P. 115 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11401-11500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvdsext 11401*	Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. x e. NN0 ( A || x <-> B || x ) ) )
Theorem	fzm1ndvds 11402	No number between 1 and M - 1 divides M. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> -. M || N )
Theorem	fzo0dvdseq 11403	Zero is the only one of the first A nonnegative integers that is divisible by A. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( B e. ( 0..^ A ) -> ( A || B <-> B = 0 ) )
Theorem	fzocongeq 11404	Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( A e. ( C..^ D ) /\ B e. ( C..^ D ) ) -> ( ( D - C ) || ( A - B ) <-> A = B ) )
Theorem	addmodlteqALT 11405	Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 10064 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) /\ S e. ZZ ) -> ( ( ( I + S ) mod N ) = ( ( J + S ) mod N ) <-> I = J ) )
Theorem	dvdsfac 11406	A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
|- ( ( K e. NN /\ N e. ( ZZ>= ` K ) ) -> K || ( ! ` N ) )
Theorem	dvdsexp 11407	A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( A e. ZZ /\ M e. NN0 /\ N e. ( ZZ>= ` M ) ) -> ( A ^ M ) || ( A ^ N ) )
Theorem	dvdsmod 11408	Any number K whose mod base N is divisible by a divisor P of the base is also divisible by P. This means that primes will also be relatively prime to the base when reduced mod N for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || K ) )
Theorem	mulmoddvds 11409	If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
|- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( N || A -> ( ( A x. B ) mod N ) = 0 ) )
Theorem	3dvdsdec 11410	A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if A and B actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers A and B, especially if A is itself a decimal number, e.g. A = ; C D. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)
|- A e. NN0   &    |- B e. NN0   =>    |- ( 3 || ; A B <-> 3 || ( A + B ) )
Theorem	3dvds2dec 11411	A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if A, B and C actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers A, B and C. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
|- A e. NN0   &    |- B e. NN0   &    |- C e. NN0   =>    |- ( 3 || ;; A B C <-> 3 || ( ( A + B ) + C ) )
5.1.2  Even and odd numbers The set ZZ of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 11415. Instead of defining new class variables Even and Odd to represent these sets, we use the idiom 2 || N to say that " N is even" (which implies N e. ZZ, see evenelz 11412) and -. 2 || N to say that " N is odd" (under the assumption that N e. ZZ). The previously proven theorems about even and odd numbers, like zneo 9056, zeo 9060, zeo2 9061, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 11434 and oddp1d2 11435. The corresponding theorems are zeneo 11416, zeo3 11413 and zeo4 11415.
Theorem	evenelz 11412	An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11346. (Contributed by AV, 22-Jun-2021.)
|- ( 2 || N -> N e. ZZ )
Theorem	zeo3 11413	An integer is even or odd. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
Theorem	zeoxor 11414	An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.)
|- ( N e. ZZ -> ( 2 || N \/_ -. 2 || N ) )
Theorem	zeo4 11415	An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> -. -. 2 || N ) )
Theorem	zeneo 11416	No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 9056 follows immediately from the fact that a contradiction implies anything, see pm2.21i 618. (Contributed by AV, 22-Jun-2021.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 || A /\ -. 2 || B ) -> A =/= B ) )
Theorem	odd2np1lem 11417*	Lemma for odd2np1 11418. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( N e. NN0 -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) )
Theorem	odd2np1 11418*	An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
Theorem	even2n 11419*	An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)
|- ( 2 || N <-> E. n e. ZZ ( 2 x. n ) = N )
Theorem	oddm1even 11420	An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( -. 2 || N <-> 2 || ( N - 1 ) ) )
Theorem	oddp1even 11421	An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( -. 2 || N <-> 2 || ( N + 1 ) ) )
Theorem	oexpneg 11422	The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)
|- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> ( -u A ^ N ) = -u ( A ^ N ) )
Theorem	mod2eq0even 11423	An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example 2 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
|- ( N e. ZZ -> ( ( N mod 2 ) = 0 <-> 2 || N ) )
Theorem	mod2eq1n2dvds 11424	An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.)
|- ( N e. ZZ -> ( ( N mod 2 ) = 1 <-> -. 2 || N ) )
Theorem	oddnn02np1 11425*	A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.)
|- ( N e. NN0 -> ( -. 2 || N <-> E. n e. NN0 ( ( 2 x. n ) + 1 ) = N ) )
Theorem	oddge22np1 11426*	An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)
|- ( N e. ( ZZ>= ` 2 ) -> ( -. 2 || N <-> E. n e. NN ( ( 2 x. n ) + 1 ) = N ) )
Theorem	evennn02n 11427*	A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)
|- ( N e. NN0 -> ( 2 || N <-> E. n e. NN0 ( 2 x. n ) = N ) )
Theorem	evennn2n 11428*	A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.)
