P. 122 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12101-12200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	phiprmpw 12101	Value of the Euler phi function at a prime power. Theorem 2.5(a) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.)
|- ( ( P e. Prime /\ K e. NN ) -> ( phi ` ( P ^ K ) ) = ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) )
Theorem	phiprm 12102	Value of the Euler phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
Theorem	crth 12103*	The Chinese Remainder Theorem: the function that maps x to its remainder classes mod M and mod N is 1-1 and onto when M and N are coprime. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-May-2016.)
|- S = ( 0..^ ( M x. N ) )   &    |- T = ( ( 0..^ M ) X. ( 0..^ N ) )   &    |- F = ( x e. S |-> <. ( x mod M ) , ( x mod N ) >. )   &    |- ( ph -> ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) )   =>    |- ( ph -> F : S -1-1-onto-> T )
Theorem	phimullem 12104*	Lemma for phimul 12105. (Contributed by Mario Carneiro, 24-Feb-2014.)
|- S = ( 0..^ ( M x. N ) )   &    |- T = ( ( 0..^ M ) X. ( 0..^ N ) )   &    |- F = ( x e. S |-> <. ( x mod M ) , ( x mod N ) >. )   &    |- ( ph -> ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) )   &    |- U = { y e. ( 0..^ M ) | ( y gcd M ) = 1 }   &    |- V = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- W = { y e. S | ( y gcd ( M x. N ) ) = 1 }   =>    |- ( ph -> ( phi ` ( M x. N ) ) = ( ( phi ` M ) x. ( phi ` N ) ) )
Theorem	phimul 12105	The Euler phi 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.)
|- ( ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( phi ` ( M x. N ) ) = ( ( phi ` M ) x. ( phi ` N ) ) )
Theorem	eulerthlem1 12106*	Lemma for eulerth 12112. (Contributed by Mario Carneiro, 8-May-2015.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- T = ( 1 ... ( phi ` N ) )   &    |- ( ph -> F : T -1-1-onto-> S )   &    |- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )   =>    |- ( ph -> G : T --> S )
Theorem	eulerthlemfi 12107*	Lemma for eulerth 12112. The set S is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   =>    |- ( ph -> S e. Fin )
Theorem	eulerthlemrprm 12108*	Lemma for eulerth 12112. N and prod_ x e. ( 1 ... ( phi ` N ) ) ( F ` x ) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   =>    |- ( ph -> ( N gcd prod_ x e. ( 1 ... ( phi ` N ) ) ( F ` x ) ) = 1 )
Theorem	eulerthlema 12109*	Lemma for eulerth 12112. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   =>    |- ( ph -> ( ( ( A ^ ( phi ` N ) ) x. prod_ x e. ( 1 ... ( phi ` N ) ) ( F ` x ) ) mod N ) = ( prod_ x e. ( 1 ... ( phi ` N ) ) ( ( A x. ( F ` x ) ) mod N ) mod N ) )
Theorem	eulerthlemh 12110*	Lemma for eulerth 12112. A permutation of ( 1 ... ( phi ` N ) ). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   &    |- H = ( `' F o. ( y e. ( 1 ... ( phi ` N ) ) |-> ( ( A x. ( F ` y ) ) mod N ) ) )   =>    |- ( ph -> H : ( 1 ... ( phi ` N ) ) -1-1-onto-> ( 1 ... ( phi ` N ) ) )
Theorem	eulerthlemth 12111*	Lemma for eulerth 12112. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   =>    |- ( ph -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
Theorem	eulerth 12112	Euler's theorem, a generalization of Fermat's little theorem. If A and N are coprime, then A ^ phi ( N ) == 1 (mod N). 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.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
Theorem	fermltl 12113	Fermat's little theorem. When P is prime, A ^ P == A (mod P) for any A, see theorem 5.19 in [ApostolNT] p. 114. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 19-Mar-2022.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
Theorem	prmdiv 12114	Show an explicit expression for the modular inverse of A mod P. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
Theorem	prmdiveq 12115	The modular inverse of A mod P is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) <-> S = R ) )
Theorem	prmdivdiv 12116	The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A = ( ( R ^ ( P - 2 ) ) mod P ) )
Theorem	hashgcdlem 12117*	A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.)
