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Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzpr 11101 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ZZ  ->  ( M ... ( M  +  1 )
 )  =  { M ,  ( M  +  1 ) } )
 
Theoremfztp 11102 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)
 |-  ( M  e.  ZZ  ->  ( M ... ( M  +  2 )
 )  =  { M ,  ( M  +  1 ) ,  ( M  +  2 ) }
 )
 
Theoremfz0tp 11103 An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  ( 0 ... 2
 )  =  { 0 ,  1 ,  2 }
 
Theoremfzsuc2 11104 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>=
 `  ( M  -  1 ) ) ) 
 ->  ( M ... ( N  +  1 )
 )  =  ( ( M ... N )  u.  { ( N  +  1 ) }
 ) )
 
Theoremfzp1disj 11105  ( M ... ( N  +  1 ) ) is the disjoint union of  ( M ... N ) with  { ( N  +  1 ) }. (Contributed by Mario Carneiro, 7-Mar-2014.)
 |-  ( ( M ... N )  i^i  { ( N  +  1 ) } )  =  (/)
 
Theoremfzprval 11106* Two ways of defining the first two values of a sequence on  NN. (Contributed by NM, 5-Sep-2011.)
 |-  ( A. x  e.  ( 1 ... 2
 ) ( F `  x )  =  if ( x  =  1 ,  A ,  B )  <-> 
 ( ( F `  1 )  =  A  /\  ( F `  2
 )  =  B ) )
 
Theoremfztpval 11107* Two ways of defining the first three values of a sequence on  NN. (Contributed by NM, 13-Sep-2011.)
 |-  ( A. x  e.  ( 1 ... 3
 ) ( F `  x )  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C )
 ) 
 <->  ( ( F `  1 )  =  A  /\  ( F `  2
 )  =  B  /\  ( F `  3 )  =  C ) )
 
Theoremfzrev 11108 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( K  e.  (
 ( J  -  N ) ... ( J  -  M ) )  <->  ( J  -  K )  e.  ( M ... N ) ) )
 
Theoremfzrev2 11109 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( K  e.  ( M ... N )  <->  ( J  -  K )  e.  (
 ( J  -  N ) ... ( J  -  M ) ) ) )
 
Theoremfzrev2i 11110 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( J  e.  ZZ  /\  K  e.  ( M ... N ) ) 
 ->  ( J  -  K )  e.  ( ( J  -  N ) ... ( J  -  M ) ) )
 
Theoremfzrev3 11111 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
 |-  ( K  e.  ZZ  ->  ( K  e.  ( M ... N )  <->  ( ( M  +  N )  -  K )  e.  ( M ... N ) ) )
 
Theoremfzrev3i 11112 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
 |-  ( K  e.  ( M ... N )  ->  ( ( M  +  N )  -  K )  e.  ( M ... N ) )
 
Theoremfznn0 11113 Finite set of sequential integers starting at 0. (Contributed by NM, 1-Aug-2005.)
 |-  ( N  e.  NN0  ->  ( K  e.  (
 0 ... N )  <->  ( K  e.  NN0  /\  K  <_  N )
 ) )
 
Theoremnn0fz0 11114 A non-negative integer is always part of its zero-based finite sequence. (Contributed by Scott Fenton, 21-Mar-2018.)
 |-  ( N  e.  NN0  <->  N  e.  ( 0 ... N ) )
 
Theoremfznn 11115 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( K  e.  (
 1 ... N )  <->  ( K  e.  NN  /\  K  <_  N ) ) )
 
Theoremelfzm11 11116 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... ( N  -  1 ) )  <-> 
 ( K  e.  ZZ  /\  M  <_  K  /\  K  <  N ) ) )
 
Theoremfzctr 11117 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)
 |-  ( N  e.  NN0  ->  N  e.  ( 0 ... ( 2  x.  N ) ) )
 
