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Theorem List for Metamath Proof Explorer - 38101-38200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxp1d2m1eqxm1d2 38101 A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)
 |-  ( X  e.  CC  ->  ( ( ( X  +  1 )  /  2
 )  -  1 )  =  ( ( X  -  1 )  / 
 2 ) )
 
Theoremelprneb 38102 An element of a proper unordered pair is the first element iff it is not the second element. (Contributed by AV, 18-Jun-2020.)
 |-  (
 ( A  e.  { B ,  C }  /\  B  =/=  C ) 
 ->  ( A  =  B  <->  A  =/=  C ) )
 
Theoremhalfge0 38103 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
 |-  0  <_  ( 1  /  2
 )
 
Theoremleltletr 38104 Transitive law, weaker form of lelttr 9723. (Contributed by AV, 14-Oct-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <_  B 
 /\  B  <  C )  ->  A  <_  C ) )
 
Theoremdeccarry 38105 Add 1 to a 2 digit number with carry. This is a special case of decsucc 11078, but in closed form. As observed by ML, this theorem allows for carrying the 1 down multiple decimal constructors, so we can carry the 1 multiple times down a multi-digit number, e.g. by applying this theorem three times we get  (;; 9 9 9  +  1 )  = ;;; 1 0 0 0. (Contributed by AV, 4-Aug-2020.) (Revised by ML, 8-Aug-2020.)
 |-  ( A  e.  NN  ->  (; A
 9  +  1 )  = ; ( A  +  1
 ) 0 )
 
Theoremnltle2tri 38106 Negated extended trichotomy law for 'less than' and 'less than or equal to'. (Contributed by AV, 18-Jul-2020.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  -.  ( A  <  B  /\  B  <_  C  /\  C  <_  A ) )
 
Theoremzgeltp1eq 38107 If an integer is between another integer and its successor, the integer is equal to the other integer. (Contributed by AV, 30-May-2020.)
 |-  (
 ( I  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( A  <_  I 
 /\  I  <  ( A  +  1 )
 )  ->  I  =  A ) )
 
Theoremsmonoord 38108* Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 12240 (except that the case  M  =  N must be excluded). Duplicate of monoords 37123? (Contributed by AV, 12-Jul-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  ( M  +  1 ) ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  k )  <  ( F `  ( k  +  1 ) ) )   =>    |-  ( ph  ->  ( F `  M )  <  ( F `  N ) )
 
Theoremfzopred 38109 Join a predecessor to the beginning of an open integer interval. Generalization of fzo0sn0fzo1 11998. (Contributed by AV, 14-Jul-2020.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ( M..^ N )  =  ( { M }  u.  ( ( M  +  1 )..^ N ) ) )
 
Theoremfzopredsuc 38110 Join a predecessor and a successor to the beginning and the end of an open integer interval. This theorem holds even if  N  =  M (then  ( M ... N )  =  { M }  =  ( { M }  u.  (/) )  u. 
{ M } ). (Contributed by AV, 14-Jul-2020.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( M
 ... N )  =  ( ( { M }  u.  ( ( M  +  1 )..^ N ) )  u.  { N } ) )
 
Theorem1fzopredsuc 38111 Join 0 and a successor to the beginning and the end of an open integer interval starting at 1. (Contributed by AV, 14-Jul-2020.)
 |-  ( N  e.  NN0  ->  (
 0 ... N )  =  ( ( { 0 }  u.  ( 1..^ N ) )  u. 
 { N } )
 )
 
Theoremel1fzopredsuc 38112 An element of an open integer interval starting at 1 joined by 0 and a successor at the beginning and the end is either 0 or an element of the open integer interval or the successor. (Contributed by AV, 14-Jul-2020.)
 |-  ( N  e.  NN0  ->  ( I  e.  ( 0 ... N )  <->  ( I  =  0  \/  I  e.  ( 1..^ N )  \/  I  =  N ) ) )
 