|- A = { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 }   &    |- B = { z e. ( 0..^ M ) | ( z gcd M ) = N }   &    |- F = ( x e. A |-> ( x x. N ) )   =>    |- ( ( M e. NN /\ N e. NN /\ N || M ) -> F : A -1-1-onto-> B )
Theorem	hashgcdeq 12118*	Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.)
|- ( ( M e. NN /\ N e. NN ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = if ( N || M , ( phi ` ( M / N ) ) , 0 ) )
Theorem	phisum 12119*	The divisor sum identity of the totient function. Theorem 2.2 in [ApostolNT] p. 26. (Contributed by Stefan O'Rear, 12-Sep-2015.)
|- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( phi ` d ) = N )
Theorem	odzval 12120*	Value of the order function. This is a function of functions; the inner argument selects the base (i.e., mod N for some N, 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 N. In order to ensure the supremum is well-defined, we only define the expression when A and N are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
Theorem	odzcllem 12121	- Lemma for odzcl 12122, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( ( odZ ` N ) ` A ) e. NN /\ N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )
Theorem	odzcl 12122	The order of a group element is an integer. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) e. NN )
Theorem	odzid 12123	Any element raised to the power of its order is 1. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) )
Theorem	odzdvds 12124	The only powers of A that are congruent to 1 are the multiples of the order of A. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)
|- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ K e. NN0 ) -> ( N || ( ( A ^ K ) - 1 ) <-> ( ( odZ ` N ) ` A ) || K ) )
Theorem	odzphi 12125	The order of any group element is a divisor of the Euler phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) || ( phi ` N ) )
5.2.6  Arithmetic modulo a prime number
Theorem	modprm1div 12126	A prime number divides an integer minus 1 iff the integer modulo the prime number is 1. (Contributed by Alexander van der Vekens, 17-May-2018.) (Proof shortened by AV, 30-May-2023.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A mod P ) = 1 <-> P || ( A - 1 ) ) )
Theorem	m1dvdsndvds 12127	If an integer minus 1 is divisible by a prime number, the integer itself is not divisible by this prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A - 1 ) -> -. P || A ) )
Theorem	modprminv 12128	Show an explicit expression for the modular inverse of A mod P. This is an application of prmdiv 12114. (Contributed by Alexander van der Vekens, 15-May-2018.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ ( ( A x. R ) mod P ) = 1 ) )
Theorem	modprminveq 12129	The modular inverse of A mod P is unique. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ ( ( A x. S ) mod P ) = 1 ) <-> S = R ) )
Theorem	vfermltl 12130	Variant of Fermat's little theorem if A is not a multiple of P, see theorem 5.18 in [ApostolNT] p. 113. (Contributed by AV, 21-Aug-2020.) (Proof shortened by AV, 5-Sep-2020.)
|- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 1 ) ) mod P ) = 1 )
Theorem	powm2modprm 12131	If an integer minus 1 is divisible by a prime number, then the integer to the power of the prime number minus 2 is 1 modulo the prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A - 1 ) -> ( ( A ^ ( P - 2 ) ) mod P ) = 1 ) )
Theorem	reumodprminv 12132*	For any prime number and for any positive integer less than this prime number, there is a unique modular inverse of this positive integer. (Contributed by Alexander van der Vekens, 12-May-2018.)
|- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E! i e. ( 1 ... ( P - 1 ) ) ( ( N x. i ) mod P ) = 1 )
Theorem	modprm0 12133*	For two positive integers less than a given prime number there is always a nonnegative integer (less than the given prime number) so that the sum of one of the two positive integers and the other of the positive integers multiplied by the nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( ( P e. Prime /\ N e. ( 1..^ P ) /\ I e. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )
Theorem	nnnn0modprm0 12134*	For a positive integer and a nonnegative integer both less than a given prime number there is always a second nonnegative integer (less than the given prime number) so that the sum of this second nonnegative integer multiplied with the positive integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 8-Nov-2018.)
|- ( ( P e. Prime /\ N e. ( 1..^ P ) /\ I e. ( 0..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )
Theorem	modprmn0modprm0 12135*	For an integer not being 0 modulo a given prime number and a nonnegative integer less than the prime number, there is always a second nonnegative integer (less than the given prime number) so that the sum of this second nonnegative integer multiplied with the integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 10-Nov-2018.)
|- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( I e. ( 0..^ P ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )
5.2.7  Pythagorean Triples
Theorem	coprimeprodsq 12136	If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of gcd and square. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) -> A = ( ( A gcd C ) ^ 2 ) ) )
Theorem	coprimeprodsq2 12137	If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of gcd and square. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) -> B = ( ( B gcd C ) ^ 2 ) ) )
Theorem	oddprm 12138	A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 10-Jul-2022.)
|- ( N e. ( Prime \ { 2 } ) -> ( ( N - 1 ) / 2 ) e. NN )
Theorem	nnoddn2prm 12139	A prime not equal to 2 is an odd positive integer. (Contributed by AV, 28-Jun-2021.)
|- ( N e. ( Prime \ { 2 } ) -> ( N e. NN /\ -. 2 || N ) )
Theorem	oddn2prm 12140	A prime not equal to 2 is odd. (Contributed by AV, 28-Jun-2021.)
|- ( N e. ( Prime \ { 2 } ) -> -. 2 || N )
Theorem	nnoddn2prmb 12141	A number is a prime number not equal to 2 iff it is an odd prime number. Conversion theorem for two representations of odd primes. (Contributed by AV, 14-Jul-2021.)
|- ( N e. ( Prime \ { 2 } ) <-> ( N e. Prime /\ -. 2 || N ) )
Theorem	prm23lt5 12142	A prime less than 5 is either 2 or 3. (Contributed by AV, 5-Jul-2021.)
|- ( ( P e. Prime /\ P < 5 ) -> ( P = 2 \/ P = 3 ) )
Theorem	prm23ge5 12143	A prime is either 2 or 3 or greater than or equal to 5. (Contributed by AV, 5-Jul-2021.)
|- ( P e. Prime -> ( P = 2 \/ P = 3 \/ P e. ( ZZ>= ` 5 ) ) )
Theorem	pythagtriplem1 12144*	Lemma for pythagtrip 12162. Prove a weaker version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( E. n e. NN E. m e. NN E. k e. NN ( A = ( k x. ( ( m ^ 2 ) - ( n ^ 2 ) ) ) /\ B = ( k x. ( 2 x. ( m x. n ) ) ) /\ C = ( k x. ( ( m ^ 2 ) + ( n ^ 2 ) ) ) ) -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) )
Theorem	pythagtriplem2 12145*	Lemma for pythagtrip 12162. Prove the full version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( E. n e. NN E. m e. NN E. k e. NN ( { A , B } = { ( k x. ( ( m ^ 2 ) - ( n ^ 2 ) ) ) , ( k x. ( 2 x. ( m x. n ) ) ) } /\ C = ( k x. ( ( m ^ 2 ) + ( n ^ 2 ) ) ) ) -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) )
Theorem	pythagtriplem3 12146	Lemma for pythagtrip 12162. Show that C and B are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( B gcd C ) = 1 )
Theorem	pythagtriplem4 12147	Lemma for pythagtrip 12162. Show that C - B and C + B are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) = 1 )
Theorem	pythagtriplem10 12148	Lemma for pythagtrip 12162. Show that C - B is positive. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> 0 < ( C - B ) )
Theorem	pythagtriplem6 12149	Lemma for pythagtrip 12162. Calculate ( sqr ` ( C - B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) = ( ( C - B ) gcd A ) )
Theorem	pythagtriplem7 12150	Lemma for pythagtrip 12162. Calculate ( sqr ` ( C + B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C + B ) ) = ( ( C + B ) gcd A ) )
Theorem	pythagtriplem8 12151	Lemma for pythagtrip 12162. Show that ( sqr ` ( C - B ) ) is a positive integer. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) e. NN )
Theorem	pythagtriplem9 12152	Lemma for pythagtrip 12162. Show that ( sqr ` ( C + B ) ) is a positive integer. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C + B ) ) e. NN )
Theorem	pythagtriplem11 12153	Lemma for pythagtrip 12162. Show that M (which will eventually be closely related to the m in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> M e. NN )
Theorem	pythagtriplem12 12154	Lemma for pythagtrip 12162. Calculate the square of M. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( M ^ 2 ) = ( ( C + A ) / 2 ) )
Theorem	pythagtriplem13 12155	Lemma for pythagtrip 12162. Show that N (which will eventually be closely related to the n in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> N e. NN )
Theorem	pythagtriplem14 12156	Lemma for pythagtrip 12162. Calculate the square of N. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( N ^ 2 ) = ( ( C - A ) / 2 ) )
Theorem	pythagtriplem15 12157	Lemma for pythagtrip 12162. Show the relationship between M, N, and A. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   &    |- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A = ( ( M ^ 2 ) - ( N ^ 2 ) ) )
Theorem	pythagtriplem16 12158	Lemma for pythagtrip 12162. Show the relationship between M, N, and B. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   &    |- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> B = ( 2 x. ( M x. N ) ) )
Theorem	pythagtriplem17 12159	Lemma for pythagtrip 12162. Show the relationship between M, N, and C. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   &    |- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> C = ( ( M ^ 2 ) + ( N ^ 2 ) ) )
Theorem	pythagtriplem18 12160*	Lemma for pythagtrip 12162. Wrap the previous M and N up in quantifiers. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> E. n e. NN E. m e. NN ( A = ( ( m ^ 2 ) - ( n ^ 2 ) ) /\ B = ( 2 x. ( m x. n ) ) /\ C = ( ( m ^ 2 ) + ( n ^ 2 ) ) ) )
Theorem	pythagtriplem19 12161*	Lemma for pythagtrip 12162. Introduce k and remove the relative primality requirement. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ -. 2 || ( A / ( A gcd B ) ) ) -> E. n e. NN E. m e. NN E. k e. NN ( A = ( k x. ( ( m ^ 2 ) - ( n ^ 2 ) ) ) /\ B = ( k x. ( 2 x. ( m x. n ) ) ) /\ C = ( k x. ( ( m ^ 2 ) + ( n ^ 2 ) ) ) ) )
Theorem	pythagtrip 12162*	Parameterize the Pythagorean triples. If A, B, and C are naturals, then they obey the Pythagorean triple formula iff they are parameterized by three naturals. This proof follows the Isabelle proof at http://afp.sourceforge.net/entries/Fermat3_4.shtml. This is Metamath 100 proof #23. (Contributed by Scott Fenton, 19-Apr-2014.)
|- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) <-> E. n e. NN E. m e. NN E. k e. NN ( { A , B } = { ( k x. ( ( m ^ 2 ) - ( n ^ 2 ) ) ) , ( k x. ( 2 x. ( m x. n ) ) ) } /\ C = ( k x. ( ( m ^ 2 ) + ( n ^ 2 ) ) ) ) ) )
5.3  Cardinality of real and complex number subsets
5.3.1  Countability of integers and rationals
Theorem	oddennn 12163	There are as many odd positive integers as there are positive integers. (Contributed by Jim Kingdon, 11-May-2022.)
|- { z e. NN | -. 2 || z } ~~ NN
Theorem	evenennn 12164	There are as many even positive integers as there are positive integers. (Contributed by Jim Kingdon, 12-May-2022.)
|- { z e. NN | 2 || z } ~~ NN
Theorem	xpnnen 12165	The Cartesian product of the set of positive integers with itself is equinumerous to the set of positive integers. (Contributed by NM, 1-Aug-2004.)
|- ( NN X. NN ) ~~ NN
Theorem	xpomen 12166	The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.)
|- ( om X. om ) ~~ om
Theorem	xpct 12167	The cartesian product of two sets dominated by om is dominated by om. (Contributed by Thierry Arnoux, 24-Sep-2017.)
|- ( ( A ~<_ om /\ B ~<_ om ) -> ( A X. B ) ~<_ om )
Theorem	unennn 12168	The union of two disjoint countably infinite sets is countably infinite. (Contributed by Jim Kingdon, 13-May-2022.)
|- ( ( A ~~ NN /\ B ~~ NN /\ ( A i^i B ) = (/) ) -> ( A u. B ) ~~ NN )
Theorem	znnen 12169	The set of integers and the set of positive integers are equinumerous. Corollary 8.1.23 of [AczelRathjen], p. 75. (Contributed by NM, 31-Jul-2004.)
