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Theorem List for Intuitionistic Logic Explorer - 10701-10800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulgcddvds 10701 One half of rpmulgcd2 10702, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) ) 
 ||  ( ( K 
 gcd  M )  x.  ( K  gcd  N ) ) )
 
Theoremrpmulgcd2 10702 If  M is relatively prime to  N, then the GCD of  K with  M  x.  N is the product of the GCDs with  M and  N respectively. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M 
 gcd  N )  =  1 )  ->  ( K  gcd  ( M  x.  N ) )  =  (
 ( K  gcd  M )  x.  ( K  gcd  N ) ) )
 
Theoremqredeq 10703 Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M 
 gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P 
 gcd  Q )  =  1 )  /\  ( M 
 /  N )  =  ( P  /  Q ) )  ->  ( M  =  P  /\  N  =  Q ) )
 
Theoremqredeu 10704* Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  ( A  e.  QQ  ->  E! x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) )
 
Theoremrpmul 10705 If  K is relatively prime to  M and to  N, it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( K 
 gcd  M )  =  1 
 /\  ( K  gcd  N )  =  1 ) 
 ->  ( K  gcd  ( M  x.  N ) )  =  1 ) )
 
Theoremrpdvds 10706 If  K is relatively prime to  N then it is also relatively prime to any divisor  M of  N. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  =  1 )
 
4.1.10  Cancellability of congruences
 
Theoremcongr 10707* Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer  A is congruent to an integer  B modulo  M if their difference is a multiple of 
M. See also the definition in [ApostolNT] p. 104: "...  a is congruent to  b modulo  m, and we write  a  ==  b (mod  m) if  m divides the difference  a  -  b", 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.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  ( ( A  mod  M )  =  ( B 
 mod  M )  <->  E. n  e.  ZZ  ( n  x.  M )  =  ( A  -  B ) ) )
 
Theoremdivgcdcoprm0 10708 Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  B  =/=  0 ) 
 ->  ( ( A  /  ( A  gcd  B ) )  gcd  ( B 
 /  ( A  gcd  B ) ) )  =  1 )
 
Theoremdivgcdcoprmex 10709* 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.)
 |-  ( ( A  e.  ZZ  /\  ( B  e.  ZZ  /\  B  =/=  0
 )  /\  M  =  ( A  gcd  B ) )  ->  E. a  e.  ZZ  E. b  e. 
 ZZ  ( A  =  ( M  x.  a
 )  /\  B  =  ( M  x.  b
 )  /\  ( a  gcd  b )  =  1 ) )
 
Theoremcncongr1 10710 One direction of the bicondition in cncongr 10712. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( N  e.  NN  /\  M  =  ( N  /  ( C  gcd  N ) ) ) )  ->  (
 ( ( A  x.  C )  mod  N )  =  ( ( B  x.  C )  mod  N )  ->  ( A  mod  M )  =  ( B  mod  M ) ) )
 
Theoremcncongr2 10711 The other direction of the bicondition in cncongr 10712. (Contributed by AV, 11-Jul-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( N  e.  NN  /\  M  =  ( N  /  ( C  gcd  N ) ) ) )  ->  (
 ( A  mod  M )  =  ( B  mod  M )  ->  (
 ( A  x.  C )  mod  N )  =  ( ( B  x.  C )  mod  N ) ) )
 
Theoremcncongr 10712 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  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( N  e.  NN  /\  M  =  ( N  /  ( C  gcd  N ) ) ) )  ->  (
 ( ( A  x.  C )  mod  N )  =  ( ( B  x.  C )  mod  N )  <->  ( A  mod  M )  =  ( B 
 mod  M ) ) )
 
Theoremcncongrcoprm 10713 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  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( N  e.  NN  /\  ( C  gcd  N )  =  1 ) )  ->  ( ( ( A  x.  C )  mod  N )  =  ( ( B  x.  C ) 
 mod  N )  <->  ( A  mod  N )  =  ( B 
 mod  N ) ) )
 
4.2  Elementary prime number theory
 
4.2.1  Elementary properties

Remark: to represent odd prime numbers, i.e., all prime numbers except  2, the idiom  P  e.  ( Prime  \  { 2 } ) is used. It is a little bit shorter than  ( P  e. 
Prime  /\  P  =/=  2
). Both representations can be converted into each other by eldifsn 3535.

