P. 125 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12401-12500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opoe 12401	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 12402	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 12403	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 12404	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 12405	Exponentiation of -1 by an even power. Variant of m1expeven 10803. (Contributed by AV, 25-Jun-2021.)
|- ( 2 || N -> ( -u 1 ^ N ) = 1 )
Theorem	m1expo 12406	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 12407	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 12408	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 12409	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 12410	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 12411	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 12412	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 12413	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 12414	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 12415	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 12416	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 12417	0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
|- 2 || 0
Theorem	n2dvds1 12418	2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -. 2 || 1
Theorem	n2dvdsm1 12419	2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
|- -. 2 || -u 1
Theorem	z2even 12420	2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
|- 2 || 2
Theorem	n2dvds3 12421	2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
|- -. 2 || 3
Theorem	z4even 12422	4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
|- 2 || 4
Theorem	4dvdseven 12423	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 12424*	Lemma for divalg 12430. 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 12425	Lemma for divalg 12430. 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 12426	Lemma for divalg 12430. 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 12427*	Lemma for divalg 12430. 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 12428*	Lemma for divalg 12430. 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 12429*	Lemma for divalg 12430. 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 12430*	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 12431*	Express the division algorithm as stated in divalg 12430 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 12432*	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 12433	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 12432 and divalgb 12431). 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 12434	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor. Variant of divalgmod 12433. (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 12435*	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 12436	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 12437	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 12438	Special case of ndvdsadd 12437. 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 12439	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	5ndvds3 12440	5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 3
Theorem	5ndvds6 12441	5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 6
Theorem	flodddiv4 12442	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 12443	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 12444	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 12445	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  Bit sequences
Syntax	cbits 12446	Define the binary bits of an integer.
class bits
Definition	df-bits 12447*	Define the binary bits of an integer. The expression M e. (bits ` N ) means that the M-th bit of N is 1 (and its negation means the bit is 0). (Contributed by Mario Carneiro, 4-Sep-2016.)
|- bits = ( n e. ZZ |-> { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) } )
Theorem	bitsfval 12448*	Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> (bits ` N ) = { m e. NN0 | -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) } )
Theorem	bitsval 12449	Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( M e. (bits ` N ) <-> ( N e. ZZ /\ M e. NN0 /\ -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) )
Theorem	bitsval2 12450	Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( M e. (bits ` N ) <-> -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) )
Theorem	bitsss 12451	The set of bits of an integer is a subset of NN0. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- (bits ` N ) C_ NN0
Theorem	bitsf 12452	The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- bits : ZZ --> ~P NN0
Theorem	bitsdc 12453	Whether a bit is set is decidable. (Contributed by Jim Kingdon, 31-Oct-2025.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> DECID M e. (bits ` N ) )
Theorem	bits0 12454	Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( 0 e. (bits ` N ) <-> -. 2 || N ) )
Theorem	bits0e 12455	The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> -. 0 e. (bits ` ( 2 x. N ) ) )
Theorem	bits0o 12456	The zeroth bit of an odd number is one. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> 0 e. (bits ` ( ( 2 x. N ) + 1 ) ) )
Theorem	bitsp1 12457	The M + 1-th bit of N is the M-th bit of |_ ( N / 2 ). (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( M + 1 ) e. (bits ` N ) <-> M e. (bits ` ( |_ ` ( N / 2 ) ) ) ) )
Theorem	bitsp1e 12458	The M + 1-th bit of 2 N is the M-th bit of N. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( M + 1 ) e. (bits ` ( 2 x. N ) ) <-> M e. (bits ` N ) ) )
Theorem	bitsp1o 12459	The M + 1-th bit of 2 N + 1 is the M-th bit of N. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( M + 1 ) e. (bits ` ( ( 2 x. N ) + 1 ) ) <-> M e. (bits ` N ) ) )
Theorem	bitsfzolem 12460*	Lemma for bitsfzo 12461. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 1-Oct-2020.)
