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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
19.23.3.13  Operations - extension
 
Theoremoprabv 28201* If a pair and a class are in a relationship given by a class abstraction of a collection of nested ordered pairs, the involved classes are sets. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
 |-  ( <. X ,  Y >. {
 <. <. x ,  y >. ,  z >.  |  ph } Z  ->  ( X  e.  _V  /\  Y  e.  _V 
 /\  Z  e.  _V ) )
 
Theoremelovmpt2rab 28202* Implications for the value of an operation defined by the maps-to notation with a class abstration as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  M  |  ph
 } )   &    |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  M  e.  _V )   =>    |-  ( Z  e.  ( X O Y )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  M )
 )
 
Theoremelovmpt2rab1 28203* Implications for the value of an operation defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  [_ x  /  m ]_ M  |  ph
 } )   &    |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  [_ X  /  m ]_ M  e.  _V )   =>    |-  ( Z  e.  ( X O Y )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  [_ X  /  m ]_ M ) )
 
Theoremovmpt3rab1 28204* The value of an operation defined by the maps-to notation with a function into a class abstraction as a result. Here, the base set of the class abstraction depends on the first operand. (Contributed by Alexander van der Vekens, 16-Jul-2018.)
 |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  [_ x  /  m ]_ M  |->  { a  e.  [_ x  /  n ]_ N  |  ph
 } ) )   &    |-  (
 ( X  e.  _V  /\  Y  e.  _V )  -> 
 [_ X  /  m ]_ M  e.  _V )   =>    |-  (
 ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  ( z  e.  [_ X  /  m ]_ M  |->  { a  e.  [_ X  /  n ]_ N  |  [. X  /  x ].
 [. Y  /  y ]. ph } ) )
 
Theoremelovmpt3rab1 28205* Implications for the value of an operation defined by the maps-to notation with a function into a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the first operand. (Contributed by Alexander van der Vekens, 16-Jul-2018.)
 |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  [_ x  /  m ]_ M  |->  { a  e.  [_ x  /  n ]_ N  |  ph
 } ) )   &    |-  (
 ( X  e.  _V  /\  Y  e.  _V )  -> 
 [_ X  /  m ]_ M  e.  _V )   &    |-  (
 ( X  e.  _V  /\  Y  e.  _V )  -> 
 [_ X  /  n ]_ N  e.  _V )   =>    |-  ( A  e.  ( ( X O Y ) `  Z )  ->  ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( Z  e.  [_ X  /  m ]_ M  /\  A  e.  [_ X  /  n ]_ N ) ) )
 
Theoremelovmpt3rab 28206* Implications for the value of an operation defined by the maps-to notation with a class abstration as a result having an element. (Contributed by Alexander van der Vekens, 17-Jul-2018.)
 |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  M  |->  { a  e.  N  |  ph
 } ) )   &    |-  (
 ( X  e.  _V  /\  Y  e.  _V )  ->  M  e.  _V )   &    |-  (
 ( X  e.  _V  /\  Y  e.  _V )  ->  N  e.  _V )   =>    |-  ( A  e.  ( ( X O Y ) `  Z )  ->  ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( Z  e.  M  /\  A  e.  N ) ) )
 
19.23.3.14  Equinumerosity - extension
 
Theoremresfnfinfin 28207 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 28208 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 )
 
19.23.3.15  Subtraction - extension
 
Theoremcnm1cn 28209 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 )
 
19.23.3.16  Multiplication - extension
 
Theoremkcnktkm1cn 28210 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 28211 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 )
 
19.23.3.17  Ordering on reals (cont.) - extension
 
Theoremleaddsuble 28212 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 28213 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 28214 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 28215 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 ) ) )
 
19.23.3.18  Nonnegative integers (as a subset of complex numbers) - extension
 
Theorem0mnnnnn0 28216 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 28217 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 28218 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 28219 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 28220 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 )
 
19.23.3.19  Upper partititions of integers - extension
 
Theorem1eluzge0 28221 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  1  e.  ( ZZ>= `  0 )
 
Theorem2eluzge0 28222 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  0 )
 
Theorem2eluzge1 28223 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  1 )
 
