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Type | Label | Description |
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Statement | ||
Theorem | lcmdvdsb 11801 | Biconditional form of lcmdvds 11796. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmass 11802 | Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm lcm lcm | ||
Theorem | 3lcm2e6woprm 11803 | The least common multiple of three and two is six. This proof does not use the property of 2 and 3 being prime. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.) |
lcm | ||
Theorem | 6lcm4e12 11804 | The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.) |
lcm ; | ||
According to Wikipedia "Coprime integers", see https://en.wikipedia.org/wiki/Coprime_integers (16-Aug-2020) "[...] two integers a and b are said to be relatively prime, mutually prime, or coprime [...] if the only positive integer (factor) that divides both of them is 1. Consequently, any prime number that divides one does not divide the other. This is equivalent to their greatest common divisor (gcd) being 1.". In the following, we use this equivalent characterization to say that and are coprime (or relatively prime) if . The equivalence of the definitions is shown by coprmgcdb 11805. The negation, i.e. two integers are not coprime, can be expressed either by , see ncoprmgcdne1b 11806, or equivalently by , see ncoprmgcdgt1b 11807. A proof of Euclid's lemma based on coprimality is provided in coprmdvds 11809 (as opposed to Euclid's lemma for primes). | ||
Theorem | coprmgcdb 11805* | Two positive integers are coprime, i.e. the only positive integer that divides both of them is 1, iff their greatest common divisor is 1. (Contributed by AV, 9-Aug-2020.) |
Theorem | ncoprmgcdne1b 11806* | Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.) |
Theorem | ncoprmgcdgt1b 11807* | Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is greater than 1. (Contributed by AV, 9-Aug-2020.) |
Theorem | coprmdvds1 11808 | If two positive integers are coprime, i.e. their greatest common divisor is 1, the only positive integer that divides both of them is 1. (Contributed by AV, 4-Aug-2021.) |
Theorem | coprmdvds 11809 | Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.) |
Theorem | coprmdvds2 11810 | If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.) |
Theorem | mulgcddvds 11811 | One half of rpmulgcd2 11812, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | rpmulgcd2 11812 | If is relatively prime to , then the GCD of with is the product of the GCDs with and respectively. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | qredeq 11813 | Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.) |
Theorem | qredeu 11814* | Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.) |
Theorem | rpmul 11815 | If is relatively prime to and to , it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.) |
Theorem | rpdvds 11816 | If is relatively prime to then it is also relatively prime to any divisor of . (Contributed by Mario Carneiro, 19-Jun-2015.) |
Theorem | congr 11817* | Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer is congruent to an integer modulo if their difference is a multiple of . See also the definition in [ApostolNT] p. 104: "... is congruent to modulo , and we write (mod ) if divides the difference ", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence, 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021.) |
Theorem | divgcdcoprm0 11818 | Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.) |
Theorem | divgcdcoprmex 11819* | Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.) |
Theorem | cncongr1 11820 | One direction of the bicondition in cncongr 11822. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) |
Theorem | cncongr2 11821 | The other direction of the bicondition in cncongr 11822. (Contributed by AV, 11-Jul-2021.) |
Theorem | cncongr 11822 | 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.) |
Theorem | cncongrcoprm 11823 | 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.) |
Remark: to represent odd prime numbers, i.e., all prime numbers except , the idiom is used. It is a little bit shorter than . Both representations can be converted into each other by eldifsn 3658. | ||
Syntax | cprime 11824 | Extend the definition of a class to include the set of prime numbers. |
Definition | df-prm 11825* | Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm 11826* | 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.) |
Theorem | prmnn 11827 | A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | prmz 11828 | A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.) |
Theorem | prmssnn 11829 | The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.) |
Theorem | prmex 11830 | The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.) |
Theorem | 1nprm 11831 | 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
Theorem | 1idssfct 11832* | The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm2lem 11833* | Lemma for isprm2 11834. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm2 11834* | 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.) |
Theorem | isprm3 11835* | 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.) |
Theorem | isprm4 11836* | 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.) |
Theorem | prmind2 11837* | A variation on prmind 11838 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | prmind 11838* | Perform induction over the multiplicative structure of . If a property holds for the primes and and is preserved under multiplication, then it holds for every positive integer. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsprime 11839 | If divides a prime, then is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.) |
Theorem | nprm 11840 | A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | nprmi 11841 | An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsnprmd 11842 | If a number is divisible by an integer greater than 1 and less then the number, the number is not prime. (Contributed by AV, 24-Jul-2021.) |
