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Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfnrnafv 38701* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 5933. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F''' x ) } )
 
Theoremafvelrnb 38702* A member of a function's range is a value of the function, analogous to fvelrnb 5934 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  V ) 
 ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F''' x )  =  B ) )
 
Theoremafvelrnb0 38703* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 5934. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
 |-  ( F  Fn  A  ->  ( B  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  B ) )
 
Theoremdfaimafn 38704* Alternate definition of the image of a function, analogous to dfimafn 5936. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  C_  dom  F )  ->  ( F " A )  =  { y  |  E. x  e.  A  ( F''' x )  =  y } )
 
Theoremdfaimafn2 38705* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5937. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  C_  dom  F )  ->  ( F " A )  =  U_ x  e.  A  { ( F''' x ) } )
 
Theoremafvelima 38706* Function value in an image, analogous to fvelima 5939. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  ( F " B ) )  ->  E. x  e.  B  ( F''' x )  =  A )
 
Theoremafvelrn 38707 A function's value belongs to its range, analogous to fvelrn 6037. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( F''' A )  e.  ran  F )
 
Theoremfnafvelrn 38708 A function's value belongs to its range, analogous to fnfvelrn 6041. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( F''' B )  e.  ran  F )
 
Theoremfafvelrn 38709 A function's value belongs to its codomain, analogous to ffvelrn 6042. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B )
 
Theoremffnafv 38710* A function maps to a class to which all values belong, analogous to ffnfv 6071. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
 
Theoremafvres 38711 The value of a restricted function, analogous to fvres 5901. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  e.  B  ->  ( ( F  |`  B )''' A )  =  ( F''' A ) )
 
Theoremtz6.12-afv 38712* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 5904. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
 |-  (
 ( <. A ,  y >.  e.  F  /\  E! y <. A ,  y >.  e.  F )  ->  ( F''' A )  =  y )
 
Theoremtz6.12-1-afv 38713* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 5903. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
 |-  (
 ( A F y 
 /\  E! y  A F y )  ->  ( F''' A )  =  y
 )
 
Theoremdmfcoafv 38714 Domains of a function composition, analogous to dmfco 5961. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  (
 ( Fun  G  /\  A  e.  dom  G ) 
 ->  ( A  e.  dom  ( F  o.  G ) 
 <->  ( G''' A )  e.  dom  F ) )
 
Theoremafvco2 38715 Value of a function composition, analogous to fvco2 5962. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  (
 ( G  Fn  A  /\  X  e.  A ) 
 ->  ( ( F  o.  G )''' X )  =  ( F''' ( G''' X ) ) )
 
Theoremrlimdmafv 38716 Two ways to express that a function has a limit, analogous to rlimdm 13663. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = +oo )   =>    |-  ( ph  ->  ( F  e.  dom  ~~> r  <->  F  ~~> r  (  ~~> r ''' F ) ) )
 
21.33.2.8  Alternative definition of the value of an operation
 
Theoremaoveq123d 38717 Equality deduction for operation value, analogous to oveq123d 6335. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  -> (( A F C))  = (( B G D))  )
 
Theoremnfaov 38718 Bound-variable hypothesis builder for operation value, analogous to nfov 6340. To prove a deduction version of this analogous to nfovd 6339 is not quickly possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of alternative operation values is based on are not available (see nfafv 38675). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x B   =>    |-  F/_ x (( A F B))
 
Theoremcsbaovg 38719 Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( A  e.  D  ->  [_ A  /  x ]_ (( B F C))  = (( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C))  )
 
Theoremaovfundmoveq 38720 If a class is a function restricted to an ordered pair of its domain, then the value of the operation on this pair is equal for both definitions. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( F defAt  <. A ,  B >.  -> (( A F B))  =  ( A F B ) )
 
Theoremaovnfundmuv 38721 If an ordered pair is not in the domain of a class or the class is not a function restricted to the ordered pair, then the operation value for this pair is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  F defAt  <. A ,  B >.  -> (( A F B))  =  _V )
 
Theoremndmaov 38722 The value of an operation outside its domain, analogous to ndmafv 38679. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  <. A ,  B >.  e.  dom  F  -> (( A F B))  =  _V )
 