|- ( N e. NN -> ( 2 || N <-> E. n e. NN ( 2 x. n ) = N ) )
Theorem	2tp1odd 11429	A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.)
|- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B )
Theorem	mulsucdiv2z 11430	An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.)
|- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ )
Theorem	sqoddm1div8z 11431	A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( ( ( N ^ 2 ) - 1 ) / 8 ) e. ZZ )
Theorem	2teven 11432	A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)
|- ( ( A e. ZZ /\ B = ( 2 x. A ) ) -> 2 || B )
Theorem	zeo5 11433	An integer is either even or odd, version of zeo3 11413 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.)
|- ( N e. ZZ -> ( 2 || N \/ 2 || ( N + 1 ) ) )
Theorem	evend2 11434	An integer is even iff its quotient with 2 is an integer. This is a representation of even numbers without using the divides relation, see zeo 9060 and zeo2 9061. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> ( N / 2 ) e. ZZ ) )
Theorem	oddp1d2 11435	An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo 9060 and zeo2 9061. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. ZZ ) )
Theorem	zob 11436	Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)
|- ( N e. ZZ -> ( ( ( N + 1 ) / 2 ) e. ZZ <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
Theorem	oddm1d2 11437	An integer is odd iff its predecessor divided by 2 is an integer. This is another representation of odd numbers without using the divides relation. (Contributed by AV, 18-Jun-2021.) (Proof shortened by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( -. 2 || N <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
Theorem	ltoddhalfle 11438	An integer is less than half of an odd number iff it is less than or equal to the half of the predecessor of the odd number (which is an even number). (Contributed by AV, 29-Jun-2021.)
|- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ ) -> ( M < ( N / 2 ) <-> M <_ ( ( N - 1 ) / 2 ) ) )
Theorem	halfleoddlt 11439	An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ ) -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) )
Theorem	opoe 11440	The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A + B ) )
Theorem	omoe 11441	The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A - B ) )
Theorem	opeo 11442	The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> -. 2 || ( A + B ) )
Theorem	omeo 11443	The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> -. 2 || ( A - B ) )
Theorem	m1expe 11444	Exponentiation of -1 by an even power. Variant of m1expeven 10233. (Contributed by AV, 25-Jun-2021.)
|- ( 2 || N -> ( -u 1 ^ N ) = 1 )
Theorem	m1expo 11445	Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( -u 1 ^ N ) = -u 1 )
Theorem	m1exp1 11446	Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.)
|- ( N e. ZZ -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )
Theorem	nn0enne 11447	A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.)
|- ( N e. NN -> ( ( N / 2 ) e. NN0 <-> ( N / 2 ) e. NN ) )
Theorem	nn0ehalf 11448	The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.)
|- ( ( N e. NN0 /\ 2 || N ) -> ( N / 2 ) e. NN0 )
Theorem	nnehalf 11449	The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.)
|- ( ( N e. NN /\ 2 || N ) -> ( N / 2 ) e. NN )
Theorem	nn0o1gt2 11450	An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)
|- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( N = 1 \/ 2 < N ) )
Theorem	nno 11451	An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.)
|- ( ( N e. ( ZZ>= ` 2 ) /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN )
Theorem	nn0o 11452	An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.)
|- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN0 )
Theorem	nn0ob 11453	Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.)
|- ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0 <-> ( ( N - 1 ) / 2 ) e. NN0 ) )
Theorem	nn0oddm1d2 11454	A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)
|- ( N e. NN0 -> ( -. 2 || N <-> ( ( N - 1 ) / 2 ) e. NN0 ) )
Theorem	nnoddm1d2 11455	A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)
|- ( N e. NN -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. NN ) )
Theorem	z0even 11456	0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
|- 2 || 0
Theorem	n2dvds1 11457	2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -. 2 || 1
Theorem	n2dvdsm1 11458	2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
|- -. 2 || -u 1
Theorem	z2even 11459	2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
|- 2 || 2
Theorem	n2dvds3 11460	2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
|- -. 2 || 3
Theorem	z4even 11461	4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
|- 2 || 4
Theorem	4dvdseven 11462	An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.)
|- ( 4 || N -> 2 || N )
5.1.3  The division algorithm
Theorem	divalglemnn 11463*	Lemma for divalg 11469. Existence for a positive denominator. (Contributed by Jim Kingdon, 30-Nov-2021.)
|- ( ( N e. ZZ /\ D e. NN ) -> E. r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalglemqt 11464	Lemma for divalg 11469. The Q = T case involved in showing uniqueness. (Contributed by Jim Kingdon, 5-Dec-2021.)
