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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-fzo 9101* 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 8976, which includes  N. Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 9134 with fzsplit 9016, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ 
 =  ( m  e. 
 ZZ ,  n  e. 
 ZZ  |->  ( m ... ( n  -  1
 ) ) )
 
Theoremfzof 9102 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ : ( ZZ  X.  ZZ ) --> ~P ZZ
 
Theoremelfzoel1 9103 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  B  e.  ZZ )
 
Theoremelfzoel2 9104 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  C  e.  ZZ )
 
Theoremelfzoelz 9105 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  A  e.  ZZ )
 
Theoremfzoval 9106 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 9107 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 9108 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 9109 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( ZZ>= `  M )
 )
 
Theoremfzolb 9110 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 9111 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 9112 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 9113 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 9114 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 9115 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 9116 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 9117 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |- 
 -.  B  e.  ( A..^ B )
 
Theoremelfzouz2 9118 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 9119 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 9120 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 ) ) )
 
Theoremfzom 9121* A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.)
 |-  ( E. x  x  e.  ( M..^ N ) 
 <->  M  e.  ( M..^ N ) )
 
Theoremfzossfz 9122 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 9123 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 )  =  (/) ) )
 
Theoremfzonlt0 9124 A half-open integer range is empty if the bounds are equal or reversed. (Contributed by AV, 20-Oct-2018.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  M  <  N  <->  ( M..^ N )  =  (/) ) )
 
Theoremfzo0 9125 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 9126 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 9127 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 9128 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 9129 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 9130 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  ( N  e.  ZZ  ->  ( 0..^ ( N  -  1 ) ) 
 C_  ( 0..^ N ) )
 
Theoremfzo0ss1 9131 Subset relationship for half-open integer ranges with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  ( 1..^ N ) 
 C_  ( 0..^ N )
 
Theoremfzossnn0 9132 A half-open integer range starting at 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 )
 
Theoremfzospliti 9133 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 9134 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 9135 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( ( A..^ B )  i^i  ( B..^ C ) )  =  (/)
 
Theoremfzouzsplit 9136 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 9137 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 9138 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 9139 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 ) )
 
Theoremfzo1fzo0n0 9140 An integer between 1 and an upper bound of a half-open integer range is not 0 and between 0 and the upper bound of the half-open integer range. (Contributed by Alexander van der Vekens, 21-Mar-2018.)
 |-  ( K  e.  (
 1..^ N )  <->  ( K  e.  ( 0..^ N )  /\  K  =/=  0 ) )
 
Theoremelfzo0z 9141 Membership in a half-open range of nonnegative integers, generalization of elfzo0 9139 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
 |-  ( A  e.  (
 0..^ B )  <->  ( A  e.  NN0  /\  B  e.  ZZ  /\  A  <  B ) )
 
Theoremelfzo0le 9142 A member in a half-open range of nonnegative integers is less than or equal to the upper bound of the range. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
 |-  ( A  e.  (
 0..^ B )  ->  A  <_  B )
 
Theoremelfzonn0 9143 A member of a half-open range of nonnegative integers is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  ( K  e.  (
 0..^ N )  ->  K  e.  NN0 )
 
Theoremfzonmapblen 9144 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 )
 
Theoremfzofzim 9145 If a nonnegative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
 |-  ( ( K  =/=  M 
 /\  K  e.  (
 0 ... M ) ) 
 ->  K  e.  ( 0..^ M ) )
 
Theoremfzossnn 9146 Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  ( 1..^ N ) 
 C_  NN
 
Theoremelfzo1 9147 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 ) )
 
Theoremfzo0m 9148* A half-open integer range based at 0 is inhabited precisely if the upper bound is a positive integer. (Contributed by Jim Kingdon, 20-Apr-2020.)
 |-  ( E. x  x  e.  ( 0..^ A ) 
 <->  A  e.  NN )
 
Theoremfzoaddel 9149 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 9150 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 9151 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 9152 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 9153 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 ) )
 
Theoremeluzgtdifelfzo 9154 Membership of the difference of integers in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( N  e.  ( ZZ>= `  A )  /\  B  <  A )  ->  ( N  -  A )  e.  (
 0..^ ( N  -  B ) ) ) )
 
Theoremige2m2fzo 9155 Membership of an integer greater than 1 decreased by 2 in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( N  -  2
 )  e.  ( 0..^ ( N  -  1
 ) ) )
 
Theoremfzocatel 9156 Translate membership in a half-open integer range. (Contributed by Thierry Arnoux, 28-Sep-2018.)
 |-  ( ( ( A  e.  ( 0..^ ( B  +  C ) )  /\  -.  A  e.  ( 0..^ B ) )  /\  ( B  e.  ZZ  /\  C  e.  ZZ ) )  ->  ( A  -  B )  e.  ( 0..^ C ) )
 