Theoremndmaovg 38723 The value of an operation outside its domain, analogous to ndmovg 6478. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( dom  F  =  ( R  X.  S ) 
 /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> (( A F B))  =  _V )
 
Theoremaovvdm 38724 If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  e.  C  -> 
 <. A ,  B >.  e. 
 dom  F )
 
Theoremnfunsnaov 38725 If the restriction of a class to a singleton is not a function, its operation value is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  Fun  ( F  |`  { <. A ,  B >. } )  -> (( A F B))  =  _V )
 
Theoremaovvfunressn 38726 If the operation value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  e.  C  ->  Fun  ( F  |`  { <. A ,  B >. } )
 )
 
Theoremaovprc 38727 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 6344. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  Rel  dom 
 F   =>    |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> (( A F B))  =  _V )
 
Theoremaovrcl 38728 Reverse closure for an operation value, analogous to afvvv 38684. In contrast to ovrcl 6347, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  Rel  dom 
 F   =>    |-  ( (( A F B))  e.  C  ->  ( A  e.  _V  /\  B  e.  _V ) )
 
Theoremaovpcov0 38729 If the alternative value of the operation on an ordered pair is the universal class, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  =  _V  ->  ( A F B )  =  (/) )
 
Theoremaovnuoveq 38730 The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  =/=  _V  -> (( A F B))  =  ( A F B ) )
 
Theoremaovvoveq 38731 The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  e.  C  -> (( A F B))  =  ( A F B ) )
 
Theoremaov0ov0 38732 If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  =  (/)  ->  ( A F B )  =  (/) )
 
Theoremaovovn0oveq 38733 If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( A F B )  =/=  (/)  -> (( A F B))  =  ( A F B ) )
 
Theoremaov0nbovbi 38734 The operation's value on an ordered pair is an element of a set if and only if the alternative value of the operation on this ordered pair is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (/)  e/  C  ->  ( (( A F B))  e.  C  <->  ( A F B )  e.  C ) )
 
Theoremaovov0bi 38735 The operation's value on an ordered pair is the empty set if and only if the alternative value of the operation on this ordered pair is either the empty set or the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( A F B )  =  (/)  <->  ( (( A F B))  =  (/)  \/ (( A F B))  =  _V ) )
 
Theoremrspceaov 38736* A frequently used special case of rspc2ev 3172 for operation values, analogous to rspceov 6353. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( C  e.  A  /\  D  e.  B  /\  S  = (( C F D))  )  ->  E. x  e.  A  E. y  e.  B  S  = (( x F y))  )
 
Theoremfnotaovb 38737 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5928. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( (( C F D))  =  R  <->  <. C ,  D ,  R >.  e.  F ) )
 
Theoremffnaov 38738* An operation maps to a class to which all values belong, analogous to ffnov 6426. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B ) 
 /\  A. x  e.  A  A. y  e.  B (( x F y))  e.  C ) )
 
Theoremfaovcl 38739 Closure law for an operation, analogous to fovcl 6427. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F : ( R  X.  S ) --> C   =>    |-  ( ( A  e.  R  /\  B  e.  S )  -> (( A F B))  e.  C )
 
Theoremaovmpt4g 38740* Value of a function given by the "maps to" notation, analogous to ovmpt4g 6445. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  -> (( x F y))  =  C )
 
Theoremaoprssdm 38741* Domain of closure of an operation. In contrast to oprssdm 6476, no additional property for S (
-.  (/)  e.  S) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( x  e.  S  /\  y  e.  S )  -> (( x F y))  e.  S )   =>    |-  ( S  X.  S )  C_  dom  F
 
Theoremndmaovcl 38742 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6480 where it is required that the domain contains the empty set ( (/) 
e.  S). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   &    |-  ( ( A  e.  S  /\  B  e.  S )  -> (( A F B))  e.  S )   &    |- (( A F B))  e.  _V   =>    |- (( A F B))  e.  S
 
Theoremndmaovrcl 38743 Reverse closure law, in contrast to ndmovrcl 6481 where it is required that the operation's domain doesn't contain the empty set ( -.  (/)  e.  S), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   =>    |-  ( (( A F B))  e.  S  ->  ( A  e.  S  /\  B  e.  S ) )
 
