P. 132 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13101-13200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pccld 13101	Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> P e. Prime )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( P pCnt N ) e. NN0 )
Theorem	pcmul 13102	Multiplication property of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt ( A x. B ) ) = ( ( P pCnt A ) + ( P pCnt B ) ) )
Theorem	pcdiv 13103	Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)
|- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt ( A / B ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) )
Theorem	pcqmul 13104	Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A x. B ) ) = ( ( P pCnt A ) + ( P pCnt B ) ) )
Theorem	pc0 13105	The value of the prime power function at zero. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( P e. Prime -> ( P pCnt 0 ) = +oo )
Theorem	pc1 13106	Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( P e. Prime -> ( P pCnt 1 ) = 0 )
Theorem	pcqcl 13107	Closure of the general prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( P pCnt N ) e. ZZ )
Theorem	pcqdiv 13108	Division property of the prime power function. (Contributed by Mario Carneiro, 10-Aug-2015.)
|- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) )
Theorem	pcrec 13109	Prime power of a reciprocal. (Contributed by Mario Carneiro, 10-Aug-2015.)
|- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( 1 / A ) ) = -u ( P pCnt A ) )
Theorem	pcexp 13110	Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)
|- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ N e. ZZ ) -> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) )
Theorem	pcxnn0cl 13111	Extended nonnegative integer closure of the general prime count function. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( P e. Prime /\ N e. ZZ ) -> ( P pCnt N ) e. NN0* )
Theorem	pcxcl 13112	Extended real closure of the general prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( P e. Prime /\ N e. QQ ) -> ( P pCnt N ) e. RR* )
Theorem	pcxqcl 13113	The general prime count function is an integer or infinite. (Contributed by Jim Kingdon, 6-Jun-2025.)
|- ( ( P e. Prime /\ N e. QQ ) -> ( ( P pCnt N ) e. ZZ \/ ( P pCnt N ) = +oo ) )
Theorem	pcge0 13114	The prime count of an integer is greater than or equal to zero. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( P e. Prime /\ N e. ZZ ) -> 0 <_ ( P pCnt N ) )
Theorem	pczdvds 13115	Defining property of the prime count function. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( P pCnt N ) ) || N )
Theorem	pcdvds 13116	Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> ( P ^ ( P pCnt N ) ) || N )
Theorem	pczndvds 13117	Defining property of the prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
Theorem	pcndvds 13118	Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
Theorem	pczndvds2 13119	The remainder after dividing out all factors of P is not divisible by P. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ ( P pCnt N ) ) ) )
Theorem	pcndvds2 13120	The remainder after dividing out all factors of P is not divisible by P. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> -. P || ( N / ( P ^ ( P pCnt N ) ) ) )
Theorem	pcdvdsb 13121	P ^ A divides N if and only if A is at most the count of P. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )
Theorem	pcelnn 13122	There are a positive number of powers of a prime P in N iff P divides N. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> ( ( P pCnt N ) e. NN <-> P || N ) )
Theorem	pceq0 13123	There are zero powers of a prime P in N iff P does not divide N. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> ( ( P pCnt N ) = 0 <-> -. P || N ) )
Theorem	pcidlem 13124	The prime count of a prime power. (Contributed by Mario Carneiro, 12-Mar-2014.)
|- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) = A )
Theorem	pcid 13125	The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt ( P ^ A ) ) = A )
Theorem	pcneg 13126	The prime count of a negative number. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt -u A ) = ( P pCnt A ) )
Theorem	pcabs 13127	The prime count of an absolute value. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
Theorem	pcdvdstr 13128	The prime count increases under the divisibility relation. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> ( P pCnt A ) <_ ( P pCnt B ) )
Theorem	pcgcd1 13129	The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt A ) )
Theorem	pcgcd 13130	The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( A gcd B ) ) = if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) )
Theorem	pc2dvds 13131*	A characterization of divisibility in terms of prime count. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A || B <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt B ) ) )
Theorem	pc11 13132*	The prime count function, viewed as a function from NN to ( NN ^m Prime ), is one-to-one. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) )
Theorem	pcz 13133*	The prime count function can be used as an indicator that a given rational number is an integer. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( A e. QQ -> ( A e. ZZ <-> A. p e. Prime 0 <_ ( p pCnt A ) ) )
Theorem	pcprmpw2 13134*	Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)
|- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )
Theorem	pcprmpw 13135*	Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)
|- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A = ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )
Theorem	dvdsprmpweq 13136*	If a positive integer divides a prime power, it is a prime power. (Contributed by AV, 25-Jul-2021.)
