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Theorem List for Intuitionistic Logic Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoddm1even 11801 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  2  ||  ( N  -  1 ) ) )
 
Theoremoddp1even 11802 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  2  ||  ( N  +  1 ) ) )
 
Theoremoexpneg 11803 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 ) )
 
Theoremmod2eq0even 11804 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 ) )
 
Theoremmod2eq1n2dvds 11805 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 )
 )
 
Theoremoddnn02np1 11806* 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 ) )
 
Theoremoddge22np1 11807* 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 ) )
 
Theoremevennn02n 11808* 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 ) )
 
Theoremevennn2n 11809* 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 ) )
 
Theorem2tp1odd 11810 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 )
 
Theoremmulsucdiv2z 11811 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 )
 
Theoremsqoddm1div8z 11812 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 )
 
Theorem2teven 11813 A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  =  ( 2  x.  A ) )  ->  2  ||  B )
 
Theoremzeo5 11814 An integer is either even or odd, version of zeo3 11794 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 ) ) )
 
Theoremevend2 11815 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 9288 and zeo2 9289. (Contributed by AV, 22-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  <->  ( N  /  2 )  e.  ZZ ) )
 
Theoremoddp1d2 11816 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 9288 and zeo2 9289. (Contributed by AV, 22-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  ( ( N  +  1 )  /  2
 )  e.  ZZ )
 )
 
Theoremzob 11817 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 ) )
 
Theoremoddm1d2 11818 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 )
 )
 
Theoremltoddhalfle 11819 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 ) ) )
 
Theoremhalfleoddlt 11820 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 )
 )
 
Theoremopoe 11821 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 ) )
 
Theoremomoe 11822 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 ) )
 
Theoremopeo 11823 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 ) )
 
Theoremomeo 11824 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 ) )
 
Theoremm1expe 11825 Exponentiation of -1 by an even power. Variant of m1expeven 10493. (Contributed by AV, 25-Jun-2021.)
 |-  ( 2  ||  N  ->  ( -u 1 ^ N )  =  1 )
 
Theoremm1expo 11826 Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)
 |-  ( ( N  e.  ZZ  /\  -.  2  ||  N )  ->  ( -u 1 ^ N )  =  -u 1 )
 
Theoremm1exp1 11827 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 )
 )
 
Theoremnn0enne 11828 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 ) )
 
Theoremnn0ehalf 11829 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 )
 
Theoremnnehalf 11830 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 )
 
Theoremnn0o1gt2 11831 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 ) )
 
Theoremnno 11832 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 )
 
Theoremnn0o 11833 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 )
 
Theoremnn0ob 11834 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 ) )
 
Theoremnn0oddm1d2 11835 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 ) )
 
Theoremnnoddm1d2 11836 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 )
 )
 
Theoremz0even 11837 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
 |-  2  ||  0
 
Theoremn2dvds1 11838 2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- 
 -.  2  ||  1
 
Theoremn2dvdsm1 11839 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
 |- 
 -.  2  ||  -u 1
 
Theoremz2even 11840 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
 |-  2  ||  2
 
Theoremn2dvds3 11841 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
 |- 
 -.  2  ||  3
 
Theoremz4even 11842 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
 |-  2  ||  4
 
Theorem4dvdseven 11843 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
 
Theoremdivalglemnn 11844* Lemma for divalg 11850. 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 )
 ) )
 
Theoremdivalglemqt 11845 Lemma for divalg 11850. 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 )
 
Theoremdivalglemnqt 11846 Lemma for divalg 11850. 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 )
 
Theoremdivalglemeunn 11847* Lemma for divalg 11850. 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
 ) ) )
 
Theoremdivalglemex 11848* Lemma for divalg 11850. 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 )
 ) )
 
Theoremdivalglemeuneg 11849* Lemma for divalg 11850. 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 )
 ) )
 
Theoremdivalg 11850* 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 )
 ) )
 
Theoremdivalgb 11851* Express the division algorithm as stated in divalg 11850 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
 ) ) ) )
 
Theoremdivalg2 11852* 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 ) ) )
 
Theoremdivalgmod 11853 The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor (compare divalg2 11852 and divalgb 11851). 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 ) ) ) ) )
 
Theoremdivalgmodcl 11854 The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor. Variant of divalgmod 11853. (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 ) ) ) )
 
Theoremmodremain 11855* 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 )
 )
 
