HomeHome Intuitionistic Logic Explorer
Theorem List (p. 84 of 106)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0nndivcl 8301 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  Integers (as a subset of complex numbers)
 
Syntaxcz 8302 Extend class notation to include the class of integers.
 class  ZZ
 
Definitiondf-z 8303 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 8304 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 8305 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  ( N  e.  NN  -> 
 -u N  e.  ZZ )
 
Theoremzre 8306 An integer is a real. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  ->  N  e.  RR )
 
Theoremzcn 8307 An integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  ->  N  e.  CC )
 
Theoremzrei 8308 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
 |-  A  e.  ZZ   =>    |-  A  e.  RR
 
Theoremzssre 8309 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  RR
 
Theoremzsscn 8310 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  CC
 
Theoremzex 8311 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 ZZ  e.  _V
 
Theoremelnnz 8312 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  <  N ) )
 
Theorem0z 8313 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  0  e.  ZZ
 
Theorem0zd 8314 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ph  ->  0  e.  ZZ )
 
Theoremelnn0z 8315 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  <->  ( N  e.  ZZ  /\  0  <_  N ) )
 
Theoremelznn0nn 8316 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 8317 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 8318 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 8319 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
 |- 
 NN  C_  ZZ
 
Theoremnn0ssz 8320 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  ZZ
 
Theoremnnz 8321 A positive integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN  ->  N  e.  ZZ )
 
Theoremnn0z 8322 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  N  e.  ZZ )
 
Theoremnnzi 8323 A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   =>    |-  N  e.  ZZ
 
Theoremnn0zi 8324 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  N  e.  ZZ
 
Theoremelnnz1 8325 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 8326 Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN  =  { x  e.  ZZ  |  1  <_  x }
 
Theoremnn0zrab 8327 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN0  =  { x  e.  ZZ  |  0  <_  x }
 
Theorem1z 8328 One is an integer. (Contributed by NM, 10-May-2004.)
 |-  1  e.  ZZ
 
Theorem1zzd 8329 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( ph  ->  1  e.  ZZ )
 
Theorem2z 8330 Two is an integer. (Contributed by NM, 10-May-2004.)
 |-  2  e.  ZZ
 
Theorem3z 8331 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  3  e.  ZZ
 
Theorem4z 8332 4 is an integer. (Contributed by BJ, 26-Mar-2020.)
 |-  4  e.  ZZ
 
Theoremznegcl 8333 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  -> 
 -u N  e.  ZZ )
 
Theoremneg1z 8334 -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  -u 1  e.  ZZ
 
Theoremznegclb 8335 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 8336 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  -u N  e.  ZZ )
 
Theoremnn0negzi 8337 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  -u N  e.  ZZ
 
Theorempeano2z 8338 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
 |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
 
Theoremzaddcllempos 8339 Lemma for zaddcl 8342. 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 8340 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
 |-  ( N  e.  ZZ  ->  ( N  -  1
 )  e.  ZZ )
 
Theoremzaddcllemneg 8341 Lemma for zaddcl 8342. 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 8342 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 8343 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N )  e.  ZZ )
 
Theoremztri3or0 8344 Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( N  e.  ZZ  ->  ( N  <  0  \/  N  =  0  \/  0  <  N ) )
 
Theoremztri3or 8345 Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
 
Theoremzletric 8346 Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <_  B  \/  B  <_  A ) )
 
Theoremzlelttric 8347 Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremzltnle 8348 '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 8349 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 8350 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 8351 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 8352 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 8353 The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8028.) (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <-> 
 ( N  -  M )  e.  NN )
 )
 
Theoremnzadd 8354 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 ) )
 
Theoremzmulcl 8355 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
 
Theoremzltp1le 8356 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <-> 
 ( M  +  1 )  <_  N )
 )
 
Theoremzleltp1 8357 Integer ordering relation. (Contributed by NM, 10-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  M  <  ( N  +  1 ) ) )
 
Theoremzlem1lt 8358 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <-> 
 ( M  -  1
 )  <  N )
 )
 
Theoremzltlem1 8359 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  M  <_  ( N  -  1 ) ) )
 
Theoremzgt0ge1 8360 An integer greater than  0 is greater than or equal to  1. (Contributed by AV, 14-Oct-2018.)
 |-  ( Z  e.  ZZ  ->  ( 0  <  Z  <->  1 
 <_  Z ) )
 
Theoremnnleltp1 8361 Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <_  B  <->  A  <  ( B  +  1 ) ) )
 
Theoremnnltp1le 8362 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  <-> 
 ( A  +  1 )  <_  B )
 )
 
Theoremnnaddm1cl 8363 Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  +  B )  -  1 )  e.  NN )
 
Theoremnn0ltp1le 8364 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <  N  <->  ( M  +  1 ) 
 <_  N ) )
 
Theoremnn0leltp1 8365 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  M  <  ( N  +  1 ) ) )
 
Theoremnn0ltlem1 8366 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <  N  <->  M 
 <_  ( N  -  1
 ) ) )
 
Theoremznn0sub 8367 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 8368.) (Contributed by NM, 14-Jul-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <-> 
 ( N  -  M )  e.  NN0 ) )
 