Theorem | prm2orodd 11843 | A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.) |
Theorem | 2prm 11844 | 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
Theorem | 3prm 11845 | 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | 4nprm 11846 | 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.) |
Theorem | prmuz2 11847 | A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | prmgt1 11848 | A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.) |
Theorem | prmm2nn0 11849 | Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.) |
Theorem | oddprmgt2 11850 | An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) |
Theorem | oddprmge3 11851 | 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.) |
Theorem | sqnprm 11852 | A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsprm 11853 | 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.) |
Theorem | exprmfct 11854* | 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.) |
Theorem | prmdvdsfz 11855* | 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.) |
Theorem | nprmdvds1 11856 | No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.) |
Theorem | divgcdodd 11857 | Either is odd or is odd. (Contributed by Scott Fenton, 19-Apr-2014.) |
This section is about coprimality with respect to primes, and a special version of Euclid's lemma for primes is provided, see euclemma 11860. | ||
Theorem | coprm 11858 | 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.) |
Theorem | prmrp 11859 | Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | euclemma 11860 | 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.) |
Theorem | isprm6 11861* | A number is prime iff it satisfies Euclid's lemma euclemma 11860. (Contributed by Mario Carneiro, 6-Sep-2015.) |
Theorem | prmdvdsexp 11862 | 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.) |
Theorem | prmdvdsexpb 11863 | 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.) |
Theorem | prmdvdsexpr 11864 | If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Theorem | prmexpb 11865 | Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.) |
Theorem | prmfac1 11866 | The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.) |
Theorem | rpexp 11867 | If two numbers and are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.) |
Theorem | rpexp1i 11868 | Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | rpexp12i 11869 | Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | prmndvdsfaclt 11870 | A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.) |
Theorem | cncongrprm 11871 | 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.) |
Theorem | isevengcd2 11872 | 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.) |
Theorem | isoddgcd1 11873 | 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.) |
Theorem | 3lcm2e6 11874 | 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.) |
lcm | ||
Theorem | sqrt2irrlem 11875 | Lemma for sqrt2irr 11876. This is the core of the proof: - if , then and are even, so and 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.) |
Theorem | sqrt2irr 11876 |
The square root of 2 is not rational. That is, for any rational number,
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, is apart from it" (the two
statements are equivalent given excluded middle). See sqrt2irrap 11894 for
the proof that the square root of two is irrational.
The proof's core is proven in sqrt2irrlem 11875, which shows that if , then and are even, so and are smaller representatives, which is absurd. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
Theorem | sqrt2re 11877 | The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.) |
Theorem | sqrt2irr0 11878 | The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.) |
Theorem | pw2dvdslemn 11879* | Lemma for pw2dvds 11880. 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.) |
Theorem | pw2dvds 11880* | A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.) |
Theorem | pw2dvdseulemle 11881 | Lemma for pw2dvdseu 11882. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | pw2dvdseu 11882* | A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemxy 11883* | Lemma for oddpwdc 11888. 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.) |
Theorem | oddpwdclemdvds 11884* | Lemma for oddpwdc 11888. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | oddpwdclemndvds 11885* | Lemma for oddpwdc 11888. 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.) |
Theorem | oddpwdclemodd 11886* | Lemma for oddpwdc 11888. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemdc 11887* | Lemma for oddpwdc 11888. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdc 11888* | The function 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.) |
Theorem | sqpweven 11889* | The greatest power of two dividing the square of an integer is an even power of two. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | 2sqpwodd 11890* | 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.) |
Theorem | sqne2sq 11891 | 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.) |
Theorem | znege1 11892 | The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.) |
Theorem | sqrt2irraplemnn 11893 | Lemma for sqrt2irrap 11894. The square root of 2 is apart from a positive rational expressed as a numerator and denominator. (Contributed by Jim Kingdon, 2-Oct-2021.) |
# | ||
Theorem | sqrt2irrap 11894 | The square root of 2 is irrational. That is, for any rational number, 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 11876. (Contributed by Jim Kingdon, 2-Oct-2021.) |
# | ||
Syntax | cnumer 11895 | Extend class notation to include canonical numerator function. |
numer | ||
Syntax | cdenom 11896 | Extend class notation to include canonical denominator function. |
denom | ||
Definition | df-numer 11897* | 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 | ||
Definition | df-denom 11898* | 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 | ||
Theorem | qnumval 11899* | Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdenval 11900* | Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom |
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