Theoremuzsplit 11118 Express an upper integer set as the disjoint (see uzdisj 11119) union of the first  N values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ZZ>= `  M )  =  ( ( M ... ( N  -  1
 ) )  u.  ( ZZ>=
 `  N ) ) )
 
Theoremuzdisj 11119 The first  N elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( M ... ( N  -  1
 ) )  i^i  ( ZZ>=
 `  N ) )  =  (/)
 
Theorem1fv 11120 A one value function. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  ( ( N  e.  V  /\  P  =  { <. 0 ,  N >. } )  ->  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )
 
Theorem4fvwrd4 11121* The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  ( ( L  e.  ( ZZ>= `  3 )  /\  P : ( 0
 ... L ) --> V ) 
 ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( P `
  0 )  =  a  /\  ( P `
  1 )  =  b )  /\  (
 ( P `  2
 )  =  c  /\  ( P `  3 )  =  d ) ) )
 
Theoremfseq1p1m1 11122 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.)
 |-  H  =  { <. ( N  +  1 ) ,  B >. }   =>    |-  ( N  e.  NN0 
 ->  ( ( F :
 ( 1 ... N )
 --> A  /\  B  e.  A  /\  G  =  ( F  u.  H ) )  <->  ( G :
 ( 1 ... ( N  +  1 )
 ) --> A  /\  ( G `  ( N  +  1 ) )  =  B  /\  F  =  ( G  |`  ( 1
 ... N ) ) ) ) )
 
Theoremfseq1m1p1 11123 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  H  =  { <. N ,  B >. }   =>    |-  ( N  e.  NN  ->  ( ( F : ( 1 ... ( N  -  1
 ) ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H ) )  <->  ( G :
 ( 1 ... N )
 --> A  /\  ( G `
  N )  =  B  /\  F  =  ( G  |`  ( 1
 ... ( N  -  1 ) ) ) ) ) )
 
Theoremfz1sbc 11124* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)
 |-  ( N  e.  ZZ  ->  ( A. k  e.  ( N ... N ) ph  <->  [. N  /  k ]. ph ) )
 
Theoremelfzm1b 11125 An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 1 ... N ) 
 <->  ( K  -  1
 )  e.  ( 0
 ... ( N  -  1 ) ) ) )
 
Theoremelfzp12 11126 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( K  e.  ( M ... N )  <->  ( K  =  M  \/  K  e.  (
 ( M  +  1 ) ... N ) ) ) )
 
Theoremfzm1 11127 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
 
Theoremfzneuz 11128 No finite set of sequential integers equals a set of upper integers. (Contributed by NM, 11-Dec-2005.)
 |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  ZZ )  ->  -.  ( M ... N )  =  ( ZZ>= `  K ) )
 
Theoremfznuz 11129 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)
 |-  ( K  e.  ( M ... N )  ->  -.  K  e.  ( ZZ>= `  ( N  +  1
 ) ) )
 
Theoremuznfz 11130 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
 |-  ( K  e.  ( ZZ>=
 `  N )  ->  -.  K  e.  ( M
 ... ( N  -  1 ) ) )
 
Theoremfzrevral 11131* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( A. j  e.  ( M ... N ) ph  <->  A. k  e.  (
 ( K  -  N ) ... ( K  -  M ) ) [. ( K  -  k
 )  /  j ]. ph ) )
 
Theoremfzrevral2 11132* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( A. j  e.  ( ( K  -  N ) ... ( K  -  M ) )
 ph 
 <-> 
 A. k  e.  ( M ... N ) [. ( K  -  k
 )  /  j ]. ph ) )
 
Theoremfzrevral3 11133* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A. j  e.  ( M ... N ) ph  <->  A. k  e.  ( M ... N ) [. ( ( M  +  N )  -  k
 )  /  j ]. ph ) )
 
Theoremfzshftral 11134* Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( A. j  e.  ( M ... N ) ph  <->  A. k  e.  (
 ( M  +  K ) ... ( N  +  K ) ) [. ( k  -  K )  /  j ]. ph )
 )
 
5.5.6  Half-open integer ranges
 
Syntaxcfzo 11135 Syntax for half-open integer ranges.
 class ..^
 
Definitiondf-fzo 11136* Define a function generating sets of integers using a half-open range. Read  ( M..^ N
) as the integers from 
M up to, but not including,  N; contrast with  ( M ... N ) df-fz 11044, which includes  N. Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 11166 with fzsplit 11077, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ 
 =  ( m  e. 
 ZZ ,  n  e. 
 ZZ  |->  ( m ... ( n  -  1
 ) ) )
 
Theoremfzof 11137 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ : ( ZZ  X.  ZZ ) --> ~P ZZ
 
Theoremelfzoel1 11138 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  B  e.  ZZ )
 
Theoremelfzoel2 11139 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  C  e.  ZZ )
 
Theoremelfzoelz 11140 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  A  e.  ZZ )
 
Theoremfzoval 11141 Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( N  e.  ZZ  ->  ( M..^ N )  =  ( M ... ( N  -  1
 ) ) )
 
Theoremelfzo 11142 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M..^ N )  <->  ( M  <_  K 
 /\  K  <  N ) ) )
 
Theoremelfzo2 11143 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  <->  ( K  e.  ( ZZ>= `  M )  /\  N  e.  ZZ  /\  K  <  N ) )
 
Theoremelfzouz 11144 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( ZZ>= `  M )
 )
 
Theoremfzolb 11145 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with  M  <  N. This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 
M  e.  ( ZZ>= `  N ). (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( M  e.  ( M..^ N )  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N ) )
 
Theoremfzolb2 11146 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with  M  <  N. This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 
M  e.  ( ZZ>= `  N ). (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ( M..^ N )  <->  M  <  N ) )
 
Theoremelfzole1 11147 A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  M  <_  K )
 
Theoremelfzolt2 11148 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  <  N )
 
Theoremelfzolt3 11149 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  M  <  N )
 
Theoremelfzolt2b 11150 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( K..^ N ) )
 
Theoremelfzolt3b 11151 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  M  e.  ( M..^ N ) )
 
Theoremfzonel 11152 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |- 
 -.  B  e.  ( A..^ B )
 
Theoremelfzouz2 11153 The upper bound of a half-open range is greater or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  N  e.  ( ZZ>= `  K )
 )
 
Theoremelfzofz 11154 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( M ... N ) )
 
Theoremelfzo3 11155 Express membership in a half-open integer interval in terms of the "less than or equal" and "less than" predicates on integers, resp.  K  e.  (
ZZ>= `  M )  <->  M  <_  K,  K  e.  ( K..^ N )  <->  K  <  N. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  <->  ( K  e.  ( ZZ>= `  M )  /\  K  e.  ( K..^ N ) ) )
 
Theoremfzon0 11156 A half-open integer interval is nonempty iff it contains its left endpoint. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( ( M..^ N )  =/=  (/)  <->  M  e.  ( M..^ N ) )
 
Theoremfzossfz 11157 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( A..^ B ) 
 C_  ( A ... B )
 
Theoremfzon 11158 A half-open set of sequential integers is empty if the bounds are equal or reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <_  M  <-> 
 ( M..^ N )  =  (/) ) )
 
Theoremfzo0 11159 Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( A..^ A )  =  (/)
 
Theoremfzonnsub 11160 If  K  <  N then 
N  -  K is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
 |-  ( K  e.  ( M..^ N )  ->  ( N  -  K )  e. 
 NN )
 
Theoremfzonnsub2 11161 If  M  <  N then 
N  -  M is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( K  e.  ( M..^ N )  ->  ( N  -  M )  e. 
 NN )
 
