P. 124 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12301-12400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	n2dvds3 12301	2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
|- -. 2 || 3
Theorem	z4even 12302	4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
|- 2 || 4
Theorem	4dvdseven 12303	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 12304*	Lemma for divalg 12310. 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 12305	Lemma for divalg 12310. 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 12306	Lemma for divalg 12310. 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 12307*	Lemma for divalg 12310. 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 12308*	Lemma for divalg 12310. 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 12309*	Lemma for divalg 12310. 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 12310*	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 12311*	Express the division algorithm as stated in divalg 12310 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 12312*	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 12313	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 12312 and divalgb 12311). 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 12314	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor. Variant of divalgmod 12313. (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 12315*	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 12316	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 12317	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 12318	Special case of ndvdsadd 12317. 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 12319	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 12320	5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 3
Theorem	5ndvds6 12321	5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 6
Theorem	flodddiv4 12322	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 12323	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 12324	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 12325	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 12326	Define the binary bits of an integer.
class bits
Definition	df-bits 12327*	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 12328*	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 12329	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 12330	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 12331	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 12332	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 12333	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 12334	Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( 0 e. (bits ` N ) <-> -. 2 || N ) )
Theorem	bits0e 12335	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 12336	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 12337	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 12338	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 12339	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 12340*	Lemma for bitsfzo 12341. (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 12341	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 12342	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 12343	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 12344	The bit complement of N is -u N - 1. (Thus, by bitsfi 12343, 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 12345	The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
|- (bits ` 0 ) = (/)
Theorem	m1bits 12346	The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- (bits ` -u 1 ) = NN0
Theorem	bitsinv1lem 12347	Lemma for bitsinv1 12348. (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 12348*	There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12344), 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 12349	Extend the definition of a class to include the greatest common divisor operator.
class gcd
Definition	df-gcd 12350*	Define the gcd operator. For example, ( -u 6 gcd 9 ) = 3 (ex-gcd 15806). (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 12351	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 12352*	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 12353*	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 12354*	Closure of the supremum used in defining gcd. A lemma for gcdval 12355 and gcdn0cl 12358. (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 12355*	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 12356	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 12357*	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 12358	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 12359	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 12360	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 12361	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 12362	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 12363	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 12364	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 12365	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 12366*	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 12367	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 12368	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 12369	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 12370	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 12371	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 12372	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 12373	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 12374	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 12375	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 12376	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 12377	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 12378	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 12379	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 12380	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 ) ) ) )
Theorem	gcdadd 12381	The GCD of two numbers is the same as the GCD of the left and their sum. (Contributed by Scott Fenton, 20-Apr-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( N + M ) ) )
Theorem	gcdid 12382	The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( N e. ZZ -> ( N gcd N ) = ( abs ` N ) )
Theorem	gcd1 12383	The gcd of a number with 1 is 1. Theorem 1.4(d)1 in [ApostolNT] p. 16. (Contributed by Mario Carneiro, 19-Feb-2014.)
|- ( M e. ZZ -> ( M gcd 1 ) = 1 )
Theorem	gcdabs 12384	The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
Theorem	gcdabs1 12385	gcd of the absolute value of the first operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( abs ` N ) gcd M ) = ( N gcd M ) )
Theorem	gcdabs2 12386	gcd of the absolute value of the second operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd ( abs ` M ) ) = ( N gcd M ) )
Theorem	modgcd 12387	The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) gcd N ) = ( M gcd N ) )
Theorem	1gcd 12388	The GCD of one and an integer is one. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( M e. ZZ -> ( 1 gcd M ) = 1 )
Theorem	gcdmultipled 12389	The greatest common divisor of a nonnegative integer M and a multiple of it is M itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> M e. NN0 )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M gcd ( N x. M ) ) = M )
Theorem	dvdsgcdidd 12390	The greatest common divisor of a positive integer and another integer it divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> M e. NN )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> M || N )   =>    |- ( ph -> ( M gcd N ) = M )
Theorem	6gcd4e2 12391	The greatest common divisor of six and four is two. To calculate this gcd, a simple form of Euclid's algorithm is used: ( 6 gcd 4 ) = ( ( 4 + 2 ) gcd 4 ) = ( 2 gcd 4 ) and ( 2 gcd 4 ) = ( 2 gcd ( 2 + 2 ) ) = ( 2 gcd 2 ) = 2. (Contributed by AV, 27-Aug-2020.)
|- ( 6 gcd 4 ) = 2
5.1.6  Bézout's identity
Theorem	bezoutlemnewy 12392*	Lemma for Bézout's identity. The is-bezout predicate holds for ( y mod W ). (Contributed by Jim Kingdon, 6-Jan-2022.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   &    |- ( th -> W e. NN )   &    |- ( th -> [ y / r ] ph )   &    |- ( th -> y e. NN0 )   &    |- ( th -> [. W / r ]. ph )   =>    |- ( th -> [. ( y mod W ) / r ]. ph )
Theorem	bezoutlemstep 12393*	Lemma for Bézout's identity. This is the induction step for the proof by induction. (Contributed by Jim Kingdon, 3-Jan-2022.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   &    |- ( th -> W e. NN )   &    |- ( th -> [ y / r ] ph )   &    |- ( th -> y e. NN0 )   &    |- ( th -> [. W / r ]. ph )   &    |- ( ps <-> A. z e. NN0 ( z || r -> ( z || x /\ z || y ) ) )   &    |- ( ( th /\ [. ( y mod W ) / r ]. ph ) -> E. r e. NN0 ( [. ( y mod W ) / x ]. [. W / y ]. ps /\ ph ) )   &    |- F/ x th   &    |- F/ r th   =>    |- ( th -> E. r e. NN0 ( [. W / x ]. ps /\ ph ) )
Theorem	bezoutlemmain 12394*	Lemma for Bézout's identity. This is the main result which we prove by induction and which represents the application of the Extended Euclidean algorithm. (Contributed by Jim Kingdon, 30-Dec-2021.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( ps <-> A. z e. NN0 ( z || r -> ( z || x /\ z || y ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   =>    |- ( th -> A. x e. NN0 ( [ x / r ] ph -> A. y e. NN0 ( [ y / r ] ph -> E. r e. NN0 ( ps /\ ph ) ) ) )
Theorem	bezoutlema 12395*	Lemma for Bézout's identity. The is-bezout condition is satisfied by A. (Contributed by Jim Kingdon, 30-Dec-2021.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   =>    |- ( th -> [. A / r ]. ph )
Theorem	bezoutlemb 12396*	Lemma for Bézout's identity. The is-bezout condition is satisfied by B. (Contributed by Jim Kingdon, 30-Dec-2021.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   =>    |- ( th -> [. B / r ]. ph )
Theorem	bezoutlemex 12397*	Lemma for Bézout's identity. Existence of a number which we will later show to be the greater common divisor and its decomposition into cofactors. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jan-2022.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> E. d e. NN0 ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
Theorem	bezoutlemzz 12398*	Lemma for Bézout's identity. Like bezoutlemex 12397 but where ' z ' is any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
Theorem	bezoutlemaz 12399*	Lemma for Bézout's identity. Like bezoutlemzz 12398 but where ' A ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
|- ( ( A e. ZZ /\ B e. NN0 ) -> E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
Theorem	bezoutlembz 12400*	Lemma for Bézout's identity. Like bezoutlemaz 12399 but where ' B ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )