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Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0resubcl 28101 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 )
 
19.22.3.19  Upper partititions of integers - extension
 
Theorem1eluzge0 28102 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  1  e.  ( ZZ>= `  0 )
 
Theorem2eluzge0 28103 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  0 )
 
Theorem2eluzge1 28104 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  1 )
 
Theoremuzletr 28105 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 ) ) )
 
19.22.3.20  Finite intervals of integers - extension
 
Theoremssfz12 28106 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 28107 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 28108 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 28109 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 28110 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 28111 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 28112 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 28113 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 28114 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 28115 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 28116 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 28117 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 28118 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 28119 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 28120 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 28121 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 28122 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 28123* 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 ) ) ) )
 
19.22.3.21  Half-open integer ranges - extension
 
Theoremelfzonn0 28124 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 28125 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 28126 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 28127 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 28128 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 28129 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 28130 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 28131 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 28132 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 28133 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 28134 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 28135 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 28136 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 28137 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 28138 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 28139 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 28140 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 28141 A half-open range of integers can represent an ordered pair, analogous to fzopth 11091. (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 28142* 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
 ) ) ) )
 
19.22.3.22  The floor (greatest integer) function - extension
 
Theoremnn0nndivcl 28143 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 28144 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 28145 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 28146 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 28147 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 28148 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 28149 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 28150 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 ) ) )
 
19.22.3.23  The modulo (remainder) operation - extension
 
Theoremmodvalr 28151 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 28152 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 28153 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 28154 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 28155 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 28156 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 28157 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 28158 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 28159 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 28160 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 28161 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 28162 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 28163 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 28164 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 )
 
19.22.3.24  The ` # ` (finite set size) function - extension
 
Theoremhashimarn 28165 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 28166 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 28167 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 28168 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 28169* 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 28170 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 ) )
 
19.22.3.25  Words over a set - extension
 
Theoremiswrd0i 28171 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 28172 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 )  =  (/) ) )
 
Theoremwrdfn 28173 A word is a function with a zero-based sequence of integers as domain. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
 |-  ( W  e. Word  S  ->  W  Fn  ( 0..^ ( # `  W ) ) )
 
Theoremwrdeq0 28174* Two words are equal iff they have the same length and the same symbol at each position. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
 |-  (
 ( U  e. Word  V  /\  W  e. Word  V )  ->  ( U  =  W  <->  ( ( # `  U )  =  ( # `  W )  /\  A. i  e.  ( 0..^ ( # `  U ) ) ( U `  i )  =  ( W `  i ) ) ) )
 
Theoremwrdsymb 28175 A symbol within a word over a set belongs to this set. (Contributed by Alexander van der Vekens, 28-Jun-2018.)
 |-  (
 ( P  e. Word  V  /\  I  e.  (
 0..^ ( # `  P ) ) )  ->  ( P `  I )  e.  V )
 
Theoremwrdsymb1 28176 The first symbol within a non-empty word over a set belongs to this set. (Contributed by Alexander van der Vekens, 28-Jun-2018.)
 |-  (
 ( P  e. Word  V  /\  1  <_  ( # `  P ) )  ->  ( P `  0 )  e.  V )
 
Theorem2wrdeq 28177* Two words are equal if and only if they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( W  e. Word  V  /\  S  e. Word  V )  ->  ( W  =  S  <->  ( ( # `  W )  =  ( # `  S )  /\  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( S `  i ) ) ) )
 
Theoremwrdlenge2n0 28178 A word with length at least 2 is not empty. (Contributed by Alexander van der Vekens, 18-Jun-2018.)
 |-  (
 ( P  e. Word  V  /\  2  <_  ( # `  P ) )  ->  P  =/=  (/) )
 
19.22.3.26  Words over a set - extension (concatenations)
 
Theoremelfzelfzccat 28179 An element of a finite set of sequential integers up to the length of a word is an element of an extended finite set of sequential integers up to the length of a concatenation of this word with another word. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
 |-  (
 ( A  e. Word  V  /\  B  e. Word  V )  ->  ( N  e.  (
 0 ... ( # `  A ) )  ->  N  e.  ( 0 ... ( # `
  ( A concat  B ) ) ) ) )
 