Theoremndvdssub 11856 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 ) ) )
 
Theoremndvdsadd 11857 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 ) ) )
 
Theoremndvdsp1 11858 Special case of ndvdsadd 11857. 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 )
 ) )
 
Theoremndvdsi 11859 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
 
Theoremflodddiv4 11860 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 ) ) )
 
Theoremfldivndvdslt 11861 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 ) )
 
Theoremflodddiv4lt 11862 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 ) )
 
Theoremflodddiv4t2lthalf 11863 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
 
Syntaxcgcd 11864 Extend the definition of a class to include the greatest common divisor operator.
 class  gcd
 
Definitiondf-gcd 11865* Define the  gcd operator. For example,  ( -u 6  gcd  9 )  =  3 (ex-gcd 13475). (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 ,  <  ) ) )
 
Theoremgcdmndc 11866 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 ) )
 
Theoremzsupcllemstep 11867* Lemma for zsupcl 11869. 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 )
 ) ) ) )
 
Theoremzsupcllemex 11868* Lemma for zsupcl 11869. 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 ) ) )
 
Theoremzsupcl 11869* 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 ) )
 
Theoremzssinfcl 11870* 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 )
 
Theoreminfssuzex 11871* 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
 ) ) )
 
Theoreminfssuzledc 11872* 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 )
 
Theoreminfssuzcldc 11873* 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 )
 
Theoremsuprzubdc 11874* The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
 |-  ( ph  ->  A  C_ 
 ZZ )   &    |-  ( ph  ->  A. x  e.  ZZ DECID  x  e.  A )   &    |-  ( ph  ->  E. x  e.  ZZ  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  B 
 <_  sup ( A ,  RR ,  <  ) )
 
Theoremnninfdcex 11875* A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.)
 |-  ( ph  ->  A  C_ 
 NN )   &    |-  ( ph  ->  A. x  e.  NN DECID  x  e.  A )   &    |-  ( ph  ->  E. y  y  e.  A )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  y  < 
 x  /\  A. y  e. 
 RR  ( x  < 
 y  ->  E. z  e.  A  z  <  y
 ) ) )
 
Theoremzsupssdc 11876* An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7866.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
 |-  ( ph  ->  A  C_ 
 ZZ )   &    |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  A. x  e.  ZZ DECID  x  e.  A )   &    |-  ( ph  ->  E. x  e.  ZZ  A. y  e.  A  y  <_  x )   =>    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  B  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )
 
Theoremsuprzcl2dc 11877* The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 7866.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.)
 |-  ( ph  ->  A  C_ 
 ZZ )   &    |-  ( ph  ->  A. x  e.  ZZ DECID  x  e.  A )   &    |-  ( ph  ->  E. x  e.  ZZ  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  E. x  x  e.  A )   =>    |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  A )
 
Theoremdvdsbnd 11878* 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 )
 
Theoremgcdsupex 11879* 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
 ) ) )
 
Theoremgcdsupcl 11880* Closure of the supremum used in defining  gcd. A lemma for gcdval 11881 and gcdn0cl 11884. (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 )
 
Theoremgcdval 11881* 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 ,  <  ) ) )
 
Theoremgcd0val 11882 The value, by convention, of the 
gcd operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( 0  gcd  0
 )  =  0
 
Theoremgcdn0val 11883* 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 ,  <  ) )
 
Theoremgcdn0cl 11884 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 )
 
Theoremgcddvds 11885 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 ) )
 
Theoremdvdslegcd 11886 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 ) ) )
 
Theoremnndvdslegcd 11887 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 ) ) )
 
Theoremgcdcl 11888 Closure of the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  NN0 )
 
Theoremgcdnncl 11889 Closure of the  gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  N )  e.  NN )
 
Theoremgcdcld 11890 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 )
 
Theoremgcd2n0cl 11891 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 )
 
Theoremzeqzmulgcd 11892* 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 ) ) )
 
Theoremdivgcdz 11893 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 )
 
Theoremgcdf 11894 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
 
Theoremgcdcom 11895 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 ) )
 
Theoremgcdcomd 11896 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 ) )
 
Theoremdivgcdnn 11897 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 )
 
Theoremdivgcdnnr 11898 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 )
 
Theoremgcdeq0 11899 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 ) ) )
 
Theoremgcdn0gt0 11900 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 ) ) )
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