Theoremnn0sub 8368 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( N  -  M )  e.  NN0 ) )
 
Theoremnn0n0n1ge2 8369 A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  -> 
 2  <_  N )
 
Theoremelz2 8370* Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  ZZ  <->  E. x  e.  NN  E. y  e.  NN  N  =  ( x  -  y ) )
 
Theoremdfz2 8371 Alternative definition of the integers, based on elz2 8370. (Contributed by Mario Carneiro, 16-May-2014.)
 |- 
 ZZ  =  (  -  " ( NN  X.  NN ) )
 
Theoremnn0sub2 8372 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0  /\  M  <_  N )  ->  ( N  -  M )  e. 
 NN0 )
 
Theoremzapne 8373 Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M #  N  <->  M  =/=  N ) )
 
Theoremzdceq 8374 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  =  B )
 
Theoremzdcle 8375 Integer  <_ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  <_  B )
 
Theoremzdclt 8376 Integer  < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  <  B )
 
Theoremzltlen 8377 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7695 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  <-> 
 ( A  <_  B  /\  B  =/=  A ) ) )
 
Theoremnn0n0n1ge2b 8378 A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
 |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1
 ) 
 <->  2  <_  N )
 )
 
Theoremnn0lt10b 8379 A nonnegative integer less than  1 is  0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN0  ->  ( N  <  1  <->  N  =  0
 ) )
 
Theoremnn0lt2 8380 A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
 |-  ( ( N  e.  NN0  /\  N  <  2 ) 
 ->  ( N  =  0  \/  N  =  1 ) )
 
Theoremnn0lem1lt 8381 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( M  -  1 )  <  N ) )
 
Theoremnnlem1lt 8382 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <_  N  <-> 
 ( M  -  1
 )  <  N )
 )
 
Theoremnnltlem1 8383 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <  N  <->  M  <_  ( N  -  1 ) ) )
 
Theoremnnm1ge0 8384 A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)
 |-  ( N  e.  NN  ->  0  <_  ( N  -  1 ) )
 
Theoremnn0ge0div 8385 Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  0  <_  ( K  /  L ) )
 
Theoremzdiv 8386* Two ways to express " M divides  N. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( M  x.  k )  =  N  <->  ( N  /  M )  e.  ZZ ) )
 
Theoremzdivadd 8387 Property of divisibility: if  D divides  A and  B then it divides  A  +  B. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( ( D  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  /  D )  e.  ZZ  /\  ( B  /  D )  e. 
 ZZ ) )  ->  ( ( A  +  B )  /  D )  e.  ZZ )
 
Theoremzdivmul 8388 Property of divisibility: if  D divides  A then it divides  B  x.  A. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( ( D  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A 
 /  D )  e. 
 ZZ )  ->  (
 ( B  x.  A )  /  D )  e. 
 ZZ )
 
Theoremzextle 8389* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\ 
 A. k  e.  ZZ  ( k  <_  M  <->  k  <_  N ) )  ->  M  =  N )
 
Theoremzextlt 8390* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\ 
 A. k  e.  ZZ  ( k  <  M  <->  k  <  N ) )  ->  M  =  N )
 
Theoremrecnz 8391 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
 |-  ( ( A  e.  RR  /\  1  <  A )  ->  -.  ( 1  /  A )  e.  ZZ )
 
Theorembtwnnz 8392 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
 |-  ( ( A  e.  ZZ  /\  A  <  B  /\  B  <  ( A  +  1 ) ) 
 ->  -.  B  e.  ZZ )
 
Theoremgtndiv 8393 A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A ) 
 ->  -.  ( B  /  A )  e.  ZZ )
 
Theoremhalfnz 8394 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
 |- 
 -.  ( 1  / 
 2 )  e.  ZZ
 
Theorem3halfnz 8395 Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
 |- 
 -.  ( 3  / 
 2 )  e.  ZZ
 
Theoremprime 8396* Two ways to express " A is a prime number (or 1)." (Contributed by NM, 4-May-2005.)
 |-  ( A  e.  NN  ->  ( A. x  e. 
 NN  ( ( A 
 /  x )  e. 
 NN  ->  ( x  =  1  \/  x  =  A ) )  <->  A. x  e.  NN  ( ( 1  < 
 x  /\  x  <_  A 
 /\  ( A  /  x )  e.  NN )  ->  x  =  A ) ) )
 
Theoremmsqznn 8397 The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  ZZ  /\  A  =/=  0
 )  ->  ( A  x.  A )  e.  NN )
 
Theoremzneo 8398 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A )  =/=  (
 ( 2  x.  B )  +  1 )
 )
 
Theoremnneoor 8399 A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
 |-  ( N  e.  NN  ->  ( ( N  / 
 2 )  e.  NN  \/  ( ( N  +  1 )  /  2
 )  e.  NN )
 )
 
Theoremnneo 8400 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
 |-  ( N  e.  NN  ->  ( ( N  / 
 2 )  e.  NN  <->  -.  ( ( N  +  1 )  /  2
 )  e.  NN )
 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10511
  Copyright terms: Public domain < Previous  Next >