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Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnnn0addcl 8801 A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  NN )
 
Theoremnn0nnaddcl 8802 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN )  ->  ( M  +  N )  e.  NN )
 
Theorem0mnnnnn0 8803 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
 |-  ( N  e.  NN  ->  ( 0  -  N )  e/  NN0 )
 
Theoremun0addcl 8804 If  S is closed under addition, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  T  =  ( S  u.  { 0 } )   &    |-  ( ( ph  /\  ( M  e.  S  /\  N  e.  S ) )  ->  ( M  +  N )  e.  S )   =>    |-  ( ( ph  /\  ( M  e.  T  /\  N  e.  T )
 )  ->  ( M  +  N )  e.  T )
 
Theoremun0mulcl 8805 If  S is closed under multiplication, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  T  =  ( S  u.  { 0 } )   &    |-  ( ( ph  /\  ( M  e.  S  /\  N  e.  S ) )  ->  ( M  x.  N )  e.  S )   =>    |-  ( ( ph  /\  ( M  e.  T  /\  N  e.  T )
 )  ->  ( M  x.  N )  e.  T )
 
Theoremnn0addcl 8806 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  NN0 )
 
Theoremnn0mulcl 8807 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  x.  N )  e.  NN0 )
 
Theoremnn0addcli 8808 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( M  +  N )  e.  NN0
 
Theoremnn0mulcli 8809 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( M  x.  N )  e.  NN0
 
Theoremnn0p1nn 8810 A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN )
 
Theorempeano2nn0 8811 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN0 )
 
Theoremnnm1nn0 8812 A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN  ->  ( N  -  1
 )  e.  NN0 )
 
Theoremelnn0nn 8813 The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  <->  ( N  e.  CC  /\  ( N  +  1 )  e.  NN ) )
 
Theoremelnnnn0 8814 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
 |-  ( N  e.  NN  <->  ( N  e.  CC  /\  ( N  -  1 )  e. 
 NN0 ) )
 
Theoremelnnnn0b 8815 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  0  <  N ) )
 
Theoremelnnnn0c 8816 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
 
Theoremnn0addge1 8817 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  A  <_  ( A  +  N )
 )
 
Theoremnn0addge2 8818 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  A  <_  ( N  +  A )
 )
 
Theoremnn0addge1i 8819 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  A  e.  RR   &    |-  N  e.  NN0   =>    |-  A  <_  ( A  +  N )
 
Theoremnn0addge2i 8820 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  A  e.  RR   &    |-  N  e.  NN0   =>    |-  A  <_  ( N  +  A )
 
Theoremnn0le2xi 8821 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  N  e.  NN0   =>    |-  N  <_  ( 2  x.  N )
 
Theoremnn0lele2xi 8822 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( N  <_  M  ->  N  <_  ( 2  x.  M ) )
 
Theoremnn0supp 8823 Two ways to write the support of a function on  NN0. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( F : I --> NN0  ->  ( `' F " ( _V  \  {
 0 } ) )  =  ( `' F " NN ) )
 
Theoremnnnn0d 8824 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  NN0 )
 
Theoremnn0red 8825 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnn0cnd 8826 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremnn0ge0d 8827 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremnn0addcld 8828 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   =>    |-  ( ph  ->  ( A  +  B )  e.  NN0 )
 
Theoremnn0mulcld 8829 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   =>    |-  ( ph  ->  ( A  x.  B )  e. 
 NN0 )
 
Theoremnn0readdcl 8830 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  +  B )  e.  RR )
 
Theoremnn0ge2m1nn 8831 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)
 |-  ( ( N  e.  NN0  /\  2  <_  N ) 
 ->  ( N  -  1
 )  e.  NN )
 
Theoremnn0ge2m1nn0 8832 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
 |-  ( ( N  e.  NN0  /\  2  <_  N ) 
 ->  ( N  -  1
 )  e.  NN0 )
 
Theoremnn0nndivcl 8833 Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( K  /  L )  e.  RR )
 
3.4.8  Extended nonnegative integers

The function values of the hash (set size) function are either nonnegative integers or positive infinity. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers  RR*, see df-xr 7623.

 
Syntaxcxnn0 8834 The set of extended nonnegative integers.
 class NN0*
 
Definitiondf-xnn0 8835 Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers  RR*, see df-xr 7623. If we assumed excluded middle, this would be essentially the same as ℕ as defined at df-nninf 6878 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
 |- NN0*  =  ( NN0  u.  { +oo } )
 
Theoremelxnn0 8836 An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  <->  ( A  e.  NN0 
 \/  A  = +oo ) )
 
Theoremnn0ssxnn0 8837 The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
 |- 
 NN0  C_ NN0*
 
Theoremnn0xnn0 8838 A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e.  NN0  ->  A  e. NN0* )
 
Theoremxnn0xr 8839 An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  A  e.  RR* )
 
Theorem0xnn0 8840 Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |-  0  e. NN0*
 
Theorempnf0xnn0 8841 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |- +oo  e. NN0*
 
Theoremnn0nepnf 8842 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e.  NN0  ->  A  =/= +oo )
 
Theoremnn0xnn0d 8843 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e. NN0* )
 
Theoremnn0nepnfd 8844 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  =/= +oo )
 
Theoremxnn0nemnf 8845 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  A  =/= -oo )
 
Theoremxnn0xrnemnf 8846 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  ( A  e.  RR*  /\  A  =/= -oo ) )
 
Theoremxnn0nnn0pnf 8847 An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( ( N  e. NN0*  /\ 
 -.  N  e.  NN0 )  ->  N  = +oo )
 
