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Type | Label | Description |
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Statement | ||
Theorem | divgcdodd 12101 | 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 12104. | ||
Theorem | coprm 12102 | 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 12103 | Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | euclemma 12104 | 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 12105* | A number is prime iff it satisfies Euclid's lemma euclemma 12104. (Contributed by Mario Carneiro, 6-Sep-2015.) |
Theorem | prmdvdsexp 12106 | 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 12107 | 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 12108 | If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Theorem | prmexpb 12109 | Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.) |
Theorem | prmfac1 12110 | The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.) |
Theorem | rpexp 12111 | 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 12112 | Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | rpexp12i 12113 | Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | prmndvdsfaclt 12114 | 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 12115 | 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 12116 | 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 12117 | 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 12118 | 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 12119 | Lemma for sqrt2irr 12120. 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 12120 |
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 12138 for
the proof that the square root of two is irrational.
The proof's core is proven in sqrt2irrlem 12119, 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 12121 | The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.) |
Theorem | sqrt2irr0 12122 | The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.) |
Theorem | pw2dvdslemn 12123* | Lemma for pw2dvds 12124. 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 12124* | A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.) |
Theorem | pw2dvdseulemle 12125 | Lemma for pw2dvdseu 12126. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | pw2dvdseu 12126* | A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemxy 12127* | Lemma for oddpwdc 12132. 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 12128* | Lemma for oddpwdc 12132. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | oddpwdclemndvds 12129* | Lemma for oddpwdc 12132. 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 12130* | Lemma for oddpwdc 12132. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemdc 12131* | Lemma for oddpwdc 12132. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdc 12132* | 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 12133* | 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 12134* | 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 12135 | 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 12136 | The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.) |
Theorem | sqrt2irraplemnn 12137 | Lemma for sqrt2irrap 12138. 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 12138 | 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 12120. (Contributed by Jim Kingdon, 2-Oct-2021.) |
# | ||
Syntax | cnumer 12139 | Extend class notation to include canonical numerator function. |
numer | ||
Syntax | cdenom 12140 | Extend class notation to include canonical denominator function. |
denom | ||
Definition | df-numer 12141* | 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 12142* | 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 12143* | Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdenval 12144* | Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdencl 12145 | Lemma for qnumcl 12146 and qdencl 12147. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qnumcl 12146 | The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdencl 12147 | The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | fnum 12148 | Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | fden 12149 | Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdenbi 12150 | 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.) |
numer denom | ||
Theorem | qnumdencoprm 12151 | The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qeqnumdivden 12152 | Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qmuldeneqnum 12153 | Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom numer | ||
Theorem | divnumden 12154 | Calculate the reduced form of a quotient using . (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | divdenle 12155 | Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumgt0 12156 | A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | qgt0numnn 12157 | A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | nn0gcdsq 12158 | Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | zgcdsq 12159 | nn0gcdsq 12158 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | numdensq 12160 | Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer denom denom | ||
Theorem | numsq 12161 | Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer | ||
Theorem | densq 12162 | Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom denom | ||
Theorem | qden1elz 12163 | A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom | ||
Theorem | nn0sqrtelqelz 12164 | If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.) |
Theorem | nonsq 12165 | Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Syntax | codz 12166 | Extend class notation with the order function on the class of integers modulo N. |
Syntax | cphi 12167 | Extend class notation with the Euler phi function. |
Definition | df-odz 12168* | Define the order function on the class of integers modulo N. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.) |
inf | ||
Definition | df-phi 12169* | Define the Euler phi function (also called "Euler totient function"), which counts the number of integers less than and coprime to it, see definition in [ApostolNT] p. 25. (Contributed by Mario Carneiro, 23-Feb-2014.) |
♯ | ||
Theorem | phivalfi 12170* | Finiteness of an expression used to define the Euler function. (Contributed by Jim Kingon, 28-May-2022.) |
Theorem | phival 12171* | Value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
♯ | ||
Theorem | phicl2 12172 | Bounds and closure for the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicl 12173 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | phibndlem 12174* | Lemma for phibnd 12175. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phibnd 12175 | A slightly tighter bound on the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicld 12176 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 29-May-2016.) |
Theorem | phi1 12177 | Value of the Euler function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | dfphi2 12178* | Alternate definition of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.) |
♯ ..^ | ||
Theorem | hashdvds 12179* | 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.) |
♯ | ||
Theorem | phiprmpw 12180 | Value of the Euler function at a prime power. Theorem 2.5(a) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.) |
Theorem | phiprm 12181 | Value of the Euler function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | crth 12182* | The Chinese Remainder Theorem: the function that maps to its remainder classes and is 1-1 and onto when and are coprime. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-May-2016.) |
..^ ..^ ..^ | ||
Theorem | phimullem 12183* | Lemma for phimul 12184. (Contributed by Mario Carneiro, 24-Feb-2014.) |
..^ ..^ ..^ ..^ ..^ | ||
Theorem | phimul 12184 | The Euler 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.) |
Theorem | eulerthlem1 12185* | Lemma for eulerth 12191. (Contributed by Mario Carneiro, 8-May-2015.) |
..^ | ||
Theorem | eulerthlemfi 12186* | Lemma for eulerth 12191. The set is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.) |
..^ | ||
Theorem | eulerthlemrprm 12187* | Lemma for eulerth 12191. and are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.) |
..^ | ||
Theorem | eulerthlema 12188* | Lemma for eulerth 12191. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.) |
..^ | ||
Theorem | eulerthlemh 12189* | Lemma for eulerth 12191. A permutation of . (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.) |
..^ | ||
Theorem | eulerthlemth 12190* | Lemma for eulerth 12191. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.) |
..^ | ||
Theorem | eulerth 12191 | Euler's theorem, a generalization of Fermat's little theorem. If and are coprime, then (mod ). This is Metamath 100 proof #10. Also called Euler-Fermat theorem, see theorem 5.17 in [ApostolNT] p. 113. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | fermltl 12192 | Fermat's little theorem. When is prime, (mod ) for any , see theorem 5.19 in [ApostolNT] p. 114. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 19-Mar-2022.) |
Theorem | prmdiv 12193 | Show an explicit expression for the modular inverse of . (Contributed by Mario Carneiro, 24-Jan-2015.) |
Theorem | prmdiveq 12194 | The modular inverse of is unique. (Contributed by Mario Carneiro, 24-Jan-2015.) |
Theorem | prmdivdiv 12195 | The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.) |
Theorem | hashgcdlem 12196* | A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
..^ ..^ | ||
Theorem | hashgcdeq 12197* | Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
♯ ..^ | ||
Theorem | phisum 12198* | The divisor sum identity of the totient function. Theorem 2.2 in [ApostolNT] p. 26. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Theorem | odzval 12199* | Value of the order function. This is a function of functions; the inner argument selects the base (i.e., mod for some , often prime) and the outer argument selects the integer or equivalence class (if you want to think about it that way) from the integers mod . In order to ensure the supremum is well-defined, we only define the expression when and are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.) |
inf | ||
Theorem | odzcllem 12200 | - Lemma for odzcl 12201, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.) |
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