|- ( ph -> D e. ZZ )   &    |- ( ph -> R e. ZZ )   &    |- ( ph -> S e. ZZ )   &    |- ( ph -> Q e. ZZ )   &    |- ( ph -> T e. ZZ )   &    |- ( ph -> Q = T )   &    |- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )   =>    |- ( ph -> R = S )
Theorem	divalglemnqt 11465	Lemma for divalg 11469. The Q < T case involved in showing uniqueness. (Contributed by Jim Kingdon, 4-Dec-2021.)
|- ( ph -> D e. NN )   &    |- ( ph -> R e. ZZ )   &    |- ( ph -> S e. ZZ )   &    |- ( ph -> Q e. ZZ )   &    |- ( ph -> T e. ZZ )   &    |- ( ph -> 0 <_ S )   &    |- ( ph -> R < D )   &    |- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )   =>    |- ( ph -> -. Q < T )
Theorem	divalglemeunn 11466*	Lemma for divalg 11469. Uniqueness for a positive denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
|- ( ( N e. ZZ /\ D e. NN ) -> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalglemex 11467*	Lemma for divalg 11469. The quotient and remainder exist. (Contributed by Jim Kingdon, 30-Nov-2021.)
|- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> E. r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalglemeuneg 11468*	Lemma for divalg 11469. Uniqueness for a negative denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
|- ( ( N e. ZZ /\ D e. ZZ /\ D < 0 ) -> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalg 11469*	The division algorithm (theorem). Dividing an integer N by a nonzero integer D produces a (unique) quotient q and a unique remainder 0 <_ r < ( abs ` D ). Theorem 1.14 in [ApostolNT] p. 19. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalgb 11470*	Express the division algorithm as stated in divalg 11469 in terms of ||. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> ( E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) <-> E! r e. NN0 ( r < ( abs ` D ) /\ D || ( N - r ) ) ) )
Theorem	divalg2 11471*	The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( N e. ZZ /\ D e. NN ) -> E! r e. NN0 ( r < D /\ D || ( N - r ) ) )
Theorem	divalgmod 11472	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 11471 and divalgb 11470). This demonstration theorem justifies the use of mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.)
|- ( ( N e. ZZ /\ D e. NN ) -> ( R = ( N mod D ) <-> ( R e. NN0 /\ ( R < D /\ D || ( N - R ) ) ) ) )
Theorem	divalgmodcl 11473	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor. Variant of divalgmod 11472. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by AV, 21-Aug-2021.)
|- ( ( N e. ZZ /\ D e. NN /\ R e. NN0 ) -> ( R = ( N mod D ) <-> ( R < D /\ D || ( N - R ) ) ) )
Theorem	modremain 11474*	The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.)
|- ( ( N e. ZZ /\ D e. NN /\ ( R e. NN0 /\ R < D ) ) -> ( ( N mod D ) = R <-> E. z e. ZZ ( ( z x. D ) + R ) = N ) )
Theorem	ndvdssub 11475	Corollary of the division algorithm. If an integer D greater than 1 divides N, then it does not divide any of N - 1, N - 2... N - ( D - 1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) -> ( D || N -> -. D || ( N - K ) ) )
Theorem	ndvdsadd 11476	Corollary of the division algorithm. If an integer D greater than 1 divides N, then it does not divide any of N + 1, N + 2... N + ( D - 1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) -> ( D || N -> -. D || ( N + K ) ) )
Theorem	ndvdsp1 11477	Special case of ndvdsadd 11476. If an integer D greater than 1 divides N, it does not divide N + 1. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( N e. ZZ /\ D e. NN /\ 1 < D ) -> ( D || N -> -. D || ( N + 1 ) ) )
Theorem	ndvdsi 11478	A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN   &    |- Q e. NN0   &    |- R e. NN   &    |- ( ( A x. Q ) + R ) = B   &    |- R < A   =>    |- -. A || B
Theorem	flodddiv4 11479	The floor of an odd integer divided by 4. (Contributed by AV, 17-Jun-2021.)
|- ( ( M e. ZZ /\ N = ( ( 2 x. M ) + 1 ) ) -> ( |_ ` ( N / 4 ) ) = if ( 2 || M , ( M / 2 ) , ( ( M - 1 ) / 2 ) ) )
Theorem	fldivndvdslt 11480	The floor of an integer divided by a nonzero integer not dividing the first integer is less than the integer divided by the positive integer. (Contributed by AV, 4-Jul-2021.)