Theoremfzoss1 11162 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( ZZ>=
 `  M )  ->  ( K..^ N )  C_  ( M..^ N ) )
 
Theoremfzoss2 11163 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( N  e.  ( ZZ>=
 `  K )  ->  ( M..^ K )  C_  ( M..^ N ) )
 
Theoremfzossrbm1 11164 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  ( N  e.  NN0  ->  ( 0..^ ( N  -  1 ) )  C_  ( 0..^ N ) )
 
Theoremfzospliti 11165 One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( ( A  e.  ( B..^ C )  /\  D  e.  ZZ )  ->  ( A  e.  ( B..^ D )  \/  A  e.  ( D..^ C ) ) )
 
Theoremfzosplit 11166 Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( D  e.  ( B ... C )  ->  ( B..^ C )  =  ( ( B..^ D )  u.  ( D..^ C ) ) )
 
Theoremfzodisj 11167 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( ( A..^ B )  i^i  ( B..^ C ) )  =  (/)
 
Theoremfzouzsplit 11168 Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( ZZ>= `  A )  =  ( ( A..^ B )  u.  ( ZZ>= `  B ) ) )
 
Theoremfzouzdisj 11169 A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.)
 |-  ( ( A..^ B )  i^i  ( ZZ>= `  B ) )  =  (/)
 
Theoremlbfzo0 11170 An integer is strictly greater than zero iff it is a member of  NN. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( 0  e.  (
 0..^ A )  <->  A  e.  NN )
 
Theoremelfzo0 11171 Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( A  e.  (
 0..^ B )  <->  ( A  e.  NN0  /\  B  e.  NN  /\  A  <  B ) )
 
Theoremfzossnn 11172 Half-opened integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  ( 1..^ N ) 
 C_  NN
 
Theoremelfzo1 11173 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( N  e.  (
 1..^ M )  <->  ( N  e.  NN  /\  M  e.  NN  /\  N  <  M ) )
 
Theoremfzo0n0 11174 A half-open integer range based at 0 is nonempty precisely if the upper bound is a positive integer. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( ( 0..^ A )  =/=  (/)  <->  A  e.  NN )
 
Theoremfzoaddel 11175 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( B..^ C )  /\  D  e.  ZZ )  ->  ( A  +  D )  e.  ( ( B  +  D )..^ ( C  +  D ) ) )
 
Theoremfzoaddel2 11176 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( 0..^ ( B  -  C ) )  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  +  C )  e.  ( C..^ B ) )
 
Theoremfzosubel 11177 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( B..^ C )  /\  D  e.  ZZ )  ->  ( A  -  D )  e.  ( ( B  -  D )..^ ( C  -  D ) ) )
 
Theoremfzosubel2 11178 Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( ( B  +  C )..^ ( B  +  D ) )  /\  ( B  e.  ZZ  /\  C  e.  ZZ  /\  D  e.  ZZ )
 )  ->  ( A  -  B )  e.  ( C..^ D ) )
 
Theoremfzosubel3 11179 Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( B..^ ( B  +  D ) )  /\  D  e.  ZZ )  ->  ( A  -  B )  e.  ( 0..^ D ) )
 
Theoremfzval3 11180 Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( N  e.  ZZ  ->  ( M ... N )  =  ( M..^ ( N  +  1
 ) ) )
 
Theoremfzosn 11181 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ZZ  ->  ( A..^ ( A  +  1 ) )  =  { A }
 )
 
Theoremfzo01 11182 Expressing the singleton of  0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( 0..^ 1 )  =  { 0 }
 
Theoremfzo12sn 11183 A 1-based half-open integer interval up to, but not including, 2 is a singleton. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( 1..^ 2 )  =  { 1 }
 
Theoremfzo0to2pr 11184 A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( 0..^ 2 )  =  { 0 ,  1 }
 
Theoremfzo0to3tp 11185 A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  ( 0..^ 3 )  =  { 0 ,  1 ,  2 }
 