|- ( ( P e. Prime /\ A e. NN /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN0 A = ( P ^ n ) ) )
Theorem	dvdsprmpweqnn 13137*	If an integer greater than 1 divides a prime power, it is a (proper) prime power. (Contributed by AV, 13-Aug-2021.)
|- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) )
Theorem	dvdsprmpweqle 13138*	If a positive integer divides a prime power, it is a prime power with a smaller exponent. (Contributed by AV, 25-Jul-2021.)
|- ( ( P e. Prime /\ A e. NN /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN0 ( n <_ N /\ A = ( P ^ n ) ) ) )
Theorem	difsqpwdvds 13139	If the difference of two squares is a power of a prime, the prime divides twice the second squared number. (Contributed by AV, 13-Aug-2021.)
|- ( ( ( A e. NN0 /\ B e. NN0 /\ ( B + 1 ) < A ) /\ ( C e. Prime /\ D e. NN0 ) ) -> ( ( C ^ D ) = ( ( A ^ 2 ) - ( B ^ 2 ) ) -> C || ( 2 x. B ) ) )
Theorem	pcaddlem 13140	Lemma for pcadd 13141. The original numbers A and B have been decomposed using the prime count function as ( P ^ M ) x. ( R / S ) where R , S are both not divisible by P and M = ( P pCnt A ), and similarly for B. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ph -> P e. Prime )   &    |- ( ph -> A = ( ( P ^ M ) x. ( R / S ) ) )   &    |- ( ph -> B = ( ( P ^ N ) x. ( T / U ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> ( R e. ZZ /\ -. P || R ) )   &    |- ( ph -> ( S e. NN /\ -. P || S ) )   &    |- ( ph -> ( T e. ZZ /\ -. P || T ) )   &    |- ( ph -> ( U e. NN /\ -. P || U ) )   =>    |- ( ph -> M <_ ( P pCnt ( A + B ) ) )
Theorem	pcadd 13141	An inequality for the prime count of a sum. This is the source of the ultrametric inequality for the p-adic metric. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ph -> P e. Prime )   &    |- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> ( P pCnt A ) <_ ( P pCnt B ) )   =>    |- ( ph -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) )
Theorem	pcadd2 13142	The inequality of pcadd 13141 becomes an equality when one of the factors has prime count strictly less than the other. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
|- ( ph -> P e. Prime )   &    |- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> ( P pCnt A ) < ( P pCnt B ) )   =>    |- ( ph -> ( P pCnt A ) = ( P pCnt ( A + B ) ) )
Theorem	pcmptcl 13143	Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.)
|- F = ( n e. NN |-> if ( n e. Prime , ( n ^ A ) , 1 ) )   &    |- ( ph -> A. n e. Prime A e. NN0 )   =>    |- ( ph -> ( F : NN --> NN /\ seq 1 ( x. , F ) : NN --> NN ) )
Theorem	pcmpt 13144*	Construct a function with given prime count characteristics. (Contributed by Mario Carneiro, 12-Mar-2014.)
|- F = ( n e. NN |-> if ( n e. Prime , ( n ^ A ) , 1 ) )   &    |- ( ph -> A. n e. Prime A e. NN0 )   &    |- ( ph -> N e. NN )   &    |- ( ph -> P e. Prime )   &    |- ( n = P -> A = B )   =>    |- ( ph -> ( P pCnt ( seq 1 ( x. , F ) ` N ) ) = if ( P <_ N , B , 0 ) )
Theorem	pcmpt2 13145*	Dividing two prime count maps yields a number with all dividing primes confined to an interval. (Contributed by Mario Carneiro, 14-Mar-2014.)