21.33.3.2  The modulo (remainder) operation (extension)
 
Theoremm1mod0mod1 38113 An integer decreased by 1 is 0 modulo a positive integer iff the integer is 1 modulo the same modulus. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( A  e.  RR  /\  N  e.  RR  /\  1  <  N )  ->  ( ( ( A  -  1 )  mod  N )  =  0  <->  ( A  mod  N )  =  1 ) )
 
Theoremelmod2 38114 An integer modulo 2 is either 0 or 1. (Contributed by AV, 24-May-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)
 |-  ( N  e.  ZZ  ->  ( N  mod  2 )  e.  { 0 ,  1 } )
 
Theoremmod2eq1n2dvds 38115 An integer is 1 modulo 2 iff it is not divisible by 2. (Contributed by AV, 24-May-2020.) (Proof shortened by AV, 5-Jul-2020.)
 |-  ( N  e.  ZZ  ->  ( ( N  mod  2
 )  =  1  <->  -.  2  ||  N ) )
 
Theoremelmod2OLD 38116 An integer modulo 2 is either 0 or 1. (Contributed by AV, 24-May-2020.) Obsolete version of elmod2 38114 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( N  e.  ZZ  ->  ( N  mod  2 )  e.  { 0 ,  1 } )
 
21.33.3.3  Partitions of real intervals

Based on the theorems of the fourierdlem* series of GS's mathbox

 
Syntaxciccp 38117 Extend class notation with the partitions of a closed interval of extended reals.
 class RePart
 
Definitiondf-iccp 38118* Define partitions of a closed interval of extended reals. Such partitions are finite increasing sequences of extended reals. (Contributed by AV, 8-Jul-2020.)
 |- RePart  =  ( m  e.  NN  |->  { p  e.  ( RR*  ^m  ( 0 ... m ) )  |  A. i  e.  ( 0..^ m ) ( p `  i
 )  <  ( p `  ( i  +  1 ) ) } )
 
Theoremiccpval 38119* Partition consisting of a fixed number  M of parts. (Contributed by AV, 9-Jul-2020.)
 |-  ( M  e.  NN  ->  (RePart `  M )  =  { p  e.  ( RR*  ^m  ( 0 ... M ) )  |  A. i  e.  ( 0..^ M ) ( p `  i
 )  <  ( p `  ( i  +  1 ) ) } )
 
Theoremiccpart 38120* A special partition. Corresponds to fourierdlem2 37540 in GS's mathbox. (Contributed by AV, 9-Jul-2020.)
 |-  ( M  e.  NN  ->  ( P  e.  (RePart `  M ) 
 <->  ( P  e.  ( RR*  ^m  ( 0 ...
 M ) )  /\  A. i  e.  ( 0..^ M ) ( P `
  i )  < 
 ( P `  (
 i  +  1 ) ) ) ) )
 
Theoremiccpartimp 38121 Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.)
 |-  (
 ( M  e.  NN  /\  P  e.  (RePart `  M )  /\  I  e.  (
 0..^ M ) ) 
 ->  ( P  e.  ( RR*  ^m  ( 0 ...
 M ) )  /\  ( P `  I )  <  ( P `  ( I  +  1
 ) ) ) )
 
Theoremiccpartres 38122 The restriction of a partition is a partition. (Contributed by AV, 16-Jul-2020.)
 |-  (
 ( M  e.  NN  /\  P  e.  (RePart `  ( M  +  1 )
 ) )  ->  ( P  |`  ( 0 ...
 M ) )  e.  (RePart `  M )
 )
 
Theoremiccpartxr 38123 If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   &    |-  ( ph  ->  I  e.  ( 0 ... M ) )   =>    |-  ( ph  ->  ( P `  I )  e.  RR* )
 
Theoremiccpartgtprec 38124 If there is a partition, then all intermediate points and the upper bound are strictly greater than the preceeding intermediate points or lower bound. (Contributed by AV, 11-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   &    |-  ( ph  ->  I  e.  ( 1 ... M ) )   =>    |-  ( ph  ->  ( P `  ( I  -  1 ) )  < 
 ( P `  I
 ) )
 
