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Theorem List for Metamath Proof Explorer - 28001-28100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremndmaovass 28001 Any operation is associative outside its domain. In contrast to ndmovass 6227 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 28002 Any operation is distributive outside its domain. In contrast to ndmovdistr 6228 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)) ))  )
 
19.22.3  Auxiliary theorems for graph theory

Additional theorems for classical first order logic with equality, ZF set theory and theory of real and complex numbers used for proving the theorems for graph theory.

 
19.22.3.1  Logical disjunction and conjunction
 
Theoremjaoi3 28003 Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018.)
 |-  ( ph  ->  ps )   &    |-  ( ( -.  ph  /\  ch )  ->  ps )   =>    |-  ( ( ph  \/  ch )  ->  ps )
 
19.22.3.2  Negated equality and membership - extension
 
Theoremeqneqall 28004 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( A  =  B  ->  ( A  =/=  B  ->  ph ) )
 
Theoremelnelall 28005 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( A  e.  B  ->  ( A  e/  B  ->  ph ) )
 
19.22.3.3  "Weak deduction theorem" for set theory - extension
 
Theoremifeqda 28006 Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
 |-  (
 ( ph  /\  ps )  ->  A  =  C )   &    |-  ( ( ph  /\  -.  ps )  ->  B  =  C )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  C )
 
Theorem2if2 28007 Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)
 |-  (
 ( ph  /\  ps )  ->  D  =  A )   &    |-  ( ( ph  /\  -.  ps 
 /\  th )  ->  D  =  B )   &    |-  ( ( ph  /\ 
 -.  ps  /\  -.  th )  ->  D  =  C )   =>    |-  ( ph  ->  D  =  if ( ps ,  A ,  if ( th ,  B ,  C ) ) )
 
19.22.3.4  Power classes - extension
 
Theorem3xpexg 28008 The cross product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
 |-  ( V  e.  W  ->  ( ( V  X.  V )  X.  V )  e. 
 _V )
 
19.22.3.5  Unordered and ordered pairs - extension
 
Theoremnelprd 28009 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  -.  A  e.  { B ,  C } )
 
Theorempr1eqbg 28010 A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =  C 
 <->  { A ,  B }  =  { B ,  C } ) )
 
Theorempr1nebg 28011 A (proper) pair is not equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =/=  C  <->  { A ,  B }  =/=  { B ,  C } ) )
 
Theoremraldifsnb 28012* Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( A. x  e.  A  ( x  =/=  Y  ->  ph )  <->  A. x  e.  ( A  \  { Y }
 ) ph )
 
Theoremrexdifpr 28013 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
 |-  ( E. x  e.  ( A  \  { B ,  C } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  x  =/=  C  /\  ph )
 )
 
Theoremel2xptp 28014* A member of a nested cross product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  ( A  e.  ( ( B  X.  C )  X.  D )  <->  E. x  e.  B  E. y  e.  C  E. z  e.  D  A  =  <. x ,  y ,  z >. )
 
Theoremel2xptp0 28015 A member of a nested cross product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  (
 ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( ( A  e.  ( ( U  X.  V )  X.  W ) 
 /\  ( ( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) )  <->  A  =  <. X ,  Y ,  Z >. ) )
 
Theoremotel3xp 28016 An ordered triple is an element of a doubled cross product. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
 |-  (
 ( T  =  <. A ,  B ,  C >.  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) )  ->  T  e.  ( ( X  X.  Y )  X.  Z ) )
 
Theoremoteqimp 28017 The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 ->  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  B  /\  ( 2nd `  T )  =  C )
 ) )
 
Theoremotthg 28018 Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  (
 ( A  e.  U  /\  B  e.  V  /\  C  e.  W )  ->  ( <. A ,  B ,  C >.  =  <. D ,  E ,  F >.  <-> 
 ( A  =  D  /\  B  =  E  /\  C  =  F )
 ) )
 