|- F = ( n e. NN |-> if ( n e. Prime , ( n ^ A ) , 1 ) )   &    |- ( ph -> A. n e. Prime A e. NN0 )   &    |- ( ph -> N e. NN )   &    |- ( ph -> P e. Prime )   &    |- ( n = P -> A = B )   &    |- ( ph -> M e. ( ZZ>= ` N ) )   =>    |- ( ph -> ( P pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) = if ( ( P <_ M /\ -. P <_ N ) , B , 0 ) )
Theorem	pcmptdvds 13146	The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)
|- F = ( n e. NN |-> if ( n e. Prime , ( n ^ A ) , 1 ) )   &    |- ( ph -> A. n e. Prime A e. NN0 )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. ( ZZ>= ` N ) )   =>    |- ( ph -> ( seq 1 ( x. , F ) ` N ) || ( seq 1 ( x. , F ) ` M ) )
Theorem	pcprod 13147*	The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)
|- F = ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( N e. NN -> ( seq 1 ( x. , F ) ` N ) = N )
Theorem	sumhashdc 13148*	The sum of 1 over a set is the size of the set. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 20-May-2014.)
|- ( ( B e. Fin /\ A C_ B /\ A. x e. B DECID x e. A ) -> sum_ k e. B if ( k e. A , 1 , 0 ) = (♯ ` A ) )
Theorem	fldivp1 13149	The difference between the floors of adjacent fractions is either 1 or 0. (Contributed by Mario Carneiro, 8-Mar-2014.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( ( |_ ` ( ( M + 1 ) / N ) ) - ( |_ ` ( M / N ) ) ) = if ( N || ( M + 1 ) , 1 , 0 ) )
Theorem	pcfaclem 13150	Lemma for pcfac 13151. (Contributed by Mario Carneiro, 20-May-2014.)
|- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( |_ ` ( N / ( P ^ M ) ) ) = 0 )
Theorem	pcfac 13151*	Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
|- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P pCnt ( ! ` N ) ) = sum_ k e. ( 1 ... M ) ( |_ ` ( N / ( P ^ k ) ) ) )
Theorem	pcbc 13152*	Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
|- ( ( N e. NN /\ K e. ( 0 ... N ) /\ P e. Prime ) -> ( P pCnt ( N _C K ) ) = sum_ k e. ( 1 ... N ) ( ( |_ ` ( N / ( P ^ k ) ) ) - ( ( |_ ` ( ( N - K ) / ( P ^ k ) ) ) + ( |_ ` ( K / ( P ^ k ) ) ) ) ) )
Theorem	qexpz 13153	If a power of a rational number is an integer, then the number is an integer. (Contributed by Mario Carneiro, 10-Aug-2015.)
|- ( ( A e. QQ /\ N e. NN /\ ( A ^ N ) e. ZZ ) -> A e. ZZ )
Theorem	expnprm 13154	A second or higher power of a rational number is not a prime number. Or by contraposition, the n-th root of a prime number is not rational. Suggested by Norm Megill. (Contributed by Mario Carneiro, 10-Aug-2015.)
|- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> -. ( A ^ N ) e. Prime )
Theorem	oddprmdvds 13155*	Every positive integer which is not a power of two is divisible by an odd prime number. (Contributed by AV, 6-Aug-2021.)
|- ( ( K e. NN /\ -. E. n e. NN0 K = ( 2 ^ n ) ) -> E. p e. ( Prime \ { 2 } ) p || K )
5.2.9  Pocklington's theorem
Theorem	prmpwdvds 13156	A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)
|- ( ( ( K e. ZZ /\ D e. ZZ ) /\ ( P e. Prime /\ N e. NN ) /\ ( D || ( K x. ( P ^ N ) ) /\ -. D || ( K x. ( P ^ ( N - 1 ) ) ) ) ) -> ( P ^ N ) || D )
Theorem	pockthlem 13157	Lemma for pockthg 13158. (Contributed by Mario Carneiro, 2-Mar-2014.)
|- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   &    |- ( ph -> B < A )   &    |- ( ph -> N = ( ( A x. B ) + 1 ) )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> P || N )   &    |- ( ph -> Q e. Prime )   &    |- ( ph -> ( Q pCnt A ) e. NN )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> ( ( C ^ ( N - 1 ) ) mod N ) = 1 )   &    |- ( ph -> ( ( ( C ^ ( ( N - 1 ) / Q ) ) - 1 ) gcd N ) = 1 )   =>    |- ( ph -> ( Q pCnt A ) <_ ( Q pCnt ( P - 1 ) ) )
Theorem	pockthg 13158*	The generalized Pocklington's theorem. If N - 1 = A x. B where B < A, then N is prime if and only if for every prime factor p of A, there is an x such that x ^ ( N - 1 ) = 1 ( mod N ) and gcd ( x ^ ( ( N - 1 ) / p ) - 1 , N ) = 1. (Contributed by Mario Carneiro, 2-Mar-2014.)
|- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   &    |- ( ph -> B < A )   &    |- ( ph -> N = ( ( A x. B ) + 1 ) )   &    |- ( ph -> A. p e. Prime ( p || A -> E. x e. ZZ ( ( ( x ^ ( N - 1 ) ) mod N ) = 1 /\ ( ( ( x ^ ( ( N - 1 ) / p ) ) - 1 ) gcd N ) = 1 ) ) )   =>    |- ( ph -> N e. Prime )
Theorem	pockthi 13159	Pocklington's theorem, which gives a sufficient criterion for a number N to be prime. This is the preferred method for verifying large primes, being much more efficient to compute than trial division. This form has been optimized for application to specific large primes; see pockthg 13158 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)
|- P e. Prime   &    |- G e. NN   &    |- M = ( G x. P )   &    |- N = ( M + 1 )   &    |- D e. NN   &    |- E e. NN   &    |- A e. NN   &    |- M = ( D x. ( P ^ E ) )   &    |- D < ( P ^ E )   &    |- ( ( A ^ M ) mod N ) = ( 1 mod N )   &    |- ( ( ( A ^ G ) - 1 ) gcd N ) = 1   =>    |- N e. Prime
5.2.10  Infinite primes theorem
Theorem	infpnlem1 13160*	Lemma for infpn 13162. The smallest divisor (greater than 1) M of N ! + 1 is a prime greater than N. (Contributed by NM, 5-May-2005.)
|- K = ( ( ! ` N ) + 1 )   =>    |- ( ( N e. NN /\ M e. NN ) -> ( ( ( 1 < M /\ ( K / M ) e. NN ) /\ A. j e. NN ( ( 1 < j /\ ( K / j ) e. NN ) -> M <_ j ) ) -> ( N < M /\ A. j e. NN ( ( M / j ) e. NN -> ( j = 1 \/ j = M ) ) ) ) )
Theorem	infpnlem2 13161*	Lemma for infpn 13162. For any positive integer N, there exists a prime number j greater than N. (Contributed by NM, 5-May-2005.)
|- K = ( ( ! ` N ) + 1 )   =>    |- ( N e. NN -> E. j e. NN ( N < j /\ A. k e. NN ( ( j / k ) e. NN -> ( k = 1 \/ k = j ) ) ) )
Theorem	infpn 13162*	There exist infinitely many prime numbers: for any positive integer N, there exists a prime number j greater than N. (See infpn2 13398 for the equinumerosity version.) (Contributed by NM, 1-Jun-2006.)
|- ( N e. NN -> E. j e. NN ( N < j /\ A. k e. NN ( ( j / k ) e. NN -> ( k = 1 \/ k = j ) ) ) )
Theorem	prmunb 13163*	The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.)
|- ( N e. NN -> E. p e. Prime N < p )
5.2.11  Fundamental theorem of arithmetic
Theorem	1arithlem1 13164*	Lemma for 1arith 13168. (Contributed by Mario Carneiro, 30-May-2014.)
|- M = ( n e. NN |-> ( p e. Prime |-> ( p pCnt n ) ) )   =>    |- ( N e. NN -> ( M ` N ) = ( p e. Prime |-> ( p pCnt N ) ) )
Theorem	1arithlem2 13165*	Lemma for 1arith 13168. (Contributed by Mario Carneiro, 30-May-2014.)
|- M = ( n e. NN |-> ( p e. Prime |-> ( p pCnt n ) ) )   =>    |- ( ( N e. NN /\ P e. Prime ) -> ( ( M ` N ) ` P ) = ( P pCnt N ) )
Theorem	1arithlem3 13166*	Lemma for 1arith 13168. (Contributed by Mario Carneiro, 30-May-2014.)
|- M = ( n e. NN |-> ( p e. Prime |-> ( p pCnt n ) ) )   =>    |- ( N e. NN -> ( M ` N ) : Prime --> NN0 )
Theorem	1arithlem4 13167*	Lemma for 1arith 13168. (Contributed by Mario Carneiro, 30-May-2014.)