|- ZZ ~~ NN
Theorem	ennnfonelemdc 12170*	Lemma for ennnfone 12196. A direct consequence of fidcenumlemrk 6899. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> P e. om )   =>    |- ( ph -> DECID ( F ` P ) e. ( F " P ) )
Theorem	ennnfonelemk 12171*	Lemma for ennnfone 12196. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> F : om -onto-> A )   &    |- ( ph -> K e. om )   &    |- ( ph -> N e. om )   &    |- ( ph -> A. j e. suc N ( F ` K ) =/= ( F ` j ) )   =>    |- ( ph -> N e. K )
Theorem	ennnfonelemj0 12172*	Lemma for ennnfone 12196. Initial state for J. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> ( J ` 0 ) e. { g e. ( A ^pm om ) | dom g e. om } )
Theorem	ennnfonelemjn 12173*	Lemma for ennnfone 12196. Non-initial state for J. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ( ph /\ f e. ( ZZ>= ` ( 0 + 1 ) ) ) -> ( J ` f ) e. om )
Theorem	ennnfonelemg 12174*	Lemma for ennnfone 12196. Closure for G. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ( ph /\ ( f e. { g e. ( A ^pm om ) | dom g e. om } /\ j e. om ) ) -> ( f G j ) e. { g e. ( A ^pm om ) | dom g e. om } )
Theorem	ennnfonelemh 12175*	Lemma for ennnfone 12196. (Contributed by Jim Kingdon, 8-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> H : NN0 --> ( A ^pm om ) )
Theorem	ennnfonelem0 12176*	Lemma for ennnfone 12196. Initial value. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> ( H ` 0 ) = (/) )
Theorem	ennnfonelemp1 12177*	Lemma for ennnfone 12196. Value of H at a successor. (Contributed by Jim Kingdon, 23-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> ( H ` ( P + 1 ) ) = if ( ( F ` ( `' N ` P ) ) e. ( F " ( `' N ` P ) ) , ( H ` P ) , ( ( H ` P ) u. { <. dom ( H ` P ) , ( F ` ( `' N ` P ) ) >. } ) ) )
Theorem	ennnfonelem1 12178*	Lemma for ennnfone 12196. Second value. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> ( H ` 1 ) = { <. (/) , ( F ` (/) ) >. } )
Theorem	ennnfonelemom 12179*	Lemma for ennnfone 12196. H yields finite sequences. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> dom ( H ` P ) e. om )
Theorem	ennnfonelemhdmp1 12180*	Lemma for ennnfone 12196. Domain at a successor where we need to add an element to the sequence. (Contributed by Jim Kingdon, 23-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   &    |- ( ph -> -. ( F ` ( `' N ` P ) ) e. ( F " ( `' N ` P ) ) )   =>    |- ( ph -> dom ( H ` ( P + 1 ) ) = suc dom ( H ` P ) )
Theorem	ennnfonelemss 12181*	Lemma for ennnfone 12196. We only add elements to H as the index increases. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> ( H ` P ) C_ ( H ` ( P + 1 ) ) )
Theorem	ennnfoneleminc 12182*	Lemma for ennnfone 12196. We only add elements to H as the index increases. (Contributed by Jim Kingdon, 21-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   &    |- ( ph -> Q e. NN0 )   &    |- ( ph -> P <_ Q )   =>    |- ( ph -> ( H ` P ) C_ ( H ` Q ) )
Theorem	ennnfonelemkh 12183*	Lemma for ennnfone 12196. Because we add zero or one entries for each new index, the length of each sequence is no greater than its index. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> dom ( H ` P ) C_ ( `' N ` P ) )
Theorem	ennnfonelemhf1o 12184*	Lemma for ennnfone 12196. Each of the functions in H is one to one and onto an image of F. (Contributed by Jim Kingdon, 17-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> ( H ` P ) : dom ( H ` P ) -1-1-onto-> ( F " ( `' N ` P ) ) )
Theorem	ennnfonelemex 12185*	Lemma for ennnfone 12196. Extending the sequence ( H ` P ) to include an additional element. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> E. i e. NN0 dom ( H ` P ) e. dom ( H ` i ) )
Theorem	ennnfonelemhom 12186*	Lemma for ennnfone 12196. The sequences in H increase in length without bound if you go out far enough. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> M e. om )   =>    |- ( ph -> E. i e. NN0 M e. dom ( H ` i ) )