Theoremndmaovcom 38744 Any operation is commutative outside its domain, analogous to ndmovcom 6482. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S )  -> (( A F B))  = (( B F A))  )
 
Theoremndmaovass 38745 Any operation is associative outside its domain. In contrast to ndmovass 6483 where it is required that the operation's domain doesn't contain the empty set ( -.  (/)  e.  S), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  -> (( (( A F B))  F C))  = (( A F (( B F C)) ))  )
 
Theoremndmaovdistr 38746 Any operation is distributive outside its domain. In contrast to ndmovdistr 6484 where it is required that the operation's domain doesn't contain the empty set (
-.  (/)  e.  S), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   &    |-  dom  G  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) 
 -> (( A G (( B F C)) ))  = (( (( A G B))  F (( A G C)) ))  )
 
21.33.3  General auxiliary theorems
 
21.33.3.1  Miscellanea
 
Theorem1t10e1p1e11 38747 11 is 1 times 10 to the power of 1, plus 1. (Contributed by AV, 4-Aug-2020.)
 |- ; 1 1  =  ( ( 1  x.  ( 10 ^ 1 ) )  +  1 )
 
Theoremxp1d2m1eqxm1d2 38748 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 38749 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 38750 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
 |-  0  <_  ( 1  /  2
 )
 
Theoremleltletr 38751 Transitive law, weaker form of lelttr 9749. (Contributed by AV, 14-Oct-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <_  B 
 /\  B  <  C )  ->  A  <_  C ) )
 
Theoremdeccarry 38752 Add 1 to a 2 digit number with carry. This is a special case of decsucc 11106, 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 38753 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 38754 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 38755* Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 12274 (except that the case  M  =  N must be excluded). Duplicate of monoords 37551? (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 38756 Join a predecessor to the beginning of an open integer interval. Generalization of fzo0sn0fzo1 12031. (Contributed by AV, 14-Jul-2020.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ( M..^ N )  =  ( { M }  u.  ( ( M  +  1 )..^ N ) ) )
 
Theoremfzopredsuc 38757 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 38758 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 38759 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 38760 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 38761 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 38762 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 38763 An integer modulo 2 is either 0 or 1. (Contributed by AV, 24-May-2020.) Obsolete version of elmod2 38761 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 38764 Extend class notation with the partitions of a closed interval of extended reals.
 class RePart
 
Definitiondf-iccp 38765* 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 38766* 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 38767* A special partition. Corresponds to fourierdlem2 38008 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 38768 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 38769 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 38770 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 38771 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 38772 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 38773* 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 38774* 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 38775 If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 38017 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 38776* 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 38777* 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 38778* 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 38779* 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 38780* 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 38781 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 38782 The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 38021 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 38783 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 38784* 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 38785* 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 38786 Lemma for icceuelpart 38787. (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 38787* 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 38788* 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 38789 A point of a partition is not an element of any open interval determined by the partition. Corresponds to fourierdlem12 38018 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 38792 and df-odd 38793. Alternate definitions resp. charaterizations are provided in dfeven2 38816, dfeven3 38824, dfeven4 38805 and in dfodd2 38803, dfodd3 38817, dfodd4 38825, dfodd5 38826, dfodd6 38804. Each characterization can be useful (and used) in an appropriate context, e.g. dfodd6 38804 in opoeALTV 38849 and dfodd3 38817 in oddprmALTV 38853. 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 38848 and divgcdodd 14701).

 
21.33.4.1  Definitions and basic properties
 
Syntaxceven 38790 Extend the definition of a class to include the set of even numbers.
 class Even
 
Syntaxcodd 38791 Extend the definition of a class to include the set of odd numbers.
 class Odd
 
Definitiondf-even 38792 Define the set of even numbers. (Contributed by AV, 14-Jun-2020.)
 |- Even  =  {
 z  e.  ZZ  |  ( z  /  2
 )  e.  ZZ }
 
Definitiondf-odd 38793 Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( ( z  +  1 )  /  2
 )  e.  ZZ }
 
Theoremiseven 38794 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 38795 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 38796 An even number is an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Even  ->  Z  e.  ZZ )
 
Theoremoddz 38797 An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Odd  ->  Z  e.  ZZ )
 
Theoremevendiv2z 38798 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 38799 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 38800 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 ) )
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