Theoremccatvalfn 28180 The concatenation of two words is a function over the half-open interval of integers having the sum of the lengths of the word as length. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  (
 ( A  e. Word  V  /\  B  e. Word  V )  ->  ( A concat  B )  Fn  ( 0..^ ( ( # `  A )  +  ( # `  B ) ) ) )
 
Theoremccatsymb 28181 The symbol at a given position in a concatenated word. (Contributed by Alexander van der Vekens, 26-May-2018.)
 |-  (
 ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ZZ )  ->  ( ( S concat  T ) `  I )  =  if ( I  < 
 ( # `  S ) ,  ( S `  I ) ,  ( T `  ( I  -  ( # `  S ) ) ) ) )
 
Theoremwrdlenccats1lenm1 28182 The length of a non-empty word is the length of the word concatenated with its first symbol minus 1. (Contributed by Alexander van der Vekens, 28-Jun-2018.)
 |-  (
 ( P  e. Word  V  /\  1  <_  ( # `  P ) )  ->  ( # `  P )  =  ( ( # `  ( P concat  <" ( P `  0 ) "> ) )  -  1
 ) )
 
19.22.3.27  Words over a set - extension (subwords)
 
Theoremswrdltnd 28183 The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( W  e. Word  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( L  <_  F  ->  ( W substr  <. F ,  L >. )  =  (/) ) )
 
Theoremswrdnd 28184 The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( W  e. Word  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( ( F  <  0  \/  L  <_  F  \/  ( # `  W )  <  L )  ->  ( W substr  <. F ,  L >. )  =  (/) ) )
 
Theoremswrd0 28185 A subword of an empty set is always the empty set. REMARK: The antecedent  ( F  e.  ZZ  /\  L  e.  ZZ ) should not be necessary! (Contributed by Alexander van der Vekens, 31-Mar-2018.)
 |-  (
 ( F  e.  ZZ  /\  L  e.  ZZ )  ->  ( (/) substr  <. F ,  L >. )  =  (/) )
 
Theoremswrdrlen 28186 Length of a right-anchored subword. (Contributed by Alexander van der Vekens, 5-Apr-2018.)
 |-  (
 ( S  e. Word  A  /\  F  e.  ( 0
 ... ( # `  S ) ) )  ->  ( # `  ( S substr  <. F ,  ( # `  S ) >. ) )  =  ( ( # `  S )  -  F ) )
 
Theoremaddlenrevswrd 28187 The sum of the lengths of two parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( # `  ( W substr 
 <. M ,  ( # `  W ) >. ) )  +  ( # `  ( W substr 
 <. 0 ,  M >. ) ) )  =  ( # `  W ) )
 
Theoremswrdvalfn 28188 Value of the subword extractor as function with domain. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
 |-  (
 ( S  e. Word  A  /\  F  e.  ( 0
 ... L )  /\  L  e.  ( 0 ... ( # `  S ) ) )  ->  ( S substr  <. F ,  L >. )  Fn  (
 0..^ ( L  -  F ) ) )
 
Theoremswrdvalodmlem1 28189 Lemma for swrdvalodm 28191. (Contributed by Alexander van der Vekens, 24-May-2018.)
 |-  (
 ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( N  <_  A  \/  B  <_  0
 )  ->  ( ( A..^ B )  C_  (
 0..^ N )  ->  B  <_  A ) ) )
 
Theoremswrdvalodm2 28190 Value of the subword extractor outside its intended domain. (Contributed by Alexander van der Vekens, 26-May-2018.)
 |-  (
 ( W  e. Word  V  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( B  <_  A  \/  ( # `  W )  <  B  \/  A  <  0 )  ->  ( W substr 
 <. A ,  B >. )  =  (/) ) )
 