Theoremotsndisj 28019* The singletons consisting of ordered triples which have distinct third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  (
 ( A  e.  X  /\  B  e.  Y ) 
 -> Disj  c  e.  V { <. A ,  B ,  c >. } )
 
Theoremotiunsndisj 28020* The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  ( B  e.  X  -> Disj  a  e.  V U_ c  e.  ( W  \  {
 a } ) { <. a ,  B ,  c >. } )
 
TheoremotiunsndisjX 28021* The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  ( B  e.  X  -> Disj  a  e.  V U_ c  e.  W  { <. a ,  B ,  c >. } )
 
19.22.3.6  Indexed union and intersection - extension
 
Theoremiunxprg 28022* A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { A ,  B } C  =  ( D  u.  E ) )
 
19.22.3.7  Introduce the Axiom of Union - extension
 
Theoremralxfrd2 28023* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. Variant of ralxfrd 4729. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  y  e.  C  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps 
 <-> 
 A. y  e.  C  ch ) )
 
Theoremrexxfrd2 28024* Transfer existence from a variable 
x to another variable  y contained in expression  A. Variant of rexxfrd 4730. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  y  e.  C  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  B  ps 
 <-> 
 E. y  e.  C  ch ) )
 
19.22.3.8  Relations - extension
 
Theoremresisresindm 28025 The restriction of a relation by a set  B is identical with the restriction by the intersection of  B with the domain of the relation. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
 |-  ( Rel  F  ->  ( F  |`  B )  =  ( F  |`  ( B  i^i  dom  F ) ) )
 
19.22.3.9  Functions - extension
 
Theoremfvifeq 28026 Equality of function values with conditional arguments, see also fvif 5735. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  ( A  =  if ( ph ,  B ,  C )  ->  ( F `  A )  =  if ( ph ,  ( F `
  B ) ,  ( F `  C ) ) )
 
Theorem2f1fvneq 28027 If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  (
 ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C )  /\  A  =/=  B ) 
 ->  ( ( ( E `
  ( F `  A ) )  =  X  /\  ( E `
  ( F `  B ) )  =  Y )  ->  X  =/=  Y ) )
 
Theoremdff14a 28028* A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  ( F `  x )  =/=  ( F `  y ) ) ) )
 
Theoremdff14b 28029* A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  ( A  \  { x } ) ( F `
  x )  =/=  ( F `  y
 ) ) )
 
Theoremf12dfv 28030 A one-to-one function with a pair as domain in terms of function values. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  A  =  { X ,  Y }   =>    |-  ( ( ( X  e.  U  /\  Y  e.  V )  /\  X  =/=  Y )  ->  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  ( F `
  X )  =/=  ( F `  Y ) ) ) )
 
Theoremf13dfv 28031 A one-to-one function with a triple as domain in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  A  =  { X ,  Y ,  Z }   =>    |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) )  ->  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  ( ( F `  X )  =/=  ( F `  Y )  /\  ( F `
  X )  =/=  ( F `  Z )  /\  ( F `  Y )  =/=  ( F `  Z ) ) ) ) )
 
Theoremf13idfv 28032 A one-to-one function with a set of 3 sequential integers as domain in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  A  =  ( 0 ... 2
 )   =>    |-  ( F : A -1-1-> B  <-> 
 ( F : A --> B  /\  ( ( F `
  0 )  =/=  ( F `  1
 )  /\  ( F `  0 )  =/=  ( F `  2 )  /\  ( F `  1 )  =/=  ( F `  2 ) ) ) )
 
Theoremrnfdmpr 28033 The range of a one-to-one function 
F of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  (
 ( X  e.  V  /\  Y  e.  W ) 
 ->  ( F  Fn  { X ,  Y }  ->  ran  F  =  {
 ( F `  X ) ,  ( F `  Y ) } )
 )
 