Theoremuzletr 28224 An upper integer is also an upper integer with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
 |-  (
 ( A  e.  ZZ  /\  A  <_  B )  ->  ( N  e.  ( ZZ>=
 `  B )  ->  N  e.  ( ZZ>= `  A ) ) )
 
19.23.3.20  Finite intervals of integers - extension
 
Theoremssfz12 28225 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 28226 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 28227 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 28228 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 28229 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 28230 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 28231 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 28232 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 28233 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 28234 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 28235 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 28236 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 28237 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 28238 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 28239 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 28240 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 28241 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 ) )
 
Theorem2ffzeq 28242* Two functions over a zero-based finite interval of integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
 |-  (
 ( M  e.  NN0  /\  F : ( 0
 ... M ) --> X  /\  P : ( 0 ...
 N ) --> Y ) 
 ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0
 ... M ) ( F `  i )  =  ( P `  i ) ) ) )
 
19.23.3.21  Half-open integer ranges - extension
 
Theoremelfzonn0 28243 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 28244 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 28245 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 28246 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 28247 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 28248 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 28249 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 28250 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 28251 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 28252 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 28253 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 28254 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 28255 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 28256 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 ) )
 
Theoremfzofzim 28257 If a non-negative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open range of integers. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
 |-  (
 ( K  =/=  M  /\  K  e.  ( 0
 ... M ) ) 
 ->  K  e.  ( 0..^ M ) )
 
Theoremfzisfzounsn 28258 A finite interval of integers as union of a half-open range of integers and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
 |-  ( B  e.  ( ZZ>= `  A )  ->  ( A
 ... B )  =  ( ( A..^ B )  u.  { B }
 ) )
 
Theoremel2fzo 28259 The lower limit of a half-open range of integers which is equal to a non-empty empty half-open range of integers is element of the half-open range. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ( ( M..^ N )  =  ( J..^ K )  ->  J  e.  ( J..^ K ) ) )
 
Theoremfzoopth 28260 A half-open range of integers can represent an ordered pair, analogous to fzopth 11127. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ( ( M..^ N )  =  ( J..^ K )  <->  ( M  =  J  /\  N  =  K ) ) )
 
Theorem2ffzoeq 28261* Two functions over a zero-based half-open integer range are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N )
 --> Y ) )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `
  i )  =  ( P `  i
 ) ) ) )
 
19.23.3.22  The floor (greatest integer) function - extension
 
Theoremnn0nndivcl 28262 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 28263 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 28264 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 28265 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 28266 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 28267 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 28268 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 28269 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 ) ) )
 
19.23.3.23  The modulo (remainder) operation - extension
 
Theoremmodvalr 28270 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 28271 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 28272 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 28273 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 28274 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 28275 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 28276 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 28277 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 28278 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 28279 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 ) )
 
Theoremmodid0 28280 A positive real number modulo itself is 0 . (Contributed by Alexander van der Vekens, 15-May-2018.)
 |-  ( N  e.  RR+  ->  ( N  mod  N )  =  0 )
 
Theoremmodidmul0 28281 The product of an integer and a positive integer is 0 modulo the positive integer. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN )  ->  ( ( A  x.  N )  mod  N )  =  0 )
 
Theoremmodifeq2int 28282 If a nonnegative integer is less than the double of a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  ->  ( A  mod  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
 
Theorem2txmodxeq0 28283 Two times a positive real number module the real number is zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( X  e.  RR+  ->  (
 ( 2  x.  X )  mod  X )  =  0 )
 
19.23.3.24  The ` # ` (finite set size) function - extension
 
Theoremhashimarn 28284 The size of the image of a one-to-one function  E under the range of a function  F which is a one-to-one function into the domain of  E equals the size of the function  F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
 |-  (
 ( E : dom  E
 -1-1-> ran  E  /\  E  e.  V )  ->  ( F : ( 0..^ ( # `  F ) )
 -1-1-> dom  E  ->  ( # `
  ( E " ran  F ) )  =  ( # `  F ) ) )
 