Theoremswrdvalodm 28191 Value of the subword extractor outside its intended domain. (Contributed by Alexander van der Vekens, 24-May-2018.)
 |-  (
 ( W  e. Word  V  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( B  <_  A  \/  ( # `  W )  <_  A  \/  B  <_  0 )  ->  ( W substr 
 <. A ,  B >. )  =  (/) ) )
 
Theoremlenrevcctswrd 28192 The length of two reversely concatenated parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( # `  ( ( W substr  <. M ,  ( # `
  W ) >. ) concat 
 ( W substr  <. 0 ,  M >. ) ) )  =  ( # `  W ) )
 
Theoremccats1swrdid 28193 A non-empty word is the prefix of the word concatenated with its first symbol. (Contributed by Alexander van der Vekens, 28-Jun-2018.)
 |-  (
 ( P  e. Word  V  /\  1  <_  ( # `  P ) )  ->  P  =  ( ( P concat  <" ( P `
  0 ) "> ) substr  <. 0 ,  ( # `
  P ) >. ) )
 
Theoremswrd0fv 28194 A symbol in an left-anchored subword, indexed using the subword's indices. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) )  /\  X  e.  ( 0..^ L ) ) 
 ->  ( ( S substr  <. 0 ,  L >. ) `  X )  =  ( S `  X ) )
 
Theoremswrd0fv0 28195 The first symbol in a left-anchored subword. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 1
 ... ( # `  S ) ) )  ->  ( ( S substr  <. 0 ,  L >. ) `  0
 )  =  ( S `
  0 ) )
 
Theoremswrdtrcfv 28196 A symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
 |-  (
 ( S  e. Word  A  /\  S  =/=  (/)  /\  X  e.  ( 0..^ ( ( # `  S )  -  1 ) ) ) 
 ->  ( ( S substr  <. 0 ,  ( ( # `  S )  -  1 ) >. ) `
  X )  =  ( S `  X ) )
 
Theoremswrdtrcfv0 28197 The first symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
 |-  (
 ( S  e. Word  A  /\  2  <_  ( # `  S ) )  ->  ( ( S substr  <. 0 ,  ( ( # `  S )  -  1 ) >. ) `
  0 )  =  ( S `  0
 ) )
 
Theoremswdeq 28198* If two words have the same prefix, their symbols are identical with this prefix. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
 |-  (
 ( ( U  e. Word  V 
 /\  W  e. Word  V )  /\  ( N  e.  NN0  /\  N  <_  ( # `  U )  /\  N  <_  ( # `
  W ) ) )  ->  ( ( U substr 
 <. 0 ,  N >. )  =  ( W substr  <. 0 ,  N >. )  ->  A. i  e.  ( 0..^ N ) ( U `  i
 )  =  ( W `
  i ) ) )
 
19.22.3.28  Words over a set - extension (subwords of subwords)
 
Theoremswrd0swrd 28199 A prefix of a subword. (Contributed by Alexander van der Vekens, 2-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  M  e.  ( 0 ... N ) )  ->  ( L  e.  ( 0 ... ( N  -  M ) )  ->  ( ( W substr  <. M ,  N >. ) substr  <. 0 ,  L >. )  =  ( W substr  <. M ,  ( M  +  L ) >. ) ) )
 
Theoremswrdswrdlem 28200 Lemma for swrdswrd 28201. (Contributed by Alexander van der Vekens, 4-Apr-2018.)
 |-  (
 ( ( W  e. Word  V 
 /\  N  e.  (
 0 ... ( # `  W ) )  /\  M  e.  ( 0 ... N ) )  /\  ( K  e.  ( 0 ... ( N  -  M ) )  /\  L  e.  ( K ... ( N  -  M ) ) ) )  ->  ( W  e. Word  V  /\  ( M  +  K )  e.  ( 0 ... ( M  +  L )
 )  /\  ( M  +  L )  e.  (
 0 ... ( # `  W ) ) ) )
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