|- ( ( K e. ZZ /\ ( L e. ZZ /\ L =/= 0 ) /\ -. L || K ) -> ( |_ ` ( K / L ) ) < ( K / L ) )
Theorem	flodddiv4lt 11481	The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( |_ ` ( N / 4 ) ) < ( N / 4 ) )
Theorem	flodddiv4t2lthalf 11482	The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( ( |_ ` ( N / 4 ) ) x. 2 ) < ( N / 2 ) )
5.1.4  The greatest common divisor operator
Syntax	cgcd 11483	Extend the definition of a class to include the greatest common divisor operator.
class gcd
Definition	df-gcd 11484*	Define the gcd operator. For example, ( -u 6 gcd 9 ) = 3 (ex-gcd 12636). (Contributed by Paul Chapman, 21-Mar-2011.)
|- gcd = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) )
Theorem	gcdmndc 11485	Decidablity lemma used in various proofs related to gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
Theorem	zsupcllemstep 11486*	Lemma for zsupcl 11488. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.)
|- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps )   =>    |- ( K e. ( ZZ>= ` M ) -> ( ( ( ph /\ A. n e. ( ZZ>= ` K ) -. ps ) -> E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) ) -> ( ( ph /\ A. n e. ( ZZ>= ` ( K + 1 ) ) -. ps ) -> E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) ) ) )
Theorem	zsupcllemex 11487*	Lemma for zsupcl 11488. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.)
|- ( ph -> M e. ZZ )   &    |- ( n = M -> ( ps <-> ch ) )   &    |- ( ph -> ch )   &    |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps )   &    |- ( ph -> E. j e. ( ZZ>= ` M ) A. n e. ( ZZ>= ` j ) -. ps )   =>    |- ( ph -> E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) )
Theorem	zsupcl 11488*	Closure of supremum for decidable integer properties. The property which defines the set we are taking the supremum of must (a) be true at M (which corresponds to the nonempty condition of classical supremum theorems), (b) decidable at each value after M, and (c) be false after j (which corresponds to the upper bound condition found in classical supremum theorems). (Contributed by Jim Kingdon, 7-Dec-2021.)
|- ( ph -> M e. ZZ )   &    |- ( n = M -> ( ps <-> ch ) )   &    |- ( ph -> ch )   &    |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps )   &    |- ( ph -> E. j e. ( ZZ>= ` M ) A. n e. ( ZZ>= ` j ) -. ps )   =>    |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) )
Theorem	zssinfcl 11489*	The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. B -. y < x /\ A. y e. RR ( x < y -> E. z e. B z < y ) ) )   &    |- ( ph -> B C_ ZZ )   &    |- ( ph -> inf ( B , RR , < ) e. ZZ )   =>    |- ( ph -> inf ( B , RR , < ) e. B )
Theorem	infssuzex 11490*	Existence of the infimum of a subset of an upper set of integers. (Contributed by Jim Kingdon, 13-Jan-2022.)
|- ( ph -> M e. ZZ )   &    |- S = { n e. ( ZZ>= ` M ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> E. x e. RR ( A. y e. S -. y < x /\ A. y e. RR ( x < y -> E. z e. S z < y ) ) )
Theorem	infssuzledc 11491*	The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-Jan-2022.)
|- ( ph -> M e. ZZ )   &    |- S = { n e. ( ZZ>= ` M ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) <_ A )
Theorem	infssuzcldc 11492*	The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.)
|- ( ph -> M e. ZZ )   &    |- S = { n e. ( ZZ>= ` M ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) e. S )
Theorem	dvdsbnd 11493*	There is an upper bound to the divisors of a nonzero integer. (Contributed by Jim Kingdon, 11-Dec-2021.)
|- ( ( A e. ZZ /\ A =/= 0 ) -> E. n e. NN A. m e. ( ZZ>= ` n ) -. m || A )
Theorem	gcdsupex 11494*	Existence of the supremum used in defining gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
|- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> E. x e. ZZ ( A. y e. { n e. ZZ | ( n || X /\ n || Y ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ( n || X /\ n || Y ) } y < z ) ) )
Theorem	gcdsupcl 11495*	Closure of the supremum used in defining gcd. A lemma for gcdval 11496 and gcdn0cl 11499. (Contributed by Jim Kingdon, 11-Dec-2021.)
|- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. NN )
Theorem	gcdval 11496*	The value of the gcd operator. ( M gcd N ) is the greatest common divisor of M and N. If M and N are both 0, the result is defined conventionally as 0. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) )
Theorem	gcd0val 11497	The value, by convention, of the gcd operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( 0 gcd 0 ) = 0
Theorem	gcdn0val 11498*	The value of the gcd operator when at least one operand is nonzero. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
Theorem	gcdn0cl 11499	Closure of the gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. NN )
Theorem	gcddvds 11500	The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )