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Theorem List for Intuitionistic Logic Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1nprm 11801 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 11802* 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 11803* Lemma for isprm2 11804. (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 11804* 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 11805* 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 11806* 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 11807* A variation on prmind 11808 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 11808* 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 11809 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 11810 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 11811 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 11812 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 11813 A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
 |-  ( P  e.  Prime  ->  ( P  =  2  \/  -.  2  ||  P ) )
 
Theorem2prm 11814 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
 |-  2  e.  Prime
 
Theorem3prm 11815 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  3  e.  Prime
 
Theorem4nprm 11816 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 11817 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 11818 A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( P  e.  Prime  -> 
 1  <  P )
 
Theoremprmm2nn0 11819 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 11820 An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
 |-  ( P  e.  ( Prime  \  { 2 } )  ->  2  <  P )
 
Theoremoddprmge3 11821 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 11822 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( A  e.  ZZ  ->  -.  ( A ^
 2 )  e.  Prime )
 
Theoremdvdsprm 11823 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 11824* 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 11825* 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 11826 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 11827 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 ) ) ) )
 
5.2.2  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 11830.

 
Theoremcoprm 11828 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 11829 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 11830 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 11831* A number is prime iff it satisfies Euclid's lemma euclemma 11830. (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 11832 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 11833 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 11834 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 11835 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 11836 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 11837 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 11838 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 11839 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 11840 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 11841 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 11842 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 11843 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 11844 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
 
5.2.3  Non-rationality of square root of 2
 
Theoremsqrt2irrlem 11845 Lemma for sqrt2irr 11846. 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 11846 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 11864 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 11845, 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 11847 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
 |-  ( sqr `  2
 )  e.  RR
 
Theoremsqrt2irr0 11848 The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
 |-  ( sqr `  2
 )  e.  ( RR  \  QQ )
 
Theorempw2dvdslemn 11849* Lemma for pw2dvds 11850. 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 11850* 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 11851 Lemma for pw2dvdseu 11852. 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 11852* 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 11853* Lemma for oddpwdc 11858. 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 11854* Lemma for oddpwdc 11858. 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 11855* Lemma for oddpwdc 11858. 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 11856* Lemma for oddpwdc 11858. 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 11857* Lemma for oddpwdc 11858. 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 11858* 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 11859* 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 11860* 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 11861 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 11862 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 11863 Lemma for sqrt2irrap 11864. 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 11864 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 11846. (Contributed by Jim Kingdon, 2-Oct-2021.)
 |-  ( Q  e.  QQ  ->  ( sqr `  2
 ) #  Q )
 
5.2.4  Properties of the canonical representation of a rational
 
Syntaxcnumer 11865 Extend class notation to include canonical numerator function.
 class numer
 
Syntaxcdenom 11866 Extend class notation to include canonical denominator function.
 class denom
 
Definitiondf-numer 11867* 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 11868* 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 11869* 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 11870* 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 11871 Lemma for qnumcl 11872 and qdencl 11873. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )
 
Theoremqnumcl 11872 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 11873 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
 
Theoremfnum 11874 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer : QQ --> ZZ
 
Theoremfden 11875 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom : QQ --> NN
 
Theoremqnumdenbi 11876 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 11877 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 11878 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 11879 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 11880 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 11881 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 )
 
Theoremqnumgt0 11882 A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( 0  <  A  <->  0  <  (numer `  A ) ) )
 
Theoremqgt0numnn 11883 A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  QQ  /\  0  <  A )  ->  (numer `  A )  e.  NN )
 
Theoremnn0gcdsq 11884 Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
 
Theoremzgcdsq 11885 nn0gcdsq 11884 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A 
 gcd  B ) ^ 2
 )  =  ( ( A ^ 2 ) 
 gcd  ( B ^
 2 ) ) )
 
Theoremnumdensq 11886 Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 )  /\  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) ) )
 
Theoremnumsq 11887 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 ) )
 
Theoremdensq 11888 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) )
 
Theoremqden1elz 11889 A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (denom `  A )  =  1  <->  A  e.  ZZ ) )
 
Theoremnn0sqrtelqelz 11890 If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.)
 |-  ( ( A  e.  NN0  /\  ( sqr `  A )  e.  QQ )  ->  ( sqr `  A )  e.  ZZ )
 
Theoremnonsq 11891 Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^ 2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2
 ) ) )  ->  -.  ( sqr `  A )  e.  QQ )
 
5.2.5  Euler's theorem
 
Syntaxcphi 11892 Extend class notation with the Euler phi function.
 class  phi
 
Definitiondf-phi 11893* 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.)
 |- 
 phi  =  ( n  e.  NN  |->  ( `  { x  e.  ( 1 ... n )  |  ( x  gcd  n )  =  1 } ) )
 
Theoremphivalfi 11894* Finiteness of an expression used to define the Euler  phi function. (Contributed by Jim Kingon, 28-May-2022.)
 |-  ( N  e.  NN  ->  { x  e.  (
 1 ... N )  |  ( x  gcd  N )  =  1 }  e.  Fin )
 
Theoremphival 11895* Value of the Euler  phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  =  ( `  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } ) )
 
Theoremphicl2 11896 Bounds and closure for the value of the Euler  phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  e.  ( 1 ... N ) )
 
Theoremphicl 11897 Closure for the value of the Euler 
phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
 
Theoremphibndlem 11898* Lemma for phibnd 11899. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  { x  e.  (
 1 ... N )  |  ( x  gcd  N )  =  1 }  C_  ( 1 ... ( N  -  1 ) ) )
 
Theoremphibnd 11899 A slightly tighter bound on the value of the Euler  phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( phi `  N )  <_  ( N  -  1
 ) )
 
Theoremphicld 11900 Closure for the value of the Euler 
phi function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( phi `  N )  e. 
 NN )
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