|- M = ( n e. NN |-> ( p e. Prime |-> ( p pCnt n ) ) )   &    |- G = ( y e. NN |-> if ( y e. Prime , ( y ^ ( F ` y ) ) , 1 ) )   &    |- ( ph -> F : Prime --> NN0 )   &    |- ( ph -> N e. NN )   &    |- ( ( ph /\ ( q e. Prime /\ N <_ q ) ) -> ( F ` q ) = 0 )   =>    |- ( ph -> E. x e. NN F = ( M ` x ) )
Theorem	1arith 13168*	Fundamental theorem of arithmetic, where a prime factorization is represented as a sequence of prime exponents, for which only finitely many primes have nonzero exponent. The function M maps the set of positive integers one-to-one onto the set of prime factorizations R. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 30-May-2014.)
|- M = ( n e. NN |-> ( p e. Prime |-> ( p pCnt n ) ) )   &    |- R = { e e. ( NN0 ^m Prime ) | ( `' e " NN ) e. Fin }   =>    |- M : NN -1-1-onto-> R
Theorem	1arith2 13169*	Fundamental theorem of arithmetic, where a prime factorization is represented as a finite monotonic 1-based sequence of primes. Every positive integer has a unique prime factorization. Theorem 1.10 in [ApostolNT] p. 17. This is Metamath 100 proof #80. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 30-May-2014.)
|- M = ( n e. NN |-> ( p e. Prime |-> ( p pCnt n ) ) )   &    |- R = { e e. ( NN0 ^m Prime ) | ( `' e " NN ) e. Fin }   =>    |- A. z e. NN E! g e. R ( M ` z ) = g
5.2.12  Lagrange's four-square theorem
Syntax	cgz 13170	Extend class notation with the set of gaussian integers.
class ZZ[_i]
Definition	df-gz 13171	Define the set of gaussian integers, which are complex numbers whose real and imaginary parts are integers. (Note that the [ _i ] is actually part of the symbol token and has no independent meaning.) (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ZZ[_i] = { x e. CC | ( ( Re ` x ) e. ZZ /\ ( Im ` x ) e. ZZ ) }
Theorem	elgz 13172	Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ[_i] <-> ( A e. CC /\ ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) )
Theorem	gzcn 13173	A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ[_i] -> A e. CC )
Theorem	zgz 13174	An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ -> A e. ZZ[_i] )
Theorem	igz 13175	_i is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- _i e. ZZ[_i]
Theorem	gznegcl 13176	The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ[_i] -> -u A e. ZZ[_i] )
Theorem	gzcjcl 13177	The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ[_i] -> ( * ` A ) e. ZZ[_i] )
Theorem	gzaddcl 13178	The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. ZZ[_i] /\ B e. ZZ[_i] ) -> ( A + B ) e. ZZ[_i] )
Theorem	gzmulcl 13179	The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. ZZ[_i] /\ B e. ZZ[_i] ) -> ( A x. B ) e. ZZ[_i] )
Theorem	gzreim 13180	Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + ( _i x. B ) ) e. ZZ[_i] )
Theorem	gzsubcl 13181	The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. ZZ[_i] /\ B e. ZZ[_i] ) -> ( A - B ) e. ZZ[_i] )
Theorem	gzabssqcl 13182	The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)
|- ( A e. ZZ[_i] -> ( ( abs ` A ) ^ 2 ) e. NN0 )
Theorem	4sqlem5 13183	Lemma for 4sq 13211. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   =>    |- ( ph -> ( B e. ZZ /\ ( ( A - B ) / M ) e. ZZ ) )
Theorem	4sqlem6 13184	Lemma for 4sq 13211. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   =>    |- ( ph -> ( -u ( M / 2 ) <_ B /\ B < ( M / 2 ) ) )
Theorem	4sqlem7 13185	Lemma for 4sq 13211. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   =>    |- ( ph -> ( B ^ 2 ) <_ ( ( ( M ^ 2 ) / 2 ) / 2 ) )
Theorem	4sqlem8 13186	Lemma for 4sq 13211. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   =>    |- ( ph -> M || ( ( A ^ 2 ) - ( B ^ 2 ) ) )
Theorem	4sqlem9 13187	Lemma for 4sq 13211. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- ( ( ph /\ ps ) -> ( B ^ 2 ) = 0 )   =>    |- ( ( ph /\ ps ) -> ( M ^ 2 ) || ( A ^ 2 ) )