Theoremhashimarni 28285 If the size of the image of a one-to-one function  E under the range of a function  F which is a one-to-one function into the domain of  E is a nonnegative integer, the size of the function  F is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
 |-  (
 ( E : dom  E
 -1-1-> ran  E  /\  E  e.  V )  ->  (
 ( F : ( 0..^ ( # `  F ) ) -1-1-> dom  E  /\  P  =  ( E
 " ran  F )  /\  ( # `  P )  =  N )  ->  ( # `  F )  =  N )
 )
 
Theoremhashfirdm 28286 The size of a function with a half-open range of integers, starting with 0, as domain equals the right border of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( N  e.  NN0  /\  F : ( 0..^ N ) --> B ) 
 ->  ( # `  F )  =  N )
 
Theoremhashfzdm 28287 The size of a function with a finite set of sequential integers, starting with 0, as domain equals the right border of this range increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
 |-  (
 ( N  e.  NN0  /\  F : ( 0
 ... N ) --> B ) 
 ->  ( # `  F )  =  ( N  +  1 ) )
 
Theoremeuhash1 28288* The size of a set is 1 in terms of existential uniqueness. (Contributed by Alexander van der Vekens, 8-Feb-2018.)
 |-  ( V  e.  W  ->  ( ( # `  V )  =  1  <->  E! a  a  e.  V ) )
 
Theoremfz0hash 28289 The value of the size function on a finite 0-based sequence. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
 |-  (
 ( N  e.  NN0  /\  F  Fn  ( 0
 ... N ) ) 
 ->  ( # `  F )  =  ( N  +  1 ) )
 
Theoremhashss 28290 The size of a subset is less then or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
 |-  (
 ( A  e.  V  /\  B  C_  A )  ->  ( # `  B )  <_  ( # `  A ) )
 
Theoremhash2prv 28291* A set of size two is an unordered pair of its elements. (Contributed by Alexander van der Vekens, 12-Jul-2017.)
 |-  (
 ( V  e.  W  /\  ( # `  V )  =  2 )  ->  E. a  e.  V  E. b  e.  V  V  =  { a ,  b } )
 
Theoremhash2sspr 28292* A subset of size two is an unordered pair of elements of its superset. (Contributed by Alexander van der Vekens, 12-Jul-2017.)
 |-  (
 ( P  e.  ~P V  /\  ( # `  P )  =  2 )  ->  E. a  e.  V  E. b  e.  V  P  =  { a ,  b } )
 
Theoremelss2pr 28293* An element of the set of subsets with two elements is an unordered pair. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
 |-  ( P  e.  { z  e.  ~P V  |  ( # `  z )  =  2 }  ->  E. x  e.  V  E. y  e.  V  P  =  { x ,  y }
 )
 
Theoremexprelprel 28294* If there is an element of the set of subsets with two elements in a set, an unordered pair of sets is in the set. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
 |-  ( E. p  e.  { e  e.  ~P V  |  ( # `  e )  =  2 } p  e.  X  ->  E. v  e.  V  E. w  e.  V  { v ,  w }  e.  X )
 
Theoremhashfz0 28295 Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
 |-  ( B  e.  NN0  ->  ( # `
  ( 0 ...
 B ) )  =  ( B  +  1 ) )
 
Theoremffzohash 28296 The size of a function on a half-open integer range. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
 |-  (
 ( N  e.  NN0  /\  F  Fn  ( 0..^ N ) )  ->  ( # `  F )  =  N )
 
19.23.3.25  Words over a set - extension
 
Theoremwrdlen1 28297* A word of length 1 starts with a symbol. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
 |-  (
 ( W  e. Word  V  /\  ( # `  W )  =  1 )  ->  E. v  e.  V  ( W `  0 )  =  v )
 
Theoremcsbwrdg 28298* Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 |-  ( S  e.  V  ->  [_ S  /  x ]_Word  x  = Word  S )
 
Theoremiswrd0i 28299 A zero-based sequence is a word. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( W : ( 0 ...
 L ) --> S  ->  W  e. Word  S )
 
Theoremwrdsymb0 28300 A symbol at a position "outside" of a word. (Contributed by Alexander van der Vekens, 26-May-2018.)
 |-  (
 ( W  e. Word  V  /\  I  e.  ZZ )  ->  ( ( I  <  0  \/  ( # `
  W )  <_  I )  ->  ( W `
  I )  =  (/) ) )
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