 
Syntaxcprime 10714 Extend the definition of a class to include the set of prime numbers.
 class  Prime
 
Definitiondf-prm 10715* Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)
 |- 
 Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
 
Theoremisprm 10716* 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  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
 
Theoremprmnn 10717 A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( P  e.  Prime  ->  P  e.  NN )
 
Theoremprmz 10718 A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.)
 |-  ( P  e.  Prime  ->  P  e.  ZZ )
 
Theoremprmssnn 10719 The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
 |- 
 Prime  C_  NN
 
Theoremprmex 10720 The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
 |- 
 Prime  e.  _V
 
Theorem1nprm 10721 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
 |- 
 -.  1  e.  Prime
 
Theorem1idssfct 10722* The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN  ->  { 1 ,  N }  C_  { n  e. 
 NN  |  n  ||  N } )
 
Theoremisprm2lem 10723* Lemma for isprm2 10724. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( P  e.  NN  /\  P  =/=  1
 )  ->  ( { n  e.  NN  |  n  ||  P }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  =  {
 1 ,  P }
 ) )
 
Theoremisprm2 10724* 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  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
 
Theoremisprm3 10725* 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  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  ( 2 ... ( P  -  1 ) )  -.  z  ||  P ) )
 
Theoremisprm4 10726* 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  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  ( ZZ>= `  2 )
 ( z  ||  P  ->  z  =  P ) ) )
 
Theoremprmind2 10727* A variation on prmind 10728 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  z  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  ( y  x.  z )  ->  ( ph 
 <->  ta ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  et ) )   &    |-  ps   &    |-  (
 ( x  e.  Prime  /\ 
 A. y  e.  (
 1 ... ( x  -  1 ) ) ch )  ->  ph )   &    |-  (
 ( y  e.  ( ZZ>=
 `  2 )  /\  z  e.  ( ZZ>= `  2 ) )  ->  ( ( ch  /\  th )  ->  ta )
 )   =>    |-  ( A  e.  NN  ->  et )
 
Theoremprmind 10728* Perform induction over the multiplicative structure of  NN. If a property  ph ( 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 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  z  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  ( y  x.  z )  ->  ( ph 
 <->  ta ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  et ) )   &    |-  ps   &    |-  ( x  e.  Prime  ->  ph )   &    |-  (
 ( y  e.  ( ZZ>=
 `  2 )  /\  z  e.  ( ZZ>= `  2 ) )  ->  ( ( ch  /\  th )  ->  ta )
 )   =>    |-  ( A  e.  NN  ->  et )
 
Theoremdvdsprime 10729 If  M divides a prime, then  M is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)
 |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  ||  P 
 <->  ( M  =  P  \/  M  =  1 ) ) )
 
Theoremnprm 10730 A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ( A  e.  ( ZZ>= `  2 )  /\  B  e.  ( ZZ>= `  2 ) )  ->  -.  ( A  x.  B )  e.  Prime )
 
Theoremnprmi 10731 An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |-  A  e.  NN   &    |-  B  e.  NN   &    |-  1  <  A   &    |-  1  <  B   &    |-  ( A  x.  B )  =  N   =>    |-  -.  N  e.  Prime
 
Theoremdvdsnprmd 10732 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.)
 |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  A  <  N )   &    |-  ( ph  ->  A  ||  N )   =>    |-  ( ph  ->  -.  N  e.  Prime )
 
Theoremprm2orodd 10733 A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
 |-  ( P  e.  Prime  ->  ( P  =  2  \/  -.  2  ||  P ) )
 
Theorem2prm 10734 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
 |-  2  e.  Prime
 
Theorem3prm 10735 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  3  e.  Prime
 
