P. 105 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	flqpmodeq 10401	Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( ( |_ ` ( A / B ) ) x. B ) + ( A mod B ) ) = A )
Theorem	modqcld 10402	Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> ( A mod B ) e. QQ )
Theorem	modq0 10403	A mod B is zero iff A is evenly divisible by B. (Contributed by Jim Kingdon, 17-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( A / B ) e. ZZ ) )
Theorem	mulqmod0 10404	The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A x. M ) mod M ) = 0 )
Theorem	negqmod0 10405	A is divisible by B iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( -u A mod B ) = 0 ) )
Theorem	modqge0 10406	The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 <_ ( A mod B ) )
Theorem	modqlt 10407	The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )
Theorem	modqelico 10408	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 10409	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 10410	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 10411	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 10412	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 10413	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 10414	An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. ZZ -> ( N mod 1 ) = 0 )
Theorem	zmod1congr 10415	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 10416	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 10417	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 10418	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 10419	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 10420	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 10421	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 10422	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 10423	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 10424	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 10425	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 10426	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 10427	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 10428	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
|- ( M e. ( 0..^ N ) -> ( M mod N ) = M )
Theorem	q0mod 10429	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 10430	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 10431	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 10432	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 10433	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 10434	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 10435	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 10436	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 10437	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 10438	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 10439	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 10440*	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 10441*	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 10442*	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 10443	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 10444	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 10445	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 10446	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 10447	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 10448	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 10449	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 10450	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 10451	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 10452	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 10453	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 10454	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 10455	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 10456	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 10457	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 10458	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 10459	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 10460	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 10461	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 10462	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 10463	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 10464	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 10465	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 10466	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 10467	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 10468	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 10469*	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 10470	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 10471	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 10472	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 10473*	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 10474*	The value of G (see frec2uz0d 10473) 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 10475*	The value of G (see frec2uz0d 10473) 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 10476*	The value G (see frec2uz0d 10473) 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 10477*	Less-than relation for G (see frec2uz0d 10473). (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 10478*	The mapping G (see frec2uz0d 10473) 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 10479*	Range of G (see frec2uz0d 10473). (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 10480*	G (see frec2uz0d 10473) 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 10481*	G (see frec2uz0d 10473) 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 10482*	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 10483*	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 10473 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 10484*	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 10485*	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 10486*	The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10473 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 10487*	Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10473 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 10488*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10473 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 10489*	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 10484 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 10490*	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 10491*	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 10492*	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 10493*	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 10494*	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 10495*	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 10496*	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 10497*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10473 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 10498*	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 10499	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> Z ~~ om )
Theorem	frecfzennn 10500	The cardinality of a finite set of sequential integers. (See frec2uz0d 10473 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 ) )