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Type | Label | Description |
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Statement | ||
Theorem | cncongr 16001 | Cancellability of Congruences (see ProofWiki "Cancellability of Congruences, https://proofwiki.org/wiki/Cancellability_of_Congruences, 10-Jul-2021): Two products with a common factor are congruent modulo a positive integer iff the other factors are congruent modulo the integer divided by the greates common divisor of the integer and the common factor. See also Theorem 5.4 "Cancellation law" in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) ↔ (𝐴 mod 𝑀) = (𝐵 mod 𝑀))) | ||
Theorem | cncongrcoprm 16002 | Corollary 1 of Cancellability of Congruences: Two products with a common factor are congruent modulo an integer being coprime to the common factor iff the other factors are congruent modulo the integer. (Contributed by AV, 13-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ (𝐶 gcd 𝑁) = 1)) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) ↔ (𝐴 mod 𝑁) = (𝐵 mod 𝑁))) | ||
Remark: to represent odd prime numbers, i.e., all prime numbers except 2, the idiom 𝑃 ∈ (ℙ ∖ {2}) is used. It is a little bit shorter than (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2). Both representations can be converted into each other by eldifsn 4711. | ||
Syntax | cprime 16003 | Extend the definition of a class to include the set of prime numbers. |
class ℙ | ||
Definition | df-prm 16004* | Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o} | ||
Theorem | isprm 16005* | The predicate "is a prime number". A prime number is a positive integer with exactly two positive divisors. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o)) | ||
Theorem | prmnn 16006 | A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | ||
Theorem | prmz 16007 | A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.) |
⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | ||
Theorem | prmssnn 16008 | The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.) |
⊢ ℙ ⊆ ℕ | ||
Theorem | prmex 16009 | The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.) |
⊢ ℙ ∈ V | ||
Theorem | 0nprm 16010 | 0 is not a prime number. Already the definition df-prm 16004 excludes 0 from being prime (ℙ = {𝑝 ∈ ℕ ∣ ...), but even if 𝑝 ∈ ℕ0 was allowed, the condition {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o would not hold for 𝑝 = 0, because {𝑛 ∈ ℕ ∣ 𝑛 ∥ 0} = ℕ, see dvds0 15613, and ¬ ℕ ≈ 2o (there are more than 2 positive integers). (Contributed by AV, 29-May-2023.) |
⊢ ¬ 0 ∈ ℙ | ||
Theorem | 1nprm 16011 | 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
⊢ ¬ 1 ∈ ℙ | ||
Theorem | 1idssfct 16012* | The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝑁 ∈ ℕ → {1, 𝑁} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁}) | ||
Theorem | isprm2lem 16013* | Lemma for isprm2 16014. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃})) | ||
Theorem | isprm2 16014* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only positive divisors are 1 and itself. Definition in [ApostolNT] p. 16. (Contributed by Paul Chapman, 26-Oct-2012.) |
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ ℕ (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃)))) | ||
Theorem | isprm3 16015* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 with no divisors strictly between 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ (2...(𝑃 − 1)) ¬ 𝑧 ∥ 𝑃)) | ||
Theorem | isprm4 16016* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only divisor greater than or equal to 2 is itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ (ℤ≥‘2)(𝑧 ∥ 𝑃 → 𝑧 = 𝑃))) | ||
Theorem | prmind2 16017* | A variation on prmind 16018 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) & ⊢ 𝜓 & ⊢ ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑) & ⊢ ((𝑦 ∈ (ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏)) ⇒ ⊢ (𝐴 ∈ ℕ → 𝜂) | ||
Theorem | prmind 16018* | Perform induction over the multiplicative structure of ℕ. If a property 𝜑(𝑥) holds for the primes and 1 and is preserved under multiplication, then it holds for every positive integer. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) & ⊢ 𝜓 & ⊢ (𝑥 ∈ ℙ → 𝜑) & ⊢ ((𝑦 ∈ (ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏)) ⇒ ⊢ (𝐴 ∈ ℕ → 𝜂) | ||
Theorem | dvdsprime 16019 | If 𝑀 divides a prime, then 𝑀 is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ) → (𝑀 ∥ 𝑃 ↔ (𝑀 = 𝑃 ∨ 𝑀 = 1))) | ||
Theorem | nprm 16020 | A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → ¬ (𝐴 · 𝐵) ∈ ℙ) | ||
Theorem | nprmi 16021 | An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ & ⊢ 1 < 𝐴 & ⊢ 1 < 𝐵 & ⊢ (𝐴 · 𝐵) = 𝑁 ⇒ ⊢ ¬ 𝑁 ∈ ℙ | ||
Theorem | dvdsnprmd 16022 | If a number is divisible by an integer greater than 1 and less than the number, the number is not prime. (Contributed by AV, 24-Jul-2021.) |
⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → 𝐴 < 𝑁) & ⊢ (𝜑 → 𝐴 ∥ 𝑁) ⇒ ⊢ (𝜑 → ¬ 𝑁 ∈ ℙ) | ||
Theorem | prm2orodd 16023 | A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.) |
⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃)) | ||
Theorem | 2prm 16024 | 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
⊢ 2 ∈ ℙ | ||
Theorem | 2mulprm 16025 | A multiple of two is prime iff the multiplier is one. (Contributed by AV, 8-Jun-2023.) |
⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ ↔ 𝐴 = 1)) | ||
Theorem | 3prm 16026 | 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ 3 ∈ ℙ | ||
Theorem | 4nprm 16027 | 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.) |
⊢ ¬ 4 ∈ ℙ | ||
Theorem | prmuz2 16028 | A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2)) | ||
Theorem | prmgt1 16029 | A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.) |
⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | ||
Theorem | prmm2nn0 16030 | 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 16031 | An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) |
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃) | ||
Theorem | oddprmge3 16032 | 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 | ge2nprmge4 16033 | A composite integer greater than or equal to 2 is greater than or equal to 4. (Contributed by AV, 5-Jun-2023.) |
⊢ ((𝑋 ∈ (ℤ≥‘2) ∧ 𝑋 ∉ ℙ) → 𝑋 ∈ (ℤ≥‘4)) | ||
Theorem | sqnprm 16034 | A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝐴 ∈ ℤ → ¬ (𝐴↑2) ∈ ℙ) | ||