Theoremimarnf1pr 28034 The image of the range of a function  F under a function  E when  F is a function of a pair into the domain of  E. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  (
 ( X  e.  V  /\  Y  e.  W ) 
 ->  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R ) 
 /\  ( ( E `
  ( F `  X ) )  =  A  /\  ( E `
  ( F `  Y ) )  =  B ) )  ->  ( E " ran  F )  =  { A ,  B } ) )
 
19.22.3.10  Equinumerosity - extension
 
Theoremresfnfinfin 28035 The restriction of a function by a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
 |-  (
 ( F  Fn  A  /\  B  e.  Fin )  ->  ( F  |`  B )  e.  Fin )
 
Theorem3xpfi 28036 The cross product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( V  e.  Fin  ->  (
 ( V  X.  V )  X.  V )  e. 
 Fin )
 
19.22.3.11  Subtraction - extension
 
Theoremcnm1cn 28037 A complex number minus 1 is a complex number. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  ( N  e.  CC  ->  ( N  -  1 )  e.  CC )
 
19.22.3.12  Multiplication - extension
 
Theoremkcnktkm1cn 28038 k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
 
Theorem2txmxeqx 28039 Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( X  e.  CC  ->  ( ( 2  x.  X )  -  X )  =  X )
 
19.22.3.13  Ordering on reals (cont.) - extension
 
Theoremleaddsuble 28040 Addition and subtraction on one side of 'less or equal'. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  ( ( A  +  B )  -  C )  <_  A ) )
 
Theorem2leaddle2 28041 If two real numbers are less than a third real number, the sum of the real numbers is less then twice the third real number. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  C 
 /\  B  <  C )  ->  ( A  +  B )  <  ( 2  x.  C ) ) )
 
Theoremleaddle0 28042 If adding a real number to a real number results in a value less then the second real number, the first real number must be not positive. (Contributed by Alexander van der Vekens, 30-May-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  <_  A  <->  B  <_  0 ) )
 
Theoremltnltne 28043 Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( -.  B  <  A  /\  -.  B  =  A ) ) )
 
19.22.3.14  Nonnegative integers (as a subset of complex numbers) - extension
 
Theorem0mnnnnn0 28044 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 )
 
Theoremlesubnn0 28045 Subtracting a nonnegative integer from a nonnegative integer which is greater than or equal to the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( B  <_  A  ->  ( A  -  B )  e.  NN0 ) )
 
Theoremltsubnn0 28046 Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( B  <  A  ->  ( A  -  B )  e.  NN0 ) )
 
Theoremnn0readdcl 28047 Closure law for addition of reals, restricted to nonnnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  +  B )  e.  RR )
 
Theoremnn0resubcl 28048 Closure law for subtraction of reals, restricted to nonnnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  -  B )  e.  RR )
 
19.22.3.15  Upper partititions of integers
 
Theorem1eluzge0 28049 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  1  e.  ( ZZ>= `  0 )
 
Theorem2eluzge0 28050 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  0 )
 
Theorem2eluzge1 28051 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  1 )
 
19.22.3.16  Finite intervals of integers - extension
 
Theoremssfz12 28052 Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( K  e.  ZZ  /\  L  e.  ZZ  /\  K  <_  L )  ->  ( ( K ... L )  C_  ( M ... N )  ->  ( M  <_  K  /\  L  <_  N ) ) )
 
Theoremelfz2z 28053 Membership of an integer in a finite set of sequential integers starting at 0. (Contributed by Alexander van der Vekens, 25-May-2018.)
 |-  (
 ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  (
 0 ... N )  <->  ( 0  <_  K  /\  K  <_  N ) ) )
 
Theoremelfzmlbm 28054 Subtracting the left border of a finite sets of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
 |-  ( K  e.  ( M ... N )  ->  ( K  -  M )  e.  ( 0 ... ( N  -  M ) ) )
 
Theoremelfzmlbp 28055 Subtracting the lower bound of a finite sets of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
 |-  (
 ( N  e.  ZZ  /\  K  e.  ( M
 ... ( M  +  N ) ) ) 
 ->  ( K  -  M )  e.  ( 0 ... N ) )
 