Theorem	4sqlem10 13188	Lemma for 4sq 13211. (Contributed by Mario Carneiro, 16-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- ( ( ph /\ ps ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( B ^ 2 ) ) = 0 )   =>    |- ( ( ph /\ ps ) -> ( M ^ 2 ) || ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) )
Theorem	4sqlem1 13189*	Lemma for 4sq 13211. The set S is the set of all numbers that are expressible as a sum of four squares. Our goal is to show that S = NN0; here we show one subset direction. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- S C_ NN0
Theorem	4sqlem2 13190*	Lemma for 4sq 13211. Change bound variables in S. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( A e. S <-> E. a e. ZZ E. b e. ZZ E. c e. ZZ E. d e. ZZ A = ( ( ( a ^ 2 ) + ( b ^ 2 ) ) + ( ( c ^ 2 ) + ( d ^ 2 ) ) ) )
Theorem	4sqlem3 13191*	Lemma for 4sq 13211. Sufficient condition to be in S. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. ZZ /\ D e. ZZ ) ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) e. S )
Theorem	4sqlem4a 13192*	Lemma for 4sqlem4 13193. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( ( A e. ZZ[_i] /\ B e. ZZ[_i] ) -> ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) e. S )
Theorem	4sqlem4 13193*	Lemma for 4sq 13211. We can express the four-square property more compactly in terms of gaussian integers, because the norms of gaussian integers are exactly sums of two squares. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( A e. S <-> E. u e. ZZ[_i] E. v e. ZZ[_i] A = ( ( ( abs ` u ) ^ 2 ) + ( ( abs ` v ) ^ 2 ) ) )
Theorem	mul4sqlem 13194*	Lemma for mul4sq 13195: algebraic manipulations. The extra assumptions involving M would let us know not just that the product is a sum of squares, but also that it preserves divisibility by M. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> A e. ZZ[_i] )   &    |- ( ph -> B e. ZZ[_i] )   &    |- ( ph -> C e. ZZ[_i] )   &    |- ( ph -> D e. ZZ[_i] )   &    |- X = ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) )   &    |- Y = ( ( ( abs ` C ) ^ 2 ) + ( ( abs ` D ) ^ 2 ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> ( ( A - C ) / M ) e. ZZ[_i] )   &    |- ( ph -> ( ( B - D ) / M ) e. ZZ[_i] )   &    |- ( ph -> ( X / M ) e. NN0 )   =>    |- ( ph -> ( ( X / M ) x. ( Y / M ) ) e. S )
Theorem	mul4sq 13195*	Euler's four-square identity: The product of two sums of four squares is also a sum of four squares. This is usually quoted as an explicit formula involving eight real variables; we save some time by working with complex numbers (gaussian integers) instead, so that we only have to work with four variables, and also hiding the actual formula for the product in the proof of mul4sqlem 13194. (For the curious, the explicit formula that is used is ( | a | ^ 2 + | b | ^ 2 ) ( | c | ^ 2 + | d | ^ 2 ) = | a * x. c + b x. d * | ^ 2 + | a * x. d - b x. c * | ^ 2.) (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( ( A e. S /\ B e. S ) -> ( A x. B ) e. S )
Theorem	4sqlemafi 13196*	Lemma for 4sq 13211. A is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   =>    |- ( ph -> A e. Fin )
Theorem	4sqlemffi 13197*	Lemma for 4sq 13211. ran F is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> ran F e. Fin )
Theorem	4sqleminfi 13198*	Lemma for 4sq 13211. A i^i ran F is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> ( A i^i ran F ) e. Fin )
Theorem	4sqexercise1 13199*	Exercise which may help in understanding the proof of 4sqlemsdc 13201. (Contributed by Jim Kingdon, 25-May-2025.)
|- S = { n | E. x e. ZZ n = ( x ^ 2 ) }   =>    |- ( A e. NN0 -> DECID A e. S )
Theorem	4sqexercise2 13200*	Exercise which may help in understanding the proof of 4sqlemsdc 13201. (Contributed by Jim Kingdon, 30-May-2025.)
|- S = { n | E. x e. ZZ E. y e. ZZ n = ( ( x ^ 2 ) + ( y ^ 2 ) ) }   =>    |- ( A e. NN0 -> DECID A e. S )