Theorem | dvdsprm 16035 | 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 16036* | 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 16037* | 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 16038 | 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 | isprm5 16039* | One need only check prime divisors of 𝑃 up to √𝑃 in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))) | ||
Theorem | isprm7 16040* | One need only check prime divisors of 𝑃 up to √𝑃 in order to ensure primality. This version of isprm5 16039 combines the primality and bound on 𝑧 into a finite interval of prime numbers. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑧 ∈ ((2...(⌊‘(√‘𝑃))) ∩ ℙ) ¬ 𝑧 ∥ 𝑃)) | ||
Theorem | maxprmfct 16041* | The set of prime factors of an integer greater than or equal to 2 satisfies the conditions to have a supremum, and that supremum is a member of the set. (Contributed by Paul Chapman, 17-Nov-2012.) |
⊢ 𝑆 = {𝑧 ∈ ℙ ∣ 𝑧 ∥ 𝑁} ⇒ ⊢ (𝑁 ∈ (ℤ≥‘2) → ((𝑆 ⊆ ℤ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) ∧ sup(𝑆, ℝ, < ) ∈ 𝑆)) | ||
Theorem | divgcdodd 16042 | 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 16045. | ||
Theorem | coprm 16043 | 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 16044 | 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 16045 | 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 16046* | A number is prime iff it satisfies Euclid's lemma euclemma 16045. (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 16047 | 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 16048 | 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 16049 | 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 16050 | 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 16051 | 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 16052 | 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 16053 | 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 16054 | 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 16055 | 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 | ncoprmlnprm 16056 | If two positive integers are not coprime, the larger of them is not a prime number. (Contributed by AV, 9-Aug-2020.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → (1 < (𝐴 gcd 𝐵) → 𝐵 ∉ ℙ)) | ||
Theorem | cncongrprm 16057 | 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 16058 | 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 16059 | 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 16060 | 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 | ||
Syntax | cnumer 16061 | Extend class notation to include canonical numerator function. |
class numer | ||
Syntax | cdenom 16062 | Extend class notation to include canonical denominator function. |
class denom | ||
Definition | df-numer 16063* | 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 16064* | 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 16065* | 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 16066* | 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 16067 | Lemma for qnumcl 16068 and qdencl 16069. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ)) | ||
Theorem | qnumcl 16068 | The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ) | ||
Theorem | qdencl 16069 | The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ) | ||
Theorem | fnum 16070 | Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ numer:ℚ⟶ℤ | ||
Theorem | fden 16071 | Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ denom:ℚ⟶ℕ | ||
Theorem | qnumdenbi 16072 | 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 16073 | 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 16074 | 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 16075 | 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 16076 | 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 16077 | 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 16078 | 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 16079 | 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 16080 | 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 16081 | nn0gcdsq 16080 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 16082 | 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 16083 | Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (numer‘(𝐴↑2)) = ((numer‘𝐴)↑2)) | ||
Theorem | densq 16084 | Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2)) | ||
Theorem | qden1elz 16085 | 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 | zsqrtelqelz 16086 | If an integer has a rational square root, that root is must be an integer. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ (√‘𝐴) ∈ ℚ) → (√‘𝐴) ∈ ℤ) | ||
Theorem | nonsq 16087 | Any integer strictly between two adjacent squares has an irrational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ((𝐵↑2) < 𝐴 ∧ 𝐴 < ((𝐵 + 1)↑2))) → ¬ (√‘𝐴) ∈ ℚ) | ||
Syntax | codz 16088 | Extend class notation with the order function on the class of integers mod N. |
class odℤ | ||
Syntax | cphi 16089 | Extend class notation with the Euler phi function. |
class ϕ | ||
Definition | df-odz 16090* | Define the order function on the class of integers mod N. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.) |
⊢ odℤ = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥↑𝑚) − 1)}, ℝ, < ))) | ||
Definition | df-phi 16091* | 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 | phival 16092* | 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 16093 | 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 16094 | Closure for the value of the Euler ϕ function. (Contributed by Mario Carneiro, 28-Feb-2014.) |
⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ) | ||
Theorem | phibndlem 16095* | Lemma for phibnd 16096. (Contributed by Mario Carneiro, 23-Feb-2014.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1))) | ||
Theorem | phibnd 16096 | 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 16097 | 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 16098 | Value of the Euler ϕ function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) |
⊢ (ϕ‘1) = 1 | ||
Theorem | dfphi2 16099* | 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 16100* | 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)))) |
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