Theoremzletr 28056 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 ) )
 
Theoremelfzelfzelfz 28057 An element of a finite set of sequential integers is an element of a finite set of sequential integers with the upper bound being an element of the finite set of sequential integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
 |-  (
 ( K  e.  (
 0 ... N )  /\  L  e.  ( K ... N ) )  ->  K  e.  ( 0 ... L ) )
 
Theoremelfzelfzadd 28058 An element of a finite set of sequential integers is an element of an extended finite set of sequential integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( N  e.  (
 0 ... A )  ->  N  e.  ( 0 ... ( A  +  B ) ) ) )
 
Theorem0elfz 28059 0 is an element of a finite set of sequential integers from 0 to a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  ( N  e.  NN0  ->  0  e.  ( 0 ... N ) )
 
Theorem2elfz3nn0 28060 If there are two elements in a finite set of sequential integers from 0, these two elements as well as the upper bound are nonnegative integers. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( A  e.  NN0  /\  B  e.  NN0  /\  N  e.  NN0 ) )
 
Theoremfz0addcom 28061 The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018.) (Revised by Alexander van der Vekens, 9-Jun-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( A  +  B )  =  ( B  +  A ) )
 
Theoremelfz0fzfz0 28062 A member of a finite set of sequential integers starting at 0 is a member of a finite set of sequential integers from 0 to a member of a finite set of sequential integers starting at the right border of the first finite set of sequential integers. (Contributed by Alexander van der Vekens, 27-May-2018.)
 |-  (
 ( M  e.  (
 0 ... L )  /\  N  e.  ( L ... X ) )  ->  M  e.  ( 0 ... N ) )
 
Theoremfzmmmeqm 28063 Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range equals the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
 |-  ( M  e.  ( L ... N )  ->  (
 ( N  -  L )  -  ( M  -  L ) )  =  ( N  -  M ) )
 
Theoremelfzubelfz 28064 If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( M ... N ) )
 
Theorem2elfz2melfz 28065 If the sum of two integers of a finite set of sequential nonnegative integers is greater than the upper bound, the difference between one of the integers and the difference between the upper bound and the other integer is in the finite set of sequential nonnegative integers right bounded by the first integer. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 31-May-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( N  <  ( A  +  B )  ->  ( B  -  ( N  -  A ) )  e.  ( 0 ...
 A ) ) )
 
Theoremfz0fzelfz0 28066 If a member of a finite set of sequential integers with a lower bound being a member of a zero based finite set of sequential integers with the same uppoer bound, this member is also a member of the zero based finite set of sequential integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.)
 |-  (
 ( N  e.  (
 0 ... R )  /\  M  e.  ( N ... R ) )  ->  M  e.  ( 0 ... R ) )
 
Theoremfz0fzdiffz0 28067 The difference of a nonnegative integer in a finite set of sequential integers and a member of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.)
 |-  (
 ( M  e.  (
 0 ... N )  /\  K  e.  ( M ... N ) )  ->  ( K  -  M )  e.  ( 0 ... N ) )
 
Theoremfz0addge0 28068 The sum of two integers in zero based finite sets of sequential integers is greater than or equal to zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  (
 ( A  e.  (
 0 ... M )  /\  B  e.  ( 0 ... N ) )  -> 
 0  <_  ( A  +  B ) )
 
19.22.3.17  Half-open integer ranges - extension
 
Theoremelfzonn0 28069 A member of a half-open integer range starting at 0 is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  ( K  e.  ( 0..^ N )  ->  K  e.  NN0 )
 
Theoremfzo0ss1 28070 Subset relationship for half-open sequences of integers with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  (
 1..^ N )  C_  ( 0..^ N )
 
