P. 111 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

exprmfct 11001*

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 11002*

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 11003

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

divgcdodd 11004

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))))

4.2.2  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 11007.

Theorem

coprm 11005

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 11006

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 11007

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 11008*

A number is prime iff it satisfies Euclid's lemma euclemma 11007. (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 11009

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 11010

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 11011

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 11012

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 11013

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 11014

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 11015

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 11016

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 11017

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 11018

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 11019

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 11020

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 11021

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

4.2.3  Non-rationality of square root of 2

Theorem

sqrt2irrlem 11022

Lemma for sqrt2irr 11023. 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 11023

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 11040 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 11022, 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 11024

The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)

⊢ (√‘2) ∈ ℝ

Theorem

pw2dvdslemn 11025*

Lemma for pw2dvds 11026. 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 11026*

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 11027

Lemma for pw2dvdseu 11028. 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 11028*

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 11029*

Lemma for oddpwdc 11034. 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 11030*

Lemma for oddpwdc 11034. 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 11031*

Lemma for oddpwdc 11034. 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 11032*

Lemma for oddpwdc 11034. 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 11033*

Lemma for oddpwdc 11034. 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 11034*

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 11035*

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 ∥ (2^nd ‘(^◡𝐹‘(𝐴↑2))))

Theorem

2sqpwodd 11036*

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 ∥ (2^nd ‘(^◡𝐹‘(2 · (𝐴↑2)))))

Theorem

sqne2sq 11037

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 11038

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 11039

Lemma for sqrt2irrap 11040. 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 11040

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 11023. (Contributed by Jim Kingdon, 2-Oct-2021.)

⊢ (𝑄 ∈ ℚ → (√‘2) # 𝑄)

4.2.4  Properties of the canonical representation of a rational

Syntax

cnumer 11041

Extend class notation to include canonical numerator function.

class numer

Syntax

cdenom 11042

Extend class notation to include canonical denominator function.

class denom

Definition

df-numer 11043*

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 ↦ (1^st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑦 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))

Definition

df-denom 11044*

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 ↦ (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑦 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))

Theorem

qnumval 11045*

Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)

⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1^st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))

Theorem

qdenval 11046*

Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)

⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))

Theorem

qnumdencl 11047

Lemma for qnumcl 11048 and qdencl 11049. (Contributed by Stefan O'Rear, 13-Sep-2014.)

⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))

Theorem

qnumcl 11048

The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)

⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ)

Theorem

qdencl 11049

The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)

⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ)

Theorem

fnum 11050

Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)

⊢ numer:ℚ⟶ℤ

Theorem

fden 11051

Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)

⊢ denom:ℚ⟶ℕ

Theorem

qnumdenbi 11052

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 11053

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 11054

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 11055

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 11056

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 11057

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 11058

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 11059

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 11060

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 11061

nn0gcdsq 11060 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 11062

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 11063

Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)

⊢ (𝐴 ∈ ℚ → (numer‘(𝐴↑2)) = ((numer‘𝐴)↑2))

Theorem

densq 11064

Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)

⊢ (𝐴 ∈ ℚ → (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2))

Theorem

qden1elz 11065

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 11066

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 11067

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)

4.2.5  Euler's theorem

Syntax

cphi 11068

Extend class notation with the Euler phi function.

class ϕ

Definition

df-phi 11069*

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 11070*

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 11071*

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 11072

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 11073

Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 28-Feb-2014.)

⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)

Theorem

phibndlem 11074*

Lemma for phibnd 11075. (Contributed by Mario Carneiro, 23-Feb-2014.)