Theoremiccpartipre 38125 If there is a partition, then all intermediate points are real numbers. (Contributed by AV, 11-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   &    |-  ( ph  ->  I  e.  ( 1..^ M ) )   =>    |-  ( ph  ->  ( P `  I )  e. 
 RR )
 
Theoremiccpartiltu 38126* If there is a partition, then all intermediate points are strictly less than the upper bound. (Contributed by AV, 12-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  A. i  e.  ( 1..^ M ) ( P `  i
 )  <  ( P `  M ) )
 
Theoremiccpartigtl 38127* If there is a partition, then all intermediate points are strictly greater than the lower bound. (Contributed by AV, 12-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  A. i  e.  ( 1..^ M ) ( P `  0
 )  <  ( P `  i ) )
 
Theoremiccpartlt 38128 If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 37549 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  ( P `  0 )  < 
 ( P `  M ) )
 
Theoremiccpartltu 38129* If there is a partition, then all intermediate points and the lower bound are strictly less than the upper bound. (Contributed by AV, 14-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( P `  i
 )  <  ( P `  M ) )
 
Theoremiccpartgtl 38130* If there is a partition, then all intermediate points and the upper bound are strictly greater than the lower bound. (Contributed by AV, 14-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  A. i  e.  ( 1 ... M ) ( P `  0 )  <  ( P `
  i ) )
 
Theoremiccpartgt 38131* If there is a partition, then all intermediate points and the bounds are strictly ordered. (Contributed by AV, 18-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  A. i  e.  ( 0 ... M ) A. j  e.  (
 0 ... M ) ( i  <  j  ->  ( P `  i )  <  ( P `  j ) ) )
 
Theoremiccpartleu 38132* If there is a partition, then all intermediate points and the lower and the upper bound are less than or equal to the upper bound. (Contributed by AV, 14-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  A. i  e.  ( 0 ... M ) ( P `  i )  <_  ( P `
  M ) )
 
Theoremiccpartgel 38133* If there is a partition, then all intermediate points and the upper and the lower bound are greater than or equal to the lower bound. (Contributed by AV, 14-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  A. i  e.  ( 0 ... M ) ( P `  0 )  <_  ( P `
  i ) )
 
Theoremiccpartrn 38134 If there is a partition, then all intermediate points and bounds are contained in an closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  ran  P  C_  ( ( P `  0 ) [,] ( P `  M ) ) )
 
Theoremiccpartf 38135 The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 37553 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 14-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  P : ( 0 ...
 M ) --> ( ( P `  0 ) [,] ( P `  M ) ) )
 
Theoremiccpartel 38136 If there is a partition, then all intermediate points and bounds are contained in an closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ( ph  /\  I  e.  ( 0 ... M ) )  ->  ( P `
  I )  e.  ( ( P `  0 ) [,] ( P `  M ) ) )
 
Theoremiccelpart 38137* An element of any partitioned half opened interval of extended reals is an element of a part of this partition. (Contributed by AV, 18-Jul-2020.)
 |-  ( M  e.  NN  ->  A. p  e.  (RePart `  M ) ( X  e.  ( ( p `  0 ) [,) ( p `  M ) ) 
 ->  E. i  e.  (
 0..^ M ) X  e.  ( ( p `
  i ) [,) ( p `  (
 i  +  1 ) ) ) ) )
 
Theoremiccpartiun 38138* A half opened interval of extended reals is the union of the parts of its partition. (Contributed by AV, 18-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  (
 ( P `  0
 ) [,) ( P `  M ) )  = 
 U_ i  e.  (
 0..^ M ) ( ( P `  i
 ) [,) ( P `  ( i  +  1
 ) ) ) )
 
