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Type | Label | Description | ||||||||||||
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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)))) | ||||||||||||||
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 ∈ ℕ_{0}) → (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 ∈ ℕ_{0} ∧ 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 ∈ ℕ_{0}) → ((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 ∈ ℕ_{0} ∧ 𝑁 ∈ ℕ_{0})) → ((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 ∈ ℕ_{0}) → (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 | ||||||||||||||
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 ∈ ℕ_{0} ((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 ∈ ℕ_{0} ((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 ∈ ℕ_{0}) & ⊢ (𝜑 → 𝐵 ∈ ℕ_{0}) & ⊢ (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 ∈ ℕ_{0} ((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 ∈ ℕ_{0}) ∧ 𝐴 = ((2↑𝑌) · 𝑋)) → (𝑋 = (𝐴 / (2↑(℩𝑧 ∈ ℕ_{0} ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ_{0} ((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 ∈ ℕ_{0} ((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 ∈ ℕ_{0} ((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 ∈ ℕ_{0} ((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 ∈ ℕ_{0}) ∧ 𝐴 = ((2↑𝑌) · 𝑋)) ↔ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (2↑(℩𝑧 ∈ ℕ_{0} ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ_{0} ((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 ∈ ℕ_{0} ↦ ((2↑𝑦) · 𝑥)) ⇒ ⊢ 𝐹:(𝐽 × ℕ_{0})–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 ∈ ℕ_{0} ↦ ((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 ∈ ℕ_{0} ↦ ((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) | ||||||||||||||
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 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ_{0}) → ((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 ∈ ℕ_{0} ∧ (√‘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 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ_{0}) ∧ ((𝐵↑2) < 𝐴 ∧ 𝐴 < ((𝐵 + 1)↑2))) → ¬ (√‘𝐴) ∈ ℚ) | ||||||||||||||
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)) | ||||||||||||||
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 | ||||||||||||||
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:
| ||||||||||||||
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.
Label naming conventions Here are a few of the label naming conventions:
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.
(Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||||||||||||||
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 |
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