⊢ (𝑁 ∈ (ℤ_≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)))

Theorem

phibnd 11075

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 11076

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 11077

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

⊢ (ϕ‘1) = 1

Theorem

dfphi2 11078*

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 11079*

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 11080

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 11081

Value of the Euler ϕ function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)

⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))

Theorem

crth 11082*

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 11083*

Lemma for phimul 11084. (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 11084

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

hashgcdlem 11085*

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 = (𝑥 ∈ 𝐴 ↦ (𝑥 · 𝑁))    ⇒   ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝐹:𝐴–1-1-onto→𝐵)

Theorem

hashgcdeq 11086*

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(𝑁 ∥ 𝑀, (ϕ‘(𝑀 / 𝑁)), 0))

4.3  Cardinality of real and complex number subsets

4.3.1  Countability of integers and rationals

Theorem

oddennn 11087

There are as many odd positive integers as there are positive integers. (Contributed by Jim Kingdon, 11-May-2022.)

⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ≈ ℕ

Theorem

evenennn 11088

There are as many even positive integers as there are positive integers. (Contributed by Jim Kingdon, 12-May-2022.)

⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ

Theorem

xpnnen 11089

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

Theorem

xpomen 11090

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

Theorem

xpct 11091

The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)

⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω)

Theorem

unennn 11092

The union of two disjoint countably infinite sets is countably infinite. (Contributed by Jim Kingdon, 13-May-2022.)

⊢ ((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ≈ ℕ)

Theorem

znnen 11093

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

PART 5  GUIDES AND MISCELLANEA

5.1  Guides (conventions, explanations, and examples)

5.1.1  Conventions

This section describes the conventions we use. However, these conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

• Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
• Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
• Theorems of propositional calculus - [Heyting].
• Theorems of pure predicate calculus - Metamath Proof Explorer.
• Theorems of equality and substitution - Metamath Proof Explorer.
• Axioms of set theory - [Crosilla].
• Development of set theory - Chapter 10 of [HoTT].
• Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
• Theorems about real numbers - [Geuvers].

Theorem

conventions 11094

Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (MPE, set.mm). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.

• Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in Metamath. You can find some development using CZF in BJ's mathbox starting at wbd 11148 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 5612, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
• Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). For this reason, we generally need to prove we are evaluating functions within their domains and avoid the reverse closure theorems of the Metamath Proof Explorer.
• Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

• Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.
• iset.mm versus set.mm names

  Theorems which are the same as in set.mm should be named the same (that is, where the statement of the theorem is the same; the proof can differ without a new name being called for). Theorems which are different should be named differently (we do have a small number of intentional exceptions to this rule but on the whole it serves us well).

  As for how to choose names so they are different between iset.mm and set.mm, when possible choose a name which reflect the difference in the theorems. For example, if a theorem in set.mm is an equality and the iset.mm analogue is a subset, add "ss" to the iset.mm name. If need be, add "i" to the iset.mm name (usually as a prefix to some portion of the name).

  As with set.mm, we welcome suggestions for better names (such as names which are more consistent with naming conventions).

  We do try to keep set.mm and iset.mm similar where we can. For example, if a theorem exists in both places but the name in set.mm isn't great, we tend to keep that name for iset.mm, or change it in both files together. This is mainly to make it easier to copy theorems, but also to generally help people browse proofs, find theorems, write proofs, etc.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

Abbreviation	Mnenomic	Source	Expression	Syntax?	Example(s)
ap	apart	df-ap 8000		Yes	apadd1 8026, apne 8041

Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/forum/#!forum/metamath.

(Contributed by Jim Kingdon, 24-Feb-2020.)

⊢ 𝜑    ⇒   ⊢ 𝜑

5.1.2  Definitional examples

Theorem

ex-or 11095

Example for ax-io 663. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

⊢ (2 = 3 ∨ 4 = 4)

Theorem

ex-an 11096

Example for ax-ia1 104. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

⊢ (2 = 2 ∧ 3 = 3)

Theorem

1kp2ke3k 11097

Example for df-dec 8810, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with (2 + 1) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 8810 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

⊢ (;;;1000 + ;;;2000) = ;;;3000

Theorem

ex-fl 11098

Example for df-fl 9605. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)

Theorem

ex-ceil 11099

Example for df-ceil 9606. (Contributed by AV, 4-Sep-2021.)

⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

Theorem

ex-fac 11100

Example for df-fac 10031. (Contributed by AV, 4-Sep-2021.)

⊢ (!‘5) = ;;120