Theoremicceuelpartlem 38139 Lemma for icceuelpart 38140. (Contributed by AV, 19-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  ->  (
 ( I  e.  (
 0..^ M )  /\  J  e.  ( 0..^ M ) )  ->  ( I  <  J  ->  ( P `  ( I  +  1 ) ) 
 <_  ( P `  J ) ) ) )
 
Theoremicceuelpart 38140* An element of a partitioned half opened interval of extended reals is an element of exactly one part of the partition. (Contributed by AV, 19-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ( ph  /\  X  e.  ( ( P `  0 ) [,) ( P `  M ) ) )  ->  E! i  e.  ( 0..^ M ) X  e.  ( ( P `  i ) [,) ( P `  ( i  +  1
 ) ) ) )
 
Theoremiccpartdisj 38141* The segments of a partitioned half opened interval of extended reals are a disjoint collection. (Contributed by AV, 19-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   =>    |-  ( ph  -> Disj  i  e.  ( 0..^ M ) ( ( P `  i ) [,) ( P `  ( i  +  1 ) ) ) )
 
Theoremiccpartnel 38142 A point of a partition is not an element of any open interval determined by the partition. Corresponds to fourierdlem12 37550 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 8-Jul-2020.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  P  e.  (RePart `  M )
 )   &    |-  ( ph  ->  X  e.  ran  P )   =>    |-  ( ( ph  /\  I  e.  ( 0..^ M ) )  ->  -.  X  e.  ( ( P `  I ) (,) ( P `  ( I  +  1
 ) ) ) )
 
21.33.4  Even and odd numbers

Even and odd numbers can be characterized in many different ways. In the following, the definition of even and odd numbers is based on the fact that dividing an even number (resp. an odd number increased by 1) by 2 is an integer, see df-even 38145 and df-odd 38146. Alternate definitions resp. charaterizations are provided in dfeven2 38169, dfeven3 38177, dfeven4 38158 and in dfodd2 38156, dfodd3 38170, dfodd4 38178, dfodd5 38179, dfodd6 38157. Each characterization can be useful (and used) in an appropriate context, e.g. dfodd6 38157 in opoeALTV 38202 and dfodd3 38170 in oddprmALTV 38206. Having a fixed definition for even and odd numbers, and alternate characterizations as theorems, advanced theorems about even and/or odd numbers can be expressed more explicitly, and the appropriate characterization can be chosen for their proof, which may become clearer and sometimes also shorter (see, for example, divgcdoddALTV 38201 and divgcdodd 14624).

 
21.33.4.1  Definitions and basic properties
 
Syntaxceven 38143 Extend the definition of a class to include the set of even numbers.
 class Even
 
Syntaxcodd 38144 Extend the definition of a class to include the set of odd numbers.
 class Odd
 
Definitiondf-even 38145 Define the set of even numbers. (Contributed by AV, 14-Jun-2020.)
 |- Even  =  {
 z  e.  ZZ  |  ( z  /  2
 )  e.  ZZ }
 
Definitiondf-odd 38146 Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( ( z  +  1 )  /  2
 )  e.  ZZ }
 
Theoremiseven 38147 The predicate "is an even number". An even number is an integer which is divisible by 2, i.e. the result of dividing the even integer by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Even  <->  ( Z  e.  ZZ  /\  ( Z  / 
 2 )  e.  ZZ ) )
 
Theoremisodd 38148 The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd integer increased by 1 and then divided by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Odd  <->  ( Z  e.  ZZ  /\  ( ( Z  +  1 )  / 
 2 )  e.  ZZ ) )
 
Theoremevenz 38149 An even number is an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Even  ->  Z  e.  ZZ )
 
Theoremoddz 38150 An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Odd  ->  Z  e.  ZZ )
 
Theoremevendiv2z 38151 The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
 |-  ( Z  e. Even  ->  ( Z 
 /  2 )  e. 
 ZZ )
 
Theoremoddp1div2z 38152 The result of dividing an odd number increased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
 |-  ( Z  e. Odd  ->  ( ( Z  +  1 ) 
 /  2 )  e. 
 ZZ )
 