Theorem	ennnfonelemrnh 12187*	Lemma for ennnfone 12196. A consequence of ennnfonelemss 12181. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> X e. ran H )   &    |- ( ph -> Y e. ran H )   =>    |- ( ph -> ( X C_ Y \/ Y C_ X ) )
Theorem	ennnfonelemfun 12188*	Lemma for ennnfone 12196. L is a function. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> Fun L )
Theorem	ennnfonelemf1 12189*	Lemma for ennnfone 12196. L is one-to-one. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> L : dom L -1-1-> A )
Theorem	ennnfonelemrn 12190*	Lemma for ennnfone 12196. L is onto A. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> ran L = A )
Theorem	ennnfonelemdm 12191*	Lemma for ennnfone 12196. The function L is defined everywhere. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> dom L = om )
Theorem	ennnfonelemen 12192*	Lemma for ennnfone 12196. The result. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> A ~~ NN )
Theorem	ennnfonelemnn0 12193*	Lemma for ennnfone 12196. A version of ennnfonelemen 12192 expressed in terms of NN0 instead of om. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : NN0 -onto-> A )   &    |- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( ph -> A ~~ NN )
Theorem	ennnfonelemr 12194*	Lemma for ennnfone 12196. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : NN0 -onto-> A )   &    |- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )   =>    |- ( ph -> A ~~ NN )
Theorem	ennnfonelemim 12195*	Lemma for ennnfone 12196. The trivial direction. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( A ~~ NN -> ( A. x e. A A. y e. A DECID x = y /\ E. f ( f : NN0 -onto-> A /\ A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( f ` k ) =/= ( f ` j ) ) ) )
Theorem	ennnfone 12196*	A condition for a set being countably infinite. Corollary 8.1.13 of [AczelRathjen], p. 73. Roughly speaking, the condition says that A is countable (that's the f : NN0 -onto-> A part, as seen in theorems like ctm 7054), infinite (that's the part about being able to find an element of A distinct from any mapping of a natural number via f), and has decidable equality. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( A ~~ NN <-> ( A. x e. A A. y e. A DECID x = y /\ E. f ( f : NN0 -onto-> A /\ A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( f ` k ) =/= ( f ` j ) ) ) )
Theorem	exmidunben 12197*	If any unbounded set of positive integers is equinumerous to NN, then the Limited Principle of Omniscience (LPO) implies excluded middle. (Contributed by Jim Kingdon, 29-Jul-2023.)
|- ( ( A. x ( ( x C_ NN /\ A. m e. NN E. n e. x m < n ) -> x ~~ NN ) /\ om e. Omni ) -> EXMID )
Theorem	ctinfomlemom 12198*	Lemma for ctinfom 12199. Converting between om and NN0. (Contributed by Jim Kingdon, 10-Aug-2023.)
|- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- G = ( F o. `' N )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om -. ( F ` k ) e. ( F " n ) )   =>    |- ( ph -> ( G : NN0 -onto-> A /\ A. m e. NN0 E. j e. NN0 A. i e. ( 0 ... m ) ( G ` j ) =/= ( G ` i ) ) )
Theorem	ctinfom 12199*	A condition for a set being countably infinite. Restates ennnfone 12196 in terms of om and function image. Like ennnfone 12196 the condition can be summarized as A being countable, infinite, and having decidable equality. (Contributed by Jim Kingdon, 7-Aug-2023.)
|- ( A ~~ NN <-> ( A. x e. A A. y e. A DECID x = y /\ E. f ( f : om -onto-> A /\ A. n e. om E. k e. om -. ( f ` k ) e. ( f " n ) ) ) )
Theorem	inffinp1 12200*	An infinite set contains an element not contained in a given finite subset. (Contributed by Jim Kingdon, 7-Aug-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> om ~<_ A )   &    |- ( ph -> B C_ A )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> E. x e. A -. x e. B )