Theorem4nprm 10736 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  4  e.  Prime
 
Theoremprmuz2 10737 A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 ) )
 
Theoremprmgt1 10738 A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( P  e.  Prime  -> 
 1  <  P )
 
Theoremprmm2nn0 10739 Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
 |-  ( P  e.  Prime  ->  ( P  -  2
 )  e.  NN0 )
 
Theoremoddprmgt2 10740 An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
 |-  ( P  e.  ( Prime  \  { 2 } )  ->  2  <  P )
 
Theoremoddprmge3 10741 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  e.  ( Prime  \  { 2 } )  ->  P  e.  ( ZZ>= `  3 )
 )
 
Theoremsqnprm 10742 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( A  e.  ZZ  ->  -.  ( A ^
 2 )  e.  Prime )
 
Theoremdvdsprm 10743 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  e.  ( ZZ>= `  2 )  /\  P  e.  Prime )  ->  ( N  ||  P  <->  N  =  P ) )
 
Theoremexprmfct 10744* 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  e.  ( ZZ>=
 `  2 )  ->  E. p  e.  Prime  p 
 ||  N )
 
Theoremprmdvdsfz 10745* 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  e.  NN  /\  I  e.  (
 2 ... N ) ) 
 ->  E. p  e.  Prime  ( p  <_  N  /\  p  ||  I ) )
 
Theoremnprmdvds1 10746 No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
 |-  ( P  e.  Prime  ->  -.  P  ||  1 )
 
Theoremdivgcdodd 10747 Either  A  /  ( A  gcd  B ) is odd or  B  /  ( A  gcd  B ) is odd. (Contributed by Scott Fenton, 19-Apr-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( -.  2  ||  ( A  /  ( A  gcd  B ) )  \/  -.  2  ||  ( B  /  ( A  gcd  B ) ) ) )
 
4.2.2  Coprimality and Euclid's lemma (cont.)

This section is about coprimality with respect to primes, and a special version of Euclid's lemma for primes is provided, see euclemma 10750.

 
Theoremcoprm 10748 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  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
 
Theoremprmrp 10749 Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  Q  e.  Prime ) 
 ->  ( ( P  gcd  Q )  =  1  <->  P  =/=  Q ) )
 
Theoremeuclemma 10750 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  e.  Prime  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( P  ||  ( M  x.  N )  <->  ( P  ||  M  \/  P  ||  N ) ) )
 
Theoremisprm6 10751* A number is prime iff it satisfies Euclid's lemma euclemma 10750. (Contributed by Mario Carneiro, 6-Sep-2015.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. x  e.  ZZ  A. y  e. 
 ZZ  ( P  ||  ( x  x.  y
 )  ->  ( P  ||  x  \/  P  ||  y ) ) ) )
 
Theoremprmdvdsexp 10752 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  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <->  P  ||  A ) )
 
Theoremprmdvdsexpb 10753 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  e.  Prime  /\  Q  e.  Prime  /\  N  e.  NN )  ->  ( P  ||  ( Q ^ N )  <->  P  =  Q ) )
 
Theoremprmdvdsexpr 10754 If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( ( P  e.  Prime  /\  Q  e.  Prime  /\  N  e.  NN0 )  ->  ( P  ||  ( Q ^ N )  ->  P  =  Q )
 )
 
Theoremprmexpb 10755 Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  ( ( P ^ M )  =  ( Q ^ N )  <->  ( P  =  Q  /\  M  =  N ) ) )
 
Theoremprmfac1 10756 The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  P  e.  Prime  /\  P  ||  ( ! `  N ) )  ->  P  <_  N )
 
Theoremrpexp 10757 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  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( ( ( A ^ N )  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 ) )
 
Theoremrpexp1i 10758 Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ M )  gcd  B )  =  1 ) )
 
Theoremrpexp12i 10759 Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 )
 )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ M )  gcd  ( B ^ N ) )  =  1 ) )
 
