P. 106 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	modqelico 10501	Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) e. ( 0 [,) B ) )
Theorem	modqdiffl 10502	The modulo operation differs from A by an integer multiple of B. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) )
Theorem	modqdifz 10503	The modulo operation differs from A by an integer multiple of B. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A - ( A mod B ) ) / B ) e. ZZ )
Theorem	modqfrac 10504	The fractional part of a number is the number modulo 1. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( A e. QQ -> ( A mod 1 ) = ( A - ( |_ ` A ) ) )
Theorem	flqmod 10505	The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) = ( A - ( A mod 1 ) ) )
Theorem	intqfrac 10506	Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( A e. QQ -> A = ( ( |_ ` A ) + ( A mod 1 ) ) )
Theorem	zmod10 10507	An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. ZZ -> ( N mod 1 ) = 0 )
Theorem	zmod1congr 10508	Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A mod 1 ) = ( B mod 1 ) )
Theorem	modqmulnn 10509	Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( N e. NN /\ A e. QQ /\ M e. NN ) -> ( ( N x. ( |_ ` A ) ) mod ( N x. M ) ) <_ ( ( |_ ` ( N x. A ) ) mod ( N x. M ) ) )
Theorem	modqvalp1 10510	The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Jim Kingdon, 20-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A + B ) - ( ( ( |_ ` ( A / B ) ) + 1 ) x. B ) ) = ( A mod B ) )
Theorem	zmodcl 10511	Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. NN0 )
Theorem	zmodcld 10512	Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. NN )   =>    |- ( ph -> ( A mod B ) e. NN0 )
Theorem	zmodfz 10513	An integer mod B lies in the first B nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. ( 0 ... ( B - 1 ) ) )
Theorem	zmodfzo 10514	An integer mod B lies in the first B nonnegative integers. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. ( 0..^ B ) )
Theorem	zmodfzp1 10515	An integer mod B lies in the first B + 1 nonnegative integers. (Contributed by AV, 27-Oct-2018.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. ( 0 ... B ) )
Theorem	modqid 10516	Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
Theorem	modqid0 10517	A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( N e. QQ /\ 0 < N ) -> ( N mod N ) = 0 )
Theorem	modqid2 10518	Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = A <-> ( 0 <_ A /\ A < B ) ) )
Theorem	zmodid2 10519	Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) = M <-> M e. ( 0 ... ( N - 1 ) ) ) )
Theorem	zmodidfzo 10520	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) = M <-> M e. ( 0..^ N ) ) )
Theorem	zmodidfzoimp 10521	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
|- ( M e. ( 0..^ N ) -> ( M mod N ) = M )
Theorem	q0mod 10522	Special case: 0 modulo a positive real number is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( N e. QQ /\ 0 < N ) -> ( 0 mod N ) = 0 )
Theorem	q1mod 10523	Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( N e. QQ /\ 1 < N ) -> ( 1 mod N ) = 1 )
Theorem	modqabs 10524	Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> 0 < B )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> B <_ C )   =>    |- ( ph -> ( ( A mod B ) mod C ) = ( A mod B ) )
Theorem	modqabs2 10525	Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) mod B ) = ( A mod B ) )
Theorem	modqcyc 10526	The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( ( A + ( N x. B ) ) mod B ) = ( A mod B ) )
Theorem	modqcyc2 10527	The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( ( A - ( B x. N ) ) mod B ) = ( A mod B ) )
Theorem	modqadd1 10528	Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> D e. QQ )   &    |- ( ph -> 0 < D )   &    |- ( ph -> ( A mod D ) = ( B mod D ) )   =>    |- ( ph -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) )
Theorem	modqaddabs 10529	Absorption law for modulo. (Contributed by Jim Kingdon, 22-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( ( A mod C ) + ( B mod C ) ) mod C ) = ( ( A + B ) mod C ) )
Theorem	modqaddmod 10530	The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the same modulus. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) + B ) mod M ) = ( ( A + B ) mod M ) )
Theorem	mulqaddmodid 10531	The sum of a positive rational number less than an upper bound and the product of an integer and the upper bound is the positive rational number modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )
Theorem	mulp1mod1 10532	The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.)
|- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) + 1 ) mod N ) = 1 )
Theorem	modqmuladd 10533*	Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> B e. ( 0 [,) M ) )   &    |- ( ph -> M e. QQ )   &    |- ( ph -> 0 < M )   =>    |- ( ph -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
Theorem	modqmuladdim 10534*	Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
Theorem	modqmuladdnn0 10535*	Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. k e. NN0 A = ( ( k x. M ) + B ) ) )
Theorem	qnegmod 10536	The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( -u A mod N ) = ( ( N - A ) mod N ) )
Theorem	m1modnnsub1 10537	Minus one modulo a positive integer is equal to the integer minus one. (Contributed by AV, 14-Jul-2021.)
|- ( M e. NN -> ( -u 1 mod M ) = ( M - 1 ) )
Theorem	m1modge3gt1 10538	Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.)
|- ( M e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod M ) )
Theorem	addmodid 10539	The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.)
|- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( ( M + A ) mod M ) = A )
Theorem	addmodidr 10540	The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by AV, 19-Mar-2021.)
|- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( ( A + M ) mod M ) = A )
Theorem	modqadd2mod 10541	The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + A ) mod M ) )
Theorem	modqm1p1mod0 10542	If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = ( M - 1 ) -> ( ( A + 1 ) mod M ) = 0 ) )
Theorem	modqltm1p1mod 10543	If a number modulo a modulus is less than the modulus decreased by 1, the first number increased by 1 modulo the modulus equals the first number modulo the modulus, increased by 1. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + 1 ) mod M ) = ( ( A mod M ) + 1 ) )
Theorem	modqmul1 10544	Multiplication property of the modulo operation. Note that the multiplier C must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. QQ )   &    |- ( ph -> 0 < D )   &    |- ( ph -> ( A mod D ) = ( B mod D ) )   =>    |- ( ph -> ( ( A x. C ) mod D ) = ( ( B x. C ) mod D ) )
Theorem	modqmul12d 10545	Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- ( ph -> E e. QQ )   &    |- ( ph -> 0 < E )   &    |- ( ph -> ( A mod E ) = ( B mod E ) )   &    |- ( ph -> ( C mod E ) = ( D mod E ) )   =>    |- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. D ) mod E ) )
Theorem	modqnegd 10546	Negation property of the modulo operation. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> 0 < C )   &    |- ( ph -> ( A mod C ) = ( B mod C ) )   =>    |- ( ph -> ( -u A mod C ) = ( -u B mod C ) )
Theorem	modqadd12d 10547	Additive property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> D e. QQ )   &    |- ( ph -> E e. QQ )   &    |- ( ph -> 0 < E )   &    |- ( ph -> ( A mod E ) = ( B mod E ) )   &    |- ( ph -> ( C mod E ) = ( D mod E ) )   =>    |- ( ph -> ( ( A + C ) mod E ) = ( ( B + D ) mod E ) )
Theorem	modqsub12d 10548	Subtraction property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> D e. QQ )   &    |- ( ph -> E e. QQ )   &    |- ( ph -> 0 < E )   &    |- ( ph -> ( A mod E ) = ( B mod E ) )   &    |- ( ph -> ( C mod E ) = ( D mod E ) )   =>    |- ( ph -> ( ( A - C ) mod E ) = ( ( B - D ) mod E ) )
Theorem	modqsubmod 10549	The difference of a number modulo a modulus and another number equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) - B ) mod M ) = ( ( A - B ) mod M ) )
Theorem	modqsubmodmod 10550	The difference of a number modulo a modulus and another number modulo the same modulus equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) - ( B mod M ) ) mod M ) = ( ( A - B ) mod M ) )
Theorem	q2txmodxeq0 10551	Two times a positive number modulo the number is zero. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( X e. QQ /\ 0 < X ) -> ( ( 2 x. X ) mod X ) = 0 )
Theorem	q2submod 10552	If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A mod B ) = ( A - B ) )
Theorem	modifeq2int 10553	If a nonnegative integer is less than twice a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
|- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) )
Theorem	modaddmodup 10554	The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
|- ( ( A e. ZZ /\ M e. NN ) -> ( B e. ( ( M - ( A mod M ) )..^ M ) -> ( ( B + ( A mod M ) ) - M ) = ( ( B + A ) mod M ) ) )
