. 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)

Theoremoddpwdclemndvds 11838* Lemma for oddpwdc 11841. 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)

Theoremoddpwdclemodd 11839* Lemma for oddpwdc 11841. 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)))))

Theoremoddpwdclemdc 11840* Lemma for oddpwdc 11841. 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)))))

Theoremoddpwdc 11841* 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 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))       𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ

Theoremsqpweven 11842* 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 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))       (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(𝐹‘(𝐴↑2))))

Theorem2sqpwodd 11843* 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 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))       (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(𝐹‘(2 · (𝐴↑2)))))

Theoremsqne2sq 11844 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)))

Theoremznege1 11845 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 ∧ 𝐴𝐵) → 1 ≤ (abs‘(𝐴𝐵)))

Theoremsqrt2irraplemnn 11846 Lemma for sqrt2irrap 11847. 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))

Theoremsqrt2irrap 11847 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 11829. (Contributed by Jim Kingdon, 2-Oct-2021.)
(𝑄 ∈ ℚ → (√‘2) # 𝑄)

5.2.4  Properties of the canonical representation of a rational

Syntaxcnumer 11848 Extend class notation to include canonical numerator function.
class numer

Syntaxcdenom 11849 Extend class notation to include canonical denominator function.
class denom

Definitiondf-numer 11850* 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 11851* 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 11852* 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 11853* 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 11854 Lemma for qnumcl 11855 and qdencl 11856. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))

Theoremqnumcl 11855 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ)

Theoremqdencl 11856 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ)

Theoremfnum 11857 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer:ℚ⟶ℤ

Theoremfden 11858 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom:ℚ⟶ℕ

Theoremqnumdenbi 11859 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 11860 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 11861 Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℚ → 𝐴 = ((numer‘𝐴) / (denom‘𝐴)))

Theoremqmuldeneqnum 11862 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 11863 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 11864 Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (denom‘(𝐴 / 𝐵)) ≤ 𝐵)

Theoremqnumgt0 11865 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 11866 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 11867 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 11868 nn0gcdsq 11867 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 11869 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 11870 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (numer‘(𝐴↑2)) = ((numer‘𝐴)↑2))

Theoremdensq 11871 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2))

Theoremqden1elz 11872 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))

Theoremnn0sqrtelqelz 11873 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)

Theoremnonsq 11874 Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) ∧ ((𝐵↑2) < 𝐴𝐴 < ((𝐵 + 1)↑2))) → ¬ (√‘𝐴) ∈ ℚ)

5.2.5  Euler's theorem

Syntaxcphi 11875 Extend class notation with the Euler phi function.
class ϕ

Definitiondf-phi 11876* 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}))

Theoremphivalfi 11877* 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)

Theoremphival 11878* 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 11879 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 11880 Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 28-Feb-2014.)
(𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)

Theoremphibndlem 11881* Lemma for phibnd 11882. (Contributed by Mario Carneiro, 23-Feb-2014.)
(𝑁 ∈ (ℤ‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))

Theoremphibnd 11882 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 11883 Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → (ϕ‘𝑁) ∈ ℕ)

Theoremphi1 11884 Value of the Euler ϕ function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
(ϕ‘1) = 1

Theoremdfphi2 11885* 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 11886* 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 11887 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 11888 Value of the Euler ϕ function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
(𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))

Theoremcrth 11889* 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 11890* Lemma for phimul 11891. (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 11891 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)))

Theoremhashgcdlem 11892* 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 11893* 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))

5.3  Cardinality of real and complex number subsets

5.3.1  Countability of integers and rationals

Theoremoddennn 11894 There are as many odd positive integers as there are positive integers. (Contributed by Jim Kingdon, 11-May-2022.)
{𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ≈ ℕ

Theoremevenennn 11895 There are as many even positive integers as there are positive integers. (Contributed by Jim Kingdon, 12-May-2022.)
{𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ

Theoremxpnnen 11896 The Cartesian product of the set of positive integers with itself is equinumerous to the set of positive integers. (Contributed by NM, 1-Aug-2004.)
(ℕ × ℕ) ≈ ℕ

Theoremxpomen 11897 The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.)
(ω × ω) ≈ ω

Theoremxpct 11898 The cartesian product of two sets dominated by ω is dominated by ω. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω)

Theoremunennn 11899 The union of two disjoint countably infinite sets is countably infinite. (Contributed by Jim Kingdon, 13-May-2022.)
((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ ℕ)

Theoremznnen 11900 The set of integers and the set of positive integers are equinumerous. Corollary 8.1.23 of [AczelRathjen], p. 75. (Contributed by NM, 31-Jul-2004.)
ℤ ≈ ℕ

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