Theoremfzossnn0 28071 A half open integer range starting from a nonnegative integer is a subset of the nonnegative integers. (Contributed by Alexander van der Vekens, 13-May-2018.)
 |-  ( M  e.  NN0  ->  ( M..^ N )  C_  NN0 )
 
Theoremfzo0sn0fzo1 28072 A half open integer range starting from 0 is the union of the singleton set containing 0 and a half open integer range starting from 1. (Contributed by Alexander van der Vekens, 18-May-2018.)
 |-  ( N  e.  NN  ->  ( 0..^ N )  =  ( { 0 }  u.  ( 1..^ N ) ) )
 
Theoremubmelfzo 28073 If an integer between 0 and an upper bound of a half open interval of integers is subtracted from this upper bound, the result is contained in this half open interval. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  ( K  e.  ( 1..^ N )  ->  ( N  -  K )  e.  ( 0..^ N ) )
 
Theoremubmelm1fzo 28074 If an integer between 0 and an upper bound of a half open interval of integers minus 1 is subtracted from this upper bound, the result is contained in this half open interval. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
 |-  ( K  e.  ( 1..^ N )  ->  ( ( N  -  K )  -  1 )  e.  ( 0..^ N ) )
 
Theoremfzo1fzo0n0 28075 An integer between 1 and an upper bound of a half open interval of integers is not 0 and between 0 and the upper bound of a half open interval of integers. (Contributed by Alexander van der Vekens, 21-Mar-2018.)
 |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( 0..^ N )  /\  K  =/=  0 ) )
 
Theoremelfzomelpfzo 28076 An integer increased by another interger is an element of a half-open range of integers if and only if the integer is contained in the half-open range of integers with bounds decreased by the other integer. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  (
 ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  L  e.  ZZ ) )  ->  ( K  e.  ( ( M  -  L )..^ ( N  -  L ) )  <->  ( K  +  L )  e.  ( M..^ N ) ) )
 
Theoremssfzo12 28077 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( K  e.  ZZ  /\  L  e.  ZZ  /\  K  <  L )  ->  ( ( K..^ L )  C_  ( M..^ N )  ->  ( M  <_  K 
 /\  L  <_  N ) ) )
 
Theoremfzosplitsnm1 28078 Removing a singleton from a half-open range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ( ZZ>= `  ( A  +  1
 ) ) )  ->  ( A..^ B )  =  ( ( A..^ ( B  -  1 ) )  u.  { ( B  -  1 ) }
 ) )
 
Theoremelfzonelfzo 28079 If an element of a half-open range of integers is not contained in the lower subrange, it must be in the upper subrange. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  ( N  e.  ZZ  ->  ( ( K  e.  ( M..^ R )  /\  -.  K  e.  ( M..^ N ) )  ->  K  e.  ( N..^ R ) ) )
 
Theoremfseq0hash 28080 The value of the size function on a finite 0-based sequence. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
 |-  (
 ( N  e.  NN0  /\  F  Fn  ( 0..^ N ) )  ->  ( # `  F )  =  N )
 
Theoremfzonmapblen 28081 The result of subtracting a nonnegative integer from a positive integer and adding another nonnegative integer which is less than the first one, is less then the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.)
 |-  (
 ( A  e.  (
 0..^ N )  /\  B  e.  ( 0..^ N )  /\  B  <  A )  ->  ( B  +  ( N  -  A ) )  <  N )
 
Theoremsubsubelfzo0 28082 Subtracting a difference from a number which is not less than the difference results in a bounded nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  (
 ( A  e.  (
 0..^ N )  /\  I  e.  ( 0..^ N )  /\  -.  I  <  ( N  -  A ) )  ->  ( I  -  ( N  -  A ) )  e.  ( 0..^ A ) )
 
19.22.3.18  The floor (greatest integer) function - extension
 
Theoremnn0nndivcl 28083 Closure law for division 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 )
 
Theoremnn0ge0div 28084 Division of a nonnegative integer by a positive number is positive. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( K  e.  NN0  /\  L  e.  NN )  ->  0  <_  ( K  /  L ) )
 