|- ( ph -> N e. NN0 )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> (bits ` N ) C_ ( 0..^ M ) )   &    |- S = inf ( { n e. NN0 | N < ( 2 ^ n ) } , RR , < )   =>    |- ( ph -> N e. ( 0..^ ( 2 ^ M ) ) )
Theorem	bitsfzo 12461	The bits of a number are all at positions less than M iff the number is nonnegative and less than 2 ^ M. (Contributed by Mario Carneiro, 5-Sep-2016.) (Proof shortened by AV, 1-Oct-2020.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( N e. ( 0..^ ( 2 ^ M ) ) <-> (bits ` N ) C_ ( 0..^ M ) ) )
Theorem	bitsmod 12462	Truncating the bit sequence after some M is equivalent to reducing the argument mod 2 ^ M. (Contributed by Mario Carneiro, 6-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> (bits ` ( N mod ( 2 ^ M ) ) ) = ( (bits ` N ) i^i ( 0..^ M ) ) )
Theorem	bitsfi 12463	Every number is associated with a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. NN0 -> (bits ` N ) e. Fin )
Theorem	bitscmp 12464	The bit complement of N is -u N - 1. (Thus, by bitsfi 12463, all negative numbers have cofinite bits representations.) (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( NN0 \ (bits ` N ) ) = (bits ` ( -u N - 1 ) ) )
Theorem	0bits 12465	The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
|- (bits ` 0 ) = (/)
Theorem	m1bits 12466	The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- (bits ` -u 1 ) = NN0
Theorem	bitsinv1lem 12467	Lemma for bitsinv1 12468. (Contributed by Mario Carneiro, 22-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( N mod ( 2 ^ ( M + 1 ) ) ) = ( ( N mod ( 2 ^ M ) ) + if ( M e. (bits ` N ) , ( 2 ^ M ) , 0 ) ) )
Theorem	bitsinv1 12468*	There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12464), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)
|- ( N e. NN0 -> sum_ n e. (bits ` N ) ( 2 ^ n ) = N )
5.1.5  The greatest common divisor operator
Syntax	cgcd 12469	Extend the definition of a class to include the greatest common divisor operator.
class gcd
Definition	df-gcd 12470*	Define the gcd operator. For example, ( -u 6 gcd 9 ) = 3 (ex-gcd 16053). (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 12471	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	dvdsbnd 12472*	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 12473*	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 12474*	Closure of the supremum used in defining gcd. A lemma for gcdval 12475 and gcdn0cl 12478. (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 12475*	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 12476	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 12477*	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 12478	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 12479	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 ) )
Theorem	dvdslegcd 12480	An integer which divides both operands of the gcd operator is bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( K || M /\ K || N ) -> K <_ ( M gcd N ) ) )
Theorem	nndvdslegcd 12481	A positive integer which divides both positive operands of the gcd operator is bounded by it. (Contributed by AV, 9-Aug-2020.)
|- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( ( K || M /\ K || N ) -> K <_ ( M gcd N ) ) )
Theorem	gcdcl 12482	Closure of the gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
Theorem	gcdnncl 12483	Closure of the gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
|- ( ( M e. NN /\ N e. NN ) -> ( M gcd N ) e. NN )
Theorem	gcdcld 12484	Closure of the gcd operator. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M gcd N ) e. NN0 )
Theorem	gcd2n0cl 12485	Closure of the gcd operator if the second operand is not 0. (Contributed by AV, 10-Jul-2021.)
|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M gcd N ) e. NN )
Theorem	zeqzmulgcd 12486*	An integer is the product of an integer and the gcd of it and another integer. (Contributed by AV, 11-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> E. n e. ZZ A = ( n x. ( A gcd B ) ) )
Theorem	divgcdz 12487	An integer divided by the gcd of it and a nonzero integer is an integer. (Contributed by AV, 11-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( A / ( A gcd B ) ) e. ZZ )
Theorem	gcdf 12488	Domain and codomain of the gcd operator. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- gcd : ( ZZ X. ZZ ) --> NN0
Theorem	gcdcom 12489	The gcd operator is commutative. Theorem 1.4(a) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( N gcd M ) )
Theorem	gcdcomd 12490	The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M gcd N ) = ( N gcd M ) )
Theorem	divgcdnn 12491	A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( A / ( A gcd B ) ) e. NN )
Theorem	divgcdnnr 12492	A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( A / ( B gcd A ) ) e. NN )
Theorem	gcdeq0 12493	The gcd of two integers is zero iff they are both zero. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) = 0 <-> ( M = 0 /\ N = 0 ) ) )
Theorem	gcdn0gt0 12494	The gcd of two integers is positive (nonzero) iff they are not both zero. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) <-> 0 < ( M gcd N ) ) )
Theorem	gcd0id 12495	The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )
Theorem	gcdid0 12496	The gcd of an integer and 0 is the integer's absolute value. Theorem 1.4(d)2 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( N e. ZZ -> ( N gcd 0 ) = ( abs ` N ) )
Theorem	nn0gcdid0 12497	The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( N e. NN0 -> ( N gcd 0 ) = N )
Theorem	gcdneg 12498	Negating one operand of the gcd operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) = ( M gcd N ) )
Theorem	neggcd 12499	Negating one operand of the gcd operator does not alter the result. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M gcd N ) = ( M gcd N ) )
Theorem	gcdaddm 12500	Adding a multiple of one operand of the gcd operator to the other does not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( N + ( K x. M ) ) ) )