Theorem	modaddmodlo 10555	The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
|- ( ( A e. ZZ /\ M e. NN ) -> ( B e. ( 0..^ ( M - ( A mod M ) ) ) -> ( B + ( A mod M ) ) = ( ( B + A ) mod M ) ) )
Theorem	modqmulmod 10556	The product of a rational number modulo a modulus and an integer equals the product of the rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) x. B ) mod M ) = ( ( A x. B ) mod M ) )
Theorem	modqmulmodr 10557	The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A x. ( B mod M ) ) mod M ) = ( ( A x. B ) mod M ) )
Theorem	modqaddmulmod 10558	The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + ( ( B mod M ) x. C ) ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
Theorem	modqdi 10559	Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. ( B mod C ) ) = ( ( A x. B ) mod ( A x. C ) ) )
Theorem	modqsubdir 10560	Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( B mod C ) <_ ( A mod C ) <-> ( ( A - B ) mod C ) = ( ( A mod C ) - ( B mod C ) ) ) )
Theorem	modqeqmodmin 10561	A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( A mod M ) = ( ( A - M ) mod M ) )
Theorem	modfzo0difsn 10562*	For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.)
|- ( ( J e. ( 0..^ N ) /\ K e. ( ( 0..^ N ) \ { J } ) ) -> E. i e. ( 1..^ N ) K = ( ( i + J ) mod N ) )
Theorem	modsumfzodifsn 10563	The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.)
|- ( ( J e. ( 0..^ N ) /\ K e. ( 1..^ N ) ) -> ( ( K + J ) mod N ) e. ( ( 0..^ N ) \ { J } ) )
Theorem	modlteq 10564	Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) ) -> ( ( I mod N ) = ( J mod N ) <-> I = J ) )
Theorem	addmodlteq 10565	Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. (Contributed by AV, 20-Mar-2021.)
|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) /\ S e. ZZ ) -> ( ( ( I + S ) mod N ) = ( ( J + S ) mod N ) <-> I = J ) )
4.6.3  Miscellaneous theorems about integers
Theorem	frec2uz0d 10566*	The mapping G is a one-to-one mapping from om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number C (normally 0 for the upper integers NN0 or 1 for the upper integers NN), 1 maps to C + 1, etc. This theorem shows the value of G at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> ( G ` (/) ) = C )
Theorem	frec2uzzd 10567*	The value of G (see frec2uz0d 10566) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   =>    |- ( ph -> ( G ` A ) e. ZZ )
Theorem	frec2uzsucd 10568*	The value of G (see frec2uz0d 10566) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   =>    |- ( ph -> ( G ` suc A ) = ( ( G ` A ) + 1 ) )
Theorem	frec2uzuzd 10569*	The value G (see frec2uz0d 10566) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   =>    |- ( ph -> ( G ` A ) e. ( ZZ>= ` C ) )
Theorem	frec2uzltd 10570*	Less-than relation for G (see frec2uz0d 10566). (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   &    |- ( ph -> B e. om )   =>    |- ( ph -> ( A e. B -> ( G ` A ) < ( G ` B ) ) )
Theorem	frec2uzlt2d 10571*	The mapping G (see frec2uz0d 10566) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   &    |- ( ph -> B e. om )   =>    |- ( ph -> ( A e. B <-> ( G ` A ) < ( G ` B ) ) )
Theorem	frec2uzrand 10572*	Range of G (see frec2uz0d 10566). (Contributed by Jim Kingdon, 17-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> ran G = ( ZZ>= ` C ) )
Theorem	frec2uzf1od 10573*	G (see frec2uz0d 10566) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
Theorem	frec2uzisod 10574*	G (see frec2uz0d 10566) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> G Isom _E , < ( om , ( ZZ>= ` C ) ) )
Theorem	frecuzrdgrrn 10575*	The function R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of S. (Contributed by Jim Kingdon, 28-Mar-2022.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ( ph /\ D e. om ) -> ( R ` D ) e. ( ( ZZ>= ` C ) X. S ) )
Theorem	frec2uzrdg 10576*	A helper lemma for the value of a recursive definition generator on upper integers (typically either NN or NN0) with characteristic function F ( x , y ) and initial value A. This lemma shows that evaluating R at an element of om gives an ordered pair whose first element is the index (translated from om to ( ZZ>= ` C )). See comment in frec2uz0d 10566 which describes G and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> B e. om )   =>    |- ( ph -> ( R ` B ) = <. ( G ` B ) , ( 2nd ` ( R ` B ) ) >. )
Theorem	frecuzrdgrcl 10577*	The function R (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 1-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
Theorem	frecuzrdglem 10578*	A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> B e. ( ZZ>= ` C ) )   =>    |- ( ph -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. ran R )
Theorem	frecuzrdgtcl 10579*	The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10566 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 26-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> T = ran R )   =>    |- ( ph -> T : ( ZZ>= ` C ) --> S )
Theorem	frecuzrdg0 10580*	Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10566 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 27-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> T = ran R )   =>    |- ( ph -> ( T ` C ) = A )
Theorem	frecuzrdgsuc 10581*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10566 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 28-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> T = ran R )   =>    |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` ( B + 1 ) ) = ( B F ( T ` B ) ) )
Theorem	frecuzrdgrclt 10582*	The function R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of S. Similar to frecuzrdgrcl 10577 except that S and T need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
Theorem	frecuzrdgg 10583*	Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating R at a natural number gives an ordered pair whose first element is the mapping of that natural number via G. (Contributed by Jim Kingdon, 23-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> N e. om )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> ( 1st ` ( R ` N ) ) = ( G ` N ) )
Theorem	frecuzrdgdomlem 10584*	The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> dom ran R = ( ZZ>= ` C ) )
Theorem	frecuzrdgdom 10585*	The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ph -> dom ran R = ( ZZ>= ` C ) )
Theorem	frecuzrdgfunlem 10586*	The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> Fun ran R )
Theorem	frecuzrdgfun 10587*	The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ph -> Fun ran R )
Theorem	frecuzrdgtclt 10588*	The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> P = ran R )   =>    |- ( ph -> P : ( ZZ>= ` C ) --> S )
Theorem	frecuzrdg0t 10589*	Initial value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 28-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> P = ran R )   =>    |- ( ph -> ( P ` C ) = A )
Theorem	frecuzrdgsuctlem 10590*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10566 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 29-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> P = ran R )   =>    |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( P ` ( B + 1 ) ) = ( B F ( P ` B ) ) )
Theorem	frecuzrdgsuct 10591*	Successor value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 29-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> P = ran R )   =>    |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( P ` ( B + 1 ) ) = ( B F ( P ` B ) ) )
Theorem	uzenom 10592	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> Z ~~ om )
Theorem	frecfzennn 10593	The cardinality of a finite set of sequential integers. (See frec2uz0d 10566 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )
Theorem	frecfzen2 10594	The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )
Theorem	frechashgf1o 10595	G maps om one-to-one onto NN0. (Contributed by Jim Kingdon, 19-May-2020.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- G : om -1-1-onto-> NN0
Theorem	frec2uzled 10596*	The mapping G (see frec2uz0d 10566) preserves order. (Contributed by Jim Kingdon, 24-Feb-2022.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   &    |- ( ph -> B e. om )   =>    |- ( ph -> ( A C_ B <-> ( G ` A ) <_ ( G ` B ) ) )
Theorem	fzfig 10597	A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) e. Fin )
Theorem	fzfigd 10598	Deduction form of fzfig 10597. (Contributed by Jim Kingdon, 21-May-2020.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M ... N ) e. Fin )
Theorem	fzofig 10599	Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M..^ N ) e. Fin )
Theorem	nn0ennn 10600	The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
|- NN0 ~~ NN