Theoremprmndvdsfaclt 10760 A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.)
 |-  ( ( P  e.  Prime  /\  N  e.  NN0 )  ->  ( N  <  P 
 ->  -.  P  ||  ( ! `  N ) ) )
 
Theoremcncongrprm 10761 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  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( P  e.  Prime  /\  -.  P  ||  C ) )  ->  ( ( ( A  x.  C )  mod  P )  =  ( ( B  x.  C ) 
 mod  P )  <->  ( A  mod  P )  =  ( B 
 mod  P ) ) )
 
Theoremisevengcd2 10762 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  e.  ZZ  ->  ( 2  ||  Z  <->  ( 2  gcd  Z )  =  2 ) )
 
Theoremisoddgcd1 10763 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  e.  ZZ  ->  ( -.  2  ||  Z 
 <->  ( 2  gcd  Z )  =  1 )
 )
 
Theorem3lcm2e6 10764 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
 
4.2.3  Non-rationality of square root of 2
 
Theoremsqrt2irrlem 10765 Lemma for sqrt2irr 10766. This is the core of the proof: - if  A  /  B  =  sqr ( 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.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  ( sqr `  2
 )  =  ( A 
 /  B ) )   =>    |-  ( ph  ->  ( ( A  /  2 )  e. 
 ZZ  /\  ( B  /  2 )  e.  NN ) )
 
Theoremsqrt2irr 10766 The square root of 2 is not rational. That is, for any rational number,  ( sqr `  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, 
( sqr `  2
) is apart from it" (the two statements are equivalent given excluded middle). See sqrt2irrap 10783 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 10765, which shows that if  A  /  B  =  sqr ( 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.)

 |-  ( sqr `  2
 )  e/  QQ
 
Theoremsqrt2re 10767 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
 |-  ( sqr `  2
 )  e.  RR
 
Theorempw2dvdslemn 10768* Lemma for pw2dvds 10769. 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  e.  NN  /\  A  e.  NN  /\ 
 -.  ( 2 ^ A )  ||  N ) 
 ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 )
 )  ||  N )
 )
 
Theorempw2dvds 10769* A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.)
 |-  ( N  e.  NN  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 )
 )  ||  N )
 )
 
Theorempw2dvdseulemle 10770 Lemma for pw2dvdseu 10771. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   &    |-  ( ph  ->  ( 2 ^ A ) 
 ||  N )   &    |-  ( ph  ->  -.  ( 2 ^ ( B  +  1 ) )  ||  N )   =>    |-  ( ph  ->  A  <_  B )
 
Theorempw2dvdseu 10771* A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.)
 |-  ( N  e.  NN  ->  E! m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 )
 )  ||  N )
 )
 
Theoremoddpwdclemxy 10772* Lemma for oddpwdc 10777. 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  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) ) 
 ->  ( X  =  ( A  /  ( 2 ^ ( iota_ z  e. 
 NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) ) ) )  /\  Y  =  ( iota_ z  e.  NN0  ( ( 2 ^
 z )  ||  A  /\  -.  ( 2 ^
 ( z  +  1 ) )  ||  A ) ) ) )
 
Theoremoddpwdclemdvds 10773* Lemma for oddpwdc 10777. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.)
 |-  ( A  e.  NN  ->  ( 2 ^ ( iota_
 z  e.  NN0  (
 ( 2 ^ z
 )  ||  A  /\  -.  ( 2 ^ (
 z  +  1 ) )  ||  A )
 ) )  ||  A )
 
Theoremoddpwdclemndvds 10774* Lemma for oddpwdc 10777. 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  e.  NN  ->  -.  ( 2 ^
 ( ( iota_ z  e. 
 NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) )  +  1 ) )  ||  A )
 
Theoremoddpwdclemodd 10775* Lemma for oddpwdc 10777. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.)
 |-  ( A  e.  NN  ->  -.  2  ||  ( A  /  ( 2 ^
 ( iota_ z  e.  NN0  ( ( 2 ^
 z )  ||  A  /\  -.  ( 2 ^
 ( z  +  1 ) )  ||  A ) ) ) ) )
 