Theoremzob 38153 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 ) )
 
Theoremoddm1div2z 38154 The result of dividing an odd number decreased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
 |-  ( Z  e. Odd  ->  ( ( Z  -  1 ) 
 /  2 )  e. 
 ZZ )
 
Theoremisodd2 38155 The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd number decreased by 1 and then divided by 2 is still an integer. (Contributed by AV, 15-Jun-2020.)
 |-  ( Z  e. Odd  <->  ( Z  e.  ZZ  /\  ( ( Z  -  1 )  / 
 2 )  e.  ZZ ) )
 
Theoremdfodd2 38156 Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( ( z  -  1 )  /  2
 )  e.  ZZ }
 
Theoremdfodd6 38157* Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  E. i  e.  ZZ  z  =  ( (
 2  x.  i )  +  1 ) }
 
Theoremdfeven4 38158* Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
 |- Even  =  {
 z  e.  ZZ  |  E. i  e.  ZZ  z  =  ( 2  x.  i ) }
 
Theoremevenm1odd 38159 The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Even  ->  ( Z  -  1 )  e. Odd 
 )
 
Theoremevenp1odd 38160 The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Even  ->  ( Z  +  1 )  e. Odd 
 )
 
Theoremoddp1eveni 38161 The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Odd  ->  ( Z  +  1 )  e. Even 
 )
 
Theoremoddm1eveni 38162 The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.)
 |-  ( Z  e. Odd  ->  ( Z  -  1 )  e. Even 
 )
 
Theoremevennodd 38163 An even number is not an odd number. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Even  ->  -.  Z  e. Odd  )
 
Theoremoddneven 38164 An odd number is not an even number. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Odd  ->  -.  Z  e. Even  )
 
Theoremenege 38165 The negative of an even number is even. (Contributed by AV, 20-Jun-2020.)
 |-  ( A  e. Even  ->  -u A  e. Even  )
 
Theoremonego 38166 The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.)
 |-  ( A  e. Odd  ->  -u A  e. Odd  )
 
Theoremm1expevenALTV 38167 Exponentiation of -1 by an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 6-Jul-2020.)
 |-  ( N  e. Even  ->  ( -u 1 ^ N )  =  1 )
 
Theoremm1expoddALTV 38168 Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.)
 |-  ( N  e. Odd  ->  ( -u 1 ^ N )  =  -u 1 )
 
21.33.4.2  Alternate definitions using the "divides" relation
 
Theoremdfeven2 38169 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
 |- Even  =  {
 z  e.  ZZ  | 
 2  ||  z }
 
Theoremdfodd3 38170 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  -.  2  ||  z }
 
Theoremiseven2 38171 The predicate "is an even number". An even number is an integer which is divisible by 2. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Even  <->  ( Z  e.  ZZ  /\  2  ||  Z ) )
 
Theoremisodd3 38172 The predicate "is an odd number". An odd number is an integer which is not divisible by 2. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Odd  <->  ( Z  e.  ZZ  /\  -.  2  ||  Z ) )
 
Theorem2dvdseven 38173 2 divides an even number. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Even  ->  2  ||  Z )
 
Theorem2ndvdsodd 38174 2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Odd  ->  -.  2  ||  Z )
 
Theorem2dvdsoddp1 38175 2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Odd  ->  2  ||  ( Z  +  1
 ) )
 
Theorem2dvdsoddm1 38176 2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Odd  ->  2  ||  ( Z  -  1
 ) )
 
21.33.4.3  Alternate definitions using the "modulo" operation
 
Theoremdfeven3 38177 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
 |- Even  =  {
 z  e.  ZZ  |  ( z  mod  2 )  =  0 }
 
Theoremdfodd4 38178 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( z  mod  2 )  =  1 }
 
Theoremdfodd5 38179 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( z  mod  2 )  =/=  0 }
 