3.4.9  Integers (as a subset of complex numbers)
 
Syntaxcz 8848 Extend class notation to include the class of integers.
 class  ZZ
 
Definitiondf-z 8849 Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.)
 |- 
 ZZ  =  { n  e.  RR  |  ( n  =  0  \/  n  e.  NN  \/  -u n  e.  NN ) }
 
Theoremelz 8850 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )
 ) )
 
Theoremnnnegz 8851 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  ( N  e.  NN  -> 
 -u N  e.  ZZ )
 
Theoremzre 8852 An integer is a real. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  ->  N  e.  RR )
 
Theoremzcn 8853 An integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  ->  N  e.  CC )
 
Theoremzrei 8854 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
 |-  A  e.  ZZ   =>    |-  A  e.  RR
 
Theoremzssre 8855 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  RR
 
Theoremzsscn 8856 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  CC
 
Theoremzex 8857 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 ZZ  e.  _V
 
Theoremelnnz 8858 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  <  N ) )
 
Theorem0z 8859 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  0  e.  ZZ
 
Theorem0zd 8860 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ph  ->  0  e.  ZZ )
 
Theoremelnn0z 8861 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  <->  ( N  e.  ZZ  /\  0  <_  N ) )
 
Theoremelznn0nn 8862 Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
 |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
 
Theoremelznn0 8863 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
 
Theoremelznn 8864 Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 )
 ) )
 
Theoremnnssz 8865 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
 |- 
 NN  C_  ZZ
 
Theoremnn0ssz 8866 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  ZZ
 
Theoremnnz 8867 A positive integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN  ->  N  e.  ZZ )
 
Theoremnn0z 8868 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  N  e.  ZZ )
 
Theoremnnzi 8869 A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   =>    |-  N  e.  ZZ
 
Theoremnn0zi 8870 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  N  e.  ZZ
 
Theoremelnnz1 8871 Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  1  <_  N ) )
 
Theoremnnzrab 8872 Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN  =  { x  e.  ZZ  |  1  <_  x }
 
Theoremnn0zrab 8873 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN0  =  { x  e.  ZZ  |  0  <_  x }
 
Theorem1z 8874 One is an integer. (Contributed by NM, 10-May-2004.)
 |-  1  e.  ZZ
 
Theorem1zzd 8875 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( ph  ->  1  e.  ZZ )
 
Theorem2z 8876 Two is an integer. (Contributed by NM, 10-May-2004.)
 |-  2  e.  ZZ
 
Theorem3z 8877 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  3  e.  ZZ
 
Theorem4z 8878 4 is an integer. (Contributed by BJ, 26-Mar-2020.)
 |-  4  e.  ZZ
 
Theoremznegcl 8879 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  -> 
 -u N  e.  ZZ )
 
Theoremneg1z 8880 -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  -u 1  e.  ZZ
 
Theoremznegclb 8881 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  CC  ->  ( A  e.  ZZ  <->  -u A  e.  ZZ ) )
 
Theoremnn0negz 8882 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  -u N  e.  ZZ )
 
Theoremnn0negzi 8883 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  -u N  e.  ZZ
 
Theorempeano2z 8884 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
 |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
 
Theoremzaddcllempos 8885 Lemma for zaddcl 8888. Special case in which  N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  +  N )  e.  ZZ )
 
Theorempeano2zm 8886 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
 |-  ( N  e.  ZZ  ->  ( N  -  1
 )  e.  ZZ )
 
Theoremzaddcllemneg 8887 Lemma for zaddcl 8888. Special case in which  -u N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  RR  /\  -u N  e.  NN )  ->  ( M  +  N )  e.  ZZ )
 
Theoremzaddcl 8888 Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N )  e.  ZZ )
 
Theoremzsubcl 8889 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N )  e.  ZZ )
 
Theoremztri3or0 8890 Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( N  e.  ZZ  ->  ( N  <  0  \/  N  =  0  \/  0  <  N ) )
 
Theoremztri3or 8891 Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
 
Theoremzletric 8892 Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <_  B  \/  B  <_  A ) )
 
Theoremzlelttric 8893 Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremzltnle 8894 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  <->  -.  B  <_  A )
 )
 
Theoremzleloe 8895 Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <_  B  <-> 
 ( A  <  B  \/  A  =  B ) ) )
 
Theoremznnnlt1 8896 An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)
 |-  ( N  e.  ZZ  ->  ( -.  N  e.  NN 
 <->  N  <  1 ) )
 
Theoremzletr 8897 Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
 |-  ( ( J  e.  ZZ  /\  K  e.  ZZ  /\  L  e.  ZZ )  ->  ( ( J  <_  K 
 /\  K  <_  L )  ->  J  <_  L ) )
 
Theoremzrevaddcl 8898 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
 |-  ( N  e.  ZZ  ->  ( ( M  e.  CC  /\  ( M  +  N )  e.  ZZ ) 
 <->  M  e.  ZZ )
 )
 
Theoremznnsub 8899 The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8559.) (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <-> 
 ( N  -  M )  e.  NN )
 )
 
Theoremnzadd 8900 The sum of a real number not being an integer and an integer is not an integer. Note that "not being an integer" in this case means "the negation of is an integer" rather than "is apart from any integer" (given excluded middle, those two would be equivalent). (Contributed by AV, 19-Jul-2021.)
 |-  ( ( A  e.  ( RR  \  ZZ )  /\  B  e.  ZZ )  ->  ( A  +  B )  e.  ( RR  \  ZZ ) )
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