Theoremfldivnn0 28085 The floor function of a divison of a nonnegative integer by a positive integer is a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( K  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  e.  NN0 )
 
Theoremrefldivcl 28086 The floor function of a divison of a real number by a positive real number is a real number. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( K  e.  RR  /\  L  e.  RR+ )  ->  ( |_ `  ( K  /  L ) )  e.  RR )
 
Theoremfldivle 28087 The floor function of a divison of a real number by a positive real number is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( K  e.  RR  /\  L  e.  RR+ )  ->  ( |_ `  ( K  /  L ) ) 
 <_  ( K  /  L ) )
 
Theoremfldivnn0le 28088 The floor function of a divison of a nonnegative integer by a positive integer is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( K  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) ) 
 <_  ( K  /  L ) )
 
Theoremflltdivnn0lt 28089 The floor function of a divison of a nonnegative integer by a positive integer is less than the division of a greater denominator by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( K  <  N  ->  ( |_ `  ( K  /  L ) )  < 
 ( N  /  L ) ) )
 
Theoremltdifltdiv 28090 If the dividend of a division is less than the difference between a real number and the divisor, the floor function of the division plus 1 is less than the division of the real number by the divisor. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  <  ( C  -  B )  ->  ( ( |_ `  ( A 
 /  B ) )  +  1 )  < 
 ( C  /  B ) ) )
 
19.22.3.19  The modulo (remainder) operation - extension
 
Theoremmodvalr 28091 The value of the modulo operation (multiplication in reversed order). (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  =  ( A  -  ( ( |_ `  ( A  /  B ) )  x.  B ) ) )
 
Theoremflpmodeq 28092 Partition of a division into its integer part and the remainder. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( ( |_ `  ( A  /  B ) )  x.  B )  +  ( A  mod  B ) )  =  A )
 
Theoremmodvalp1 28093 The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  +  B )  -  (
 ( ( |_ `  ( A  /  B ) )  +  1 )  x.  B ) )  =  ( A  mod  B ) )
 
Theorem2submod 28094 If a real number is between a positive real number and the double of the positive real number, the real number modulo the positive real number equals the real number minus the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A 
 /\  A  <  (
 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )
 
Theoremmodaddmod 28095 The sum of a real number modulo a positive real number and another real number equals the sum of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  M  e.  RR+ )  ->  ( ( ( A 
 mod  M )  +  B )  mod  M )  =  ( ( A  +  B )  mod  M ) )
 
Theoremmodadd2mod 28096 The sum of a real number modulo a positive real number and another real number equals the sum of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  M  e.  RR+ )  ->  ( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod  M ) )
 
Theoremmodsubmod 28097 The difference of a real number modulo a positive real number and another real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  M  e.  RR+ )  ->  ( ( ( A 
 mod  M )  -  B )  mod  M )  =  ( ( A  -  B )  mod  M ) )
 
Theoremmodsubmodmod 28098 The difference of a real number modulo a positive real number and another real number modulo this positive real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  M  e.  RR+ )  ->  ( ( ( A 
 mod  M )  -  ( B  mod  M ) ) 
 mod  M )  =  ( ( A  -  B )  mod  M ) )
 
Theoremmodmulmod 28099 The product of a real number modulo a positive real number and an integer equals the product of the real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  ZZ  /\  M  e.  RR+ )  ->  ( ( ( A 
 mod  M )  x.  B )  mod  M )  =  ( ( A  x.  B )  mod  M ) )
 
Theoremmodaddmulmod 28100 The sum of a real number and the product of a second real number modulo a positive real number and an integer equals the sum of the real number and the product of the other real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  ZZ )  /\  M  e.  RR+ )  ->  ( ( A  +  ( ( B  mod  M )  x.  C ) )  mod  M )  =  ( ( A  +  ( B  x.  C ) )  mod  M ) )
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