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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | prmgt1 11801 | A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.) |
⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | ||
Theorem | prmm2nn0 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) | ||
Theorem | oddprmgt2 11803 | An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) |
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃) | ||
Theorem | oddprmge3 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)) | ||
Theorem | sqnprm 11805 | A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ) | ||
Theorem | dvdsprm 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) → (𝑁 ∥ 𝑃 ↔ 𝑁 = 𝑃)) | ||
Theorem | exprmfct 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 𝑝 ∥ 𝑁) | ||
Theorem | prmdvdsfz 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 (𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼)) | ||
Theorem | nprmdvds1 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) | ||
Theorem | divgcdodd 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)))) | ||
This section is about coprimality with respect to primes, and a special version of Euclid's lemma for primes is provided, see euclemma 11813. | ||
Theorem | coprm 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) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1)) | ||
Theorem | prmrp 11812 | 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 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) ↔ (𝑃 ∥ 𝑀 ∨ 𝑃 ∥ 𝑁))) | ||
Theorem | isprm6 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) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦)))) | ||
Theorem | prmdvdsexp 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 ∥ (𝐴↑𝑁) ↔ 𝑃 ∥ 𝐴)) | ||
Theorem | prmdvdsexpb 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 ∥ (𝑄↑𝑁) ↔ 𝑃 = 𝑄)) | ||
Theorem | prmdvdsexpr 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 ∥ (𝑄↑𝑁) → 𝑃 = 𝑄)) | ||
Theorem | prmexpb 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)) → ((𝑃↑𝑀) = (𝑄↑𝑁) ↔ (𝑃 = 𝑄 ∧ 𝑀 = 𝑁))) | ||
Theorem | prmfac1 11819 | 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 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) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)) | ||
Theorem | rpexp1i 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 → ((𝐴↑𝑀) gcd 𝐵) = 1)) | ||
Theorem | rpexp12i 11822 | 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 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))) | ||
Theorem | cncongrprm 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 ∧ ¬ 𝑃 ∥ 𝐶)) → (((𝐴 · 𝐶) mod 𝑃) = ((𝐵 · 𝐶) mod 𝑃) ↔ (𝐴 mod 𝑃) = (𝐵 mod 𝑃))) | ||
Theorem | isevengcd2 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)) | ||
Theorem | isoddgcd1 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)) | ||
Theorem | 3lcm2e6 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 | ||
Theorem | sqrt2irrlem 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.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) ⇒ ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) | ||
Theorem | sqrt2irr 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 | ||
Theorem | sqrt2re 11830 | The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.) |
⊢ (√‘2) ∈ ℝ | ||
Theorem | sqrt2irr0 11831 | The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.) |
⊢ (√‘2) ∈ (ℝ ∖ ℚ) | ||
Theorem | pw2dvdslemn 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)) | ||
Theorem | pw2dvds 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)) | ||
Theorem | pw2dvdseulemle 11834 | Lemma for pw2dvdseu 11835. 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 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)) | ||
Theorem | oddpwdclemxy 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)))) | ||
Theorem | oddpwdclemdvds 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) | ||
Theorem | oddpwdclemndvds 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) | ||
Theorem | oddpwdclemodd 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))))) | ||
Theorem | oddpwdclemdc 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))))) | ||
Theorem | oddpwdc 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 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) ⇒ ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ | ||
Theorem | sqpweven 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 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) ⇒ ⊢ (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(◡𝐹‘(𝐴↑2)))) | ||
Theorem | 2sqpwodd 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 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) ⇒ ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2))))) | ||
Theorem | sqne2sq 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))) | ||
Theorem | znege1 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 ∧ 𝐴 ≠ 𝐵) → 1 ≤ (abs‘(𝐴 − 𝐵))) | ||
Theorem | sqrt2irraplemnn 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)) | ||
Theorem | sqrt2irrap 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) | ||
Syntax | cnumer 11848 | Extend class notation to include canonical numerator function. |
class numer | ||
Syntax | cdenom 11849 | Extend class notation to include canonical denominator function. |
class denom | ||
Definition | df-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)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) | ||
Definition | df-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)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) | ||
Theorem | qnumval 11852* | 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 11853* | 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 11854 | Lemma for qnumcl 11855 and qdencl 11856. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ)) | ||
Theorem | qnumcl 11855 | The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ) | ||
Theorem | qdencl 11856 | The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ) | ||
Theorem | fnum 11857 | Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ numer:ℚ⟶ℤ | ||
Theorem | fden 11858 | Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ denom:ℚ⟶ℕ | ||
Theorem | qnumdenbi 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))) | ||
Theorem | qnumdencoprm 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) | ||
Theorem | qeqnumdivden 11861 | 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 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)) | ||
Theorem | divnumden 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)))) | ||
Theorem | divdenle 11864 | 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 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))) | ||
Theorem | qgt0numnn 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) | ||
Theorem | nn0gcdsq 11867 | 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 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))) | ||
Theorem | numdensq 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))) | ||
Theorem | numsq 11870 | Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (numer‘(𝐴↑2)) = ((numer‘𝐴)↑2)) | ||
Theorem | densq 11871 | Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2)) | ||
Theorem | qden1elz 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)) | ||
Theorem | nn0sqrtelqelz 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) | ||
Theorem | nonsq 11874 | 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) | ||
Syntax | cphi 11875 | Extend class notation with the Euler phi function. |
class ϕ | ||
Definition | df-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})) | ||
Theorem | phivalfi 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) | ||
Theorem | phival 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})) | ||
Theorem | phicl2 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)) | ||
Theorem | phicl 11880 | Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 28-Feb-2014.) |
⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ) | ||
Theorem | phibndlem 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))) | ||
Theorem | phibnd 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)) | ||
Theorem | phicld 11883 | 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 11884 | Value of the Euler ϕ function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) |
⊢ (ϕ‘1) = 1 | ||
Theorem | dfphi2 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})) | ||
Theorem | hashdvds 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.) |
⊢ (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 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‘(𝑃↑𝐾)) = ((𝑃↑(𝐾 − 1)) · (𝑃 − 1))) | ||
Theorem | phiprm 11888 | Value of the Euler ϕ function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.) |
⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1)) | ||
Theorem | crth 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 = (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 11890* | Lemma for phimul 11891. (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 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))) | ||
Theorem | hashgcdlem 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 = (𝑥 ∈ 𝐴 ↦ (𝑥 · 𝑁)) ⇒ ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝐹:𝐴–1-1-onto→𝐵) | ||
Theorem | hashgcdeq 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(𝑁 ∥ 𝑀, (ϕ‘(𝑀 / 𝑁)), 0)) | ||
Theorem | oddennn 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 | ||
Theorem | evenennn 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 | ||
Theorem | xpnnen 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 | ||
Theorem | xpomen 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 | ||
Theorem | xpct 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) | ||
Theorem | unennn 11899 | 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 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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