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Type | Label | Description |
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Statement | ||
Theorem | rpdvds 12101 |
If ![]() ![]() ![]() ![]() |
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Theorem | congr 12102* |
Definition of congruence by integer multiple (see ProofWiki "Congruence
(Number Theory)", 11-Jul-2021,
https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)):
An integer ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | divgcdcoprm0 12103 | Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.) |
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Theorem | divgcdcoprmex 12104* | 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.) |
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Theorem | cncongr1 12105 | One direction of the bicondition in cncongr 12107. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) |
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Theorem | cncongr2 12106 | The other direction of the bicondition in cncongr 12107. (Contributed by AV, 11-Jul-2021.) |
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Theorem | cncongr 12107 | 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.) |
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Theorem | cncongrcoprm 12108 | 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.) |
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Remark: to represent odd prime numbers, i.e., all prime numbers except | ||
Syntax | cprime 12109 | Extend the definition of a class to include the set of prime numbers. |
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Definition | df-prm 12110* | Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | isprm 12111* | 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.) |
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Theorem | prmnn 12112 | A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | prmz 12113 | A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.) |
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Theorem | prmssnn 12114 | The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.) |
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Theorem | prmex 12115 | The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.) |
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Theorem | 1nprm 12116 | 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
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Theorem | 1idssfct 12117* | The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | isprm2lem 12118* | Lemma for isprm2 12119. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | isprm2 12119* | 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.) |
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Theorem | isprm3 12120* | 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.) |
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Theorem | isprm4 12121* | 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.) |
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Theorem | prmind2 12122* | A variation on prmind 12123 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.) |
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Theorem | prmind 12123* |
Perform induction over the multiplicative structure of ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvdsprime 12124 |
If ![]() ![]() |
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Theorem | nprm 12125 | A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.) |
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Theorem | nprmi 12126 | An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) |
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Theorem | dvdsnprmd 12127 | 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.) |
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Theorem | prm2orodd 12128 | A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.) |
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Theorem | 2prm 12129 | 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
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Theorem | 3prm 12130 | 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | 4nprm 12131 | 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.) |
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Theorem | prmdc 12132 | Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.) |
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Theorem | prmuz2 12133 | A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
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Theorem | prmgt1 12134 | A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.) |
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Theorem | prmm2nn0 12135 | Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.) |
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Theorem | oddprmgt2 12136 | An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) |
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Theorem | oddprmge3 12137 | 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.) |
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Theorem | sqnprm 12138 | A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) |
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Theorem | dvdsprm 12139 | 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.) |
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Theorem | exprmfct 12140* | 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.) |
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Theorem | prmdvdsfz 12141* | 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.) |
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Theorem | nprmdvds1 12142 | No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.) |
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Theorem | isprm5lem 12143* |
Lemma for isprm5 12144. The interesting direction (showing that
one only
needs to check prime divisors up to the square root of ![]() |
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Theorem | isprm5 12144* |
One need only check prime divisors of ![]() ![]() ![]() |
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Theorem | divgcdodd 12145 |
Either ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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This section is about coprimality with respect to primes, and a special version of Euclid's lemma for primes is provided, see euclemma 12148. | ||
Theorem | coprm 12146 | 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.) |
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Theorem | prmrp 12147 | Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.) |
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Theorem | euclemma 12148 | 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.) |
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Theorem | isprm6 12149* | A number is prime iff it satisfies Euclid's lemma euclemma 12148. (Contributed by Mario Carneiro, 6-Sep-2015.) |
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Theorem | prmdvdsexp 12150 | 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.) |
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Theorem | prmdvdsexpb 12151 | 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.) |
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Theorem | prmdvdsexpr 12152 | If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.) |
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Theorem | prmexpb 12153 | Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.) |
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Theorem | prmfac1 12154 | The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.) |
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Theorem | rpexp 12155 |
If two numbers ![]() ![]() |
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Theorem | rpexp1i 12156 | Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
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Theorem | rpexp12i 12157 | Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
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Theorem | prmndvdsfaclt 12158 | A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.) |
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Theorem | cncongrprm 12159 | 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.) |
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Theorem | isevengcd2 12160 | 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.) |
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Theorem | isoddgcd1 12161 | 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.) |
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Theorem | 3lcm2e6 12162 | 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.) |
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Theorem | sqrt2irrlem 12163 |
Lemma for sqrt2irr 12164. This is the core of the proof: - if
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Theorem | sqrt2irr 12164 |
The square root of 2 is not rational. That is, for any rational number,
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The proof's core is proven in sqrt2irrlem 12163, which shows that if
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Theorem | sqrt2re 12165 | The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.) |
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Theorem | sqrt2irr0 12166 | The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.) |
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Theorem | pw2dvdslemn 12167* | Lemma for pw2dvds 12168. 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.) |
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Theorem | pw2dvds 12168* | A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.) |
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Theorem | pw2dvdseulemle 12169 | Lemma for pw2dvdseu 12170. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) |
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Theorem | pw2dvdseu 12170* | A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) |
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Theorem | oddpwdclemxy 12171* | Lemma for oddpwdc 12176. 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.) |
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Theorem | oddpwdclemdvds 12172* | Lemma for oddpwdc 12176. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) |
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Theorem | oddpwdclemndvds 12173* | Lemma for oddpwdc 12176. 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.) |
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Theorem | oddpwdclemodd 12174* | Lemma for oddpwdc 12176. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.) |
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Theorem | oddpwdclemdc 12175* | Lemma for oddpwdc 12176. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.) |
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Theorem | oddpwdc 12176* |
The function ![]() |
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Theorem | sqpweven 12177* | The greatest power of two dividing the square of an integer is an even power of two. (Contributed by Jim Kingdon, 17-Nov-2021.) |
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Theorem | 2sqpwodd 12178* | 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.) |
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Theorem | sqne2sq 12179 | 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.) |
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Theorem | znege1 12180 | The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.) |
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Theorem | sqrt2irraplemnn 12181 | Lemma for sqrt2irrap 12182. The square root of 2 is apart from a positive rational expressed as a numerator and denominator. (Contributed by Jim Kingdon, 2-Oct-2021.) |
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Theorem | sqrt2irrap 12182 |
The square root of 2 is irrational. That is, for any rational number,
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Syntax | cnumer 12183 | Extend class notation to include canonical numerator function. |
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Syntax | cdenom 12184 | Extend class notation to include canonical denominator function. |
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Definition | df-numer 12185* | 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.) |
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Definition | df-denom 12186* | 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.) |
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Theorem | qnumval 12187* | Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | qdenval 12188* | Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | qnumdencl 12189 | Lemma for qnumcl 12190 and qdencl 12191. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | qnumcl 12190 | The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | qdencl 12191 | The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | fnum 12192 | Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | fden 12193 | Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | qnumdenbi 12194 | 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.) |
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Theorem | qnumdencoprm 12195 | The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | qeqnumdivden 12196 | Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | qmuldeneqnum 12197 | Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | divnumden 12198 |
Calculate the reduced form of a quotient using ![]() |
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Theorem | divdenle 12199 | Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | qnumgt0 12200 | A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
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