Theoremzefldiv2ALTV 38180 The floor of an even number divided by 2 is equal to the even number divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
 |-  ( N  e. Even  ->  ( |_ `  ( N  /  2
 ) )  =  ( N  /  2 ) )
 
Theoremzofldiv2ALTV 38181 The floor of an odd numer divided by 2 is equal to the odd number first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
 |-  ( N  e. Odd  ->  ( |_ `  ( N  /  2
 ) )  =  ( ( N  -  1
 )  /  2 )
 )
 
TheoremoddflALTV 38182 Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 18-Jun-2020.)
 |-  ( K  e. Odd  ->  K  =  ( ( 2  x.  ( |_ `  ( K  /  2 ) ) )  +  1 ) )
 
21.33.4.4  Alternate definitions using the "gcd" operation
 
Theoremgcdzeq 38183 A positive integer  A is equal to its gcd with an integer  B if and only if  A divides  B. Generalization of gcdeq 14491. (Contributed by AV, 1-Jul-2020.)
 |-  (
 ( A  e.  NN  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  =  A  <->  A  ||  B ) )
 
Theoremiseven5 38184 The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
 |-  ( Z  e. Even  <->  ( Z  e.  ZZ  /\  ( 2  gcd 
 Z )  =  2 ) )
 
Theoremisodd7 38185 The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
 |-  ( Z  e. Odd  <->  ( Z  e.  ZZ  /\  ( 2  gcd 
 Z )  =  1 ) )
 
Theoremdfeven5 38186 Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.)
 |- Even  =  {
 z  e.  ZZ  |  ( 2  gcd  z
 )  =  2 }
 
Theoremdfodd7 38187 Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( 2  gcd  z
 )  =  1 }
 
21.33.4.5  Theorems of part 5 revised
 
TheoremzneoALTV 38188 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.) (Revised by AV, 16-Jun-2020.)
 |-  (
 ( A  e. Even  /\  B  e. Odd  )  ->  A  =/=  B )
 
TheoremzeoALTV 38189 An integer is even or odd. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 16-Jun-2020.)
 |-  ( Z  e.  ZZ  ->  ( Z  e. Even  \/  Z  e. Odd  ) )
 
Theoremzeo2ALTV 38190 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) (Revised by AV, 16-Jun-2020.)
 |-  ( Z  e.  ZZ  ->  ( Z  e. Even  <->  -.  Z  e. Odd  )
 )
 
TheoremnneoALTV 38191 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  NN  ->  ( N  e. Even  <->  -.  N  e. Odd  )
 )
 
TheoremnneoiALTV 38192 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.) (Revised by AV, 19-Jun-2020.)
 |-  N  e.  NN   =>    |-  ( N  e. Even  <->  -.  N  e. Odd  )
 
21.33.4.6  Theorems of part 6 revised
 
Theoremodd2np1ALTV 38193* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( N  e. Odd  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
 
Theoremoddm1evenALTV 38194 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( N  e. Odd  <->  ( N  -  1 )  e. Even  ) )
 
Theoremoddp1evenALTV 38195 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( N  e. Odd  <->  ( N  +  1 )  e. Even  ) )
 
TheoremoexpnegALTV 38196 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) (Revised by AV, 19-Jun-2020.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN  /\  N  e. Odd  )  ->  (
 -u A ^ N )  =  -u ( A ^ N ) )
 
Theoremoexpnegnz 38197 The exponential of the negative of a number not being 0, when the exponent is odd. (Contributed by AV, 19-Jun-2020.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  N  e. Odd  )  ->  (
 -u A ^ N )  =  -u ( A ^ N ) )
 
Theorembits0ALTV 38198 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( 0  e.  (bits `  N )  <->  N  e. Odd  ) )
 
Theorembits0eALTV 38199 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e. Even  ->  -.  0  e.  (bits `  N )
 )
 
Theorembits0oALTV 38200 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e. Odd  ->  0  e.  (bits `  N )
 )
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