Theoremoddpwdclemdc 10776* Lemma for oddpwdc 10777. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.)
 |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  <-> 
 ( A  e.  NN  /\  ( X  =  ( A  /  ( 2 ^ ( iota_ z  e. 
 NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) ) ) )  /\  Y  =  ( iota_ z  e.  NN0  ( ( 2 ^
 z )  ||  A  /\  -.  ( 2 ^
 ( z  +  1 ) )  ||  A ) ) ) ) )
 
Theoremoddpwdc 10777* 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  e.  NN  |  -.  2  ||  z }   &    |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x ) )   =>    |-  F : ( J  X.  NN0 ) -1-1-onto-> NN
 
Theoremsqpweven 10778* 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  e.  NN  |  -.  2  ||  z }   &    |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x ) )   =>    |-  ( A  e.  NN  ->  2  ||  ( 2nd `  ( `' F `  ( A ^ 2 ) ) ) )
 
Theorem2sqpwodd 10779* 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  e.  NN  |  -.  2  ||  z }   &    |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x ) )   =>    |-  ( A  e.  NN  ->  -.  2  ||  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) ) )
 
Theoremsqne2sq 10780 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  e.  NN  /\  B  e.  NN )  ->  ( A ^
 2 )  =/=  (
 2  x.  ( B ^ 2 ) ) )
 
Theoremznege1 10781 The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B ) 
 ->  1  <_  ( abs `  ( A  -  B ) ) )
 
Theoremsqrt2irraplemnn 10782 Lemma for sqrt2irrap 10783. 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  e.  NN  /\  B  e.  NN )  ->  ( sqr `  2
 ) #  ( A  /  B ) )
 
Theoremsqrt2irrap 10783 The square root of 2 is irrational. That is, for any rational number,  ( sqr `  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 10766. (Contributed by Jim Kingdon, 2-Oct-2021.)
 |-  ( Q  e.  QQ  ->  ( sqr `  2
 ) #  Q )
 
4.2.4  Properties of the canonical representation of a rational
 
Syntaxcnumer 10784 Extend class notation to include canonical numerator function.
 class numer
 
Syntaxcdenom 10785 Extend class notation to include canonical denominator function.
 class denom
 
Definitiondf-numer 10786* 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  e.  QQ  |->  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  (
 ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Definitiondf-denom 10787* 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  e.  QQ  |->  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  (
 ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqnumval 10788* Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  A )  =  ( 1st `  ( iota_ x  e.  ( ZZ  X. 
 NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqdenval 10789* Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  =  ( 2nd `  ( iota_ x  e.  ( ZZ  X. 
 NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqnumdencl 10790 Lemma for qnumcl 10791 and qdencl 10792. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )
 
Theoremqnumcl 10791 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  A )  e.  ZZ )
 
Theoremqdencl 10792 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
 
Theoremfnum 10793 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer : QQ --> ZZ
 
Theoremfden 10794 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom : QQ --> NN
 
Theoremqnumdenbi 10795 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  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( ( ( B 
 gcd  C )  =  1 
 /\  A  =  ( B  /  C ) )  <->  ( (numer `  A )  =  B  /\  (denom `  A )  =  C ) ) )
 
Theoremqnumdencoprm 10796 The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  gcd  (denom `  A ) )  =  1
 )
 
Theoremqeqnumdivden 10797 Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  A  =  ( (numer `  A )  /  (denom `  A ) ) )
 
Theoremqmuldeneqnum 10798 Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( A  x.  (denom `  A ) )  =  (numer `  A )
 )
 
Theoremdivnumden 10799 Calculate the reduced form of a quotient using  gcd. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( (numer `  ( A  /  B ) )  =  ( A 
 /  ( A  gcd  B ) )  /\  (denom `  ( A  /  B ) )  =  ( B  /  ( A  gcd  B ) ) ) )
 
Theoremdivdenle 10800 Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  (denom `  ( A  /  B ) )  <_  B )
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