Theoremfzo0to42pr 11186 A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
 |-  ( 0..^ 4 )  =  ( { 0 ,  1 }  u.  { 2 ,  3 } )
 
Theoremfzoend 11187 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( A  e.  ( A..^ B )  ->  ( B  -  1 )  e.  ( A..^ B ) )
 
Theoremfzo0end 11188 The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( B  e.  NN  ->  ( B  -  1
 )  e.  ( 0..^ B ) )
 
Theoremfzofzp1 11189 If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( C  e.  ( A..^ B )  ->  ( C  +  1 )  e.  ( A ... B ) )
 
Theoremfzofzp1b 11190 If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( C  e.  ( ZZ>=
 `  A )  ->  ( C  e.  ( A..^ B )  <->  ( C  +  1 )  e.  ( A ... B ) ) )
 
Theoremelfzom1b 11191 An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 1..^ N )  <->  ( K  -  1 )  e.  (
 0..^ ( N  -  1 ) ) ) )
 
Theoremelfznelfzo 11192 A value in a finite set of sequential integers is a border value if it is not contained in the half-open integer range contained in the finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( ( y  e.  ( 0 ... K )  /\  -.  y  e.  ( 1..^ K ) )  ->  ( y  =  0  \/  y  =  K ) )
 
Theoremelfznelfzob 11193 A value in a finite set of sequential integers is a border value if and only if it is not contained in the half-open integer range contained in the finite set of sequential integerss. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
 |-  ( y  e.  (
 0 ... K )  ->  ( -.  y  e.  (
 1..^ K )  <->  ( y  =  0  \/  y  =  K ) ) )
 
Theorempeano2fzor 11194 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  ( K  +  1 )  e.  ( M..^ N ) )  ->  K  e.  ( M..^ N ) )
 
Theoremfzosplitsn 11195 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( A..^ ( B  +  1 ) )  =  ( ( A..^ B )  u.  { B }
 ) )
 
Theoremfzosplitsni 11196 Membership in a half-open range extende by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( C  e.  ( A..^ ( B  +  1 ) )  <->  ( C  e.  ( A..^ B )  \/  C  =  B ) ) )
 
Theoremfzostep1 11197 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  (
 ( A  +  1 )  e.  ( B..^ C )  \/  ( A  +  1 )  =  C ) )
 
Theoremfzind2 11198* Induction on the integers from  M to  N inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. Version of fzind 10368 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
 |-  ( x  =  M  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  K  ->  (
 ph 
 <->  ta ) )   &    |-  ( N  e.  ( ZZ>= `  M )  ->  ps )   &    |-  (
 y  e.  ( M..^ N )  ->  ( ch  ->  th ) )   =>    |-  ( K  e.  ( M ... N ) 
 ->  ta )
 
Theoreminjresinjlem 11199 Lemma for injresinj 11200. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( -.  y  e.  ( 1..^ K ) 
 ->  ( ( F `  0 )  =/=  ( F `  K )  ->  ( ( F :
 ( 0 ... K )
 --> V  /\  K  e.  NN0 )  ->  ( (
 ( F " {
 0 ,  K }
 )  i^i  ( F " ( 1..^ K ) ) )  =  (/)  ->  ( ( x  e.  ( 0 ... K )  /\  y  e.  (
 0 ... K ) ) 
 ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
 ) ) ) ) )
 
Theoreminjresinj 11200 A function whose restriction is injective and the values of the remaining arguments are different from all other values is injective itself. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( K  e.  NN0  ->  ( ( F :
 ( 0 ... K )
 --> V  /\  Fun  `' ( F  |`  ( 1..^ K ) )  /\  ( F `  0 )  =/=  ( F `  K ) )  ->  ( ( ( F
 " { 0 ,  K } )  i^i  ( F " (
 1..^ K ) ) )  =  (/)  ->  Fun  `' F ) ) )
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