Theoremubmelfzo 9157 If an integer in a 1 based finite set of sequential integers is subtracted from the upper bound of this finite set of sequential integers, the result is contained in a half-open range of nonnegative integers with the same upper bound. (Contributed by AV, 18-Mar-2018.) (Revised by AV, 30-Oct-2018.)
 |-  ( K  e.  (
 1 ... N )  ->  ( N  -  K )  e.  ( 0..^ N ) )
 
Theoremelfzodifsumelfzo 9158 If an integer is in a half-open range of nonnegative integers with a difference as upper bound, the sum of the integer with the subtrahend of the difference is in the a half-open range of nonnegative integers containing the minuend of the difference. (Contributed by AV, 13-Nov-2018.)
 |-  ( ( M  e.  ( 0 ... N )  /\  N  e.  (
 0 ... P ) ) 
 ->  ( I  e.  (
 0..^ ( N  -  M ) )  ->  ( I  +  M )  e.  ( 0..^ P ) ) )
 
Theoremelfzom1elp1fzo 9159 Membership of an integer incremented by one in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.)
 |-  ( ( N  e.  ZZ  /\  I  e.  (
 0..^ ( N  -  1 ) ) ) 
 ->  ( I  +  1 )  e.  ( 0..^ N ) )
 
Theoremelfzom1elfzo 9160 Membership in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.)
 |-  ( ( N  e.  ZZ  /\  I  e.  (
 0..^ ( N  -  1 ) ) ) 
 ->  I  e.  (
 0..^ N ) )
 
Theoremfzval3 9161 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 9162 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ZZ  ->  ( A..^ ( A  +  1 ) )  =  { A }
 )
 
Theoremelfzomin 9163 Membership of an integer in the smallest open range of integers. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  ( Z  e.  ZZ  ->  Z  e.  ( Z..^ ( Z  +  1 ) ) )
 
Theoremzpnn0elfzo 9164 Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  ( ( Z  e.  ZZ  /\  N  e.  NN0 )  ->  ( Z  +  N )  e.  ( Z..^ ( ( Z  +  N )  +  1
 ) ) )
 
Theoremzpnn0elfzo1 9165 Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  ( ( Z  e.  ZZ  /\  N  e.  NN0 )  ->  ( Z  +  N )  e.  ( Z..^ ( Z  +  ( N  +  1 )
 ) ) )
 
Theoremfzosplitsnm1 9166 Removing a singleton from a half-open integer 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
 ) } ) )
 
Theoremelfzonlteqm1 9167 If an element of a half-open integer range is not less than the upper bound of the range decreased by 1, it must be equal to the upper bound of the range decreased by 1. (Contributed by AV, 3-Nov-2018.)
 |-  ( ( A  e.  ( 0..^ B )  /\  -.  A  <  ( B  -  1 ) ) 
 ->  A  =  ( B  -  1 ) )
 
Theoremfzonn0p1 9168 A nonnegative integer is element of the half-open range of nonnegative integers with the element increased by one as an upper bound. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
 |-  ( N  e.  NN0  ->  N  e.  ( 0..^ ( N  +  1
 ) ) )
 
Theoremfzossfzop1 9169 A half-open range of nonnegative integers is a subset of a half-open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
 |-  ( N  e.  NN0  ->  ( 0..^ N )  C_  ( 0..^ ( N  +  1 ) ) )
 
Theoremfzonn0p1p1 9170 If a nonnegative integer is element of a half-open range of nonnegative integers, increasing this integer by one results in an element of a half- open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
 |-  ( I  e.  (
 0..^ N )  ->  ( I  +  1
 )  e.  ( 0..^ ( N  +  1 ) ) )
 
Theoremelfzom1p1elfzo 9171 Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
 |-  ( ( N  e.  NN  /\  X  e.  (
 0..^ ( N  -  1 ) ) ) 
 ->  ( X  +  1 )  e.  ( 0..^ N ) )
 
Theoremfzo0ssnn0 9172 Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
 |-  ( 0..^ N ) 
 C_  NN0
 
Theoremfzo01 9173 Expressing the singleton of  0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( 0..^ 1 )  =  { 0 }
 
Theoremfzo12sn 9174 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 9175 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 9176 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 9177 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 } )
 
Theoremfzo0sn0fzo1 9178 A half-open range of nonnegative integers is the union of the singleton set containing 0 and a half-open range of positive integers. (Contributed by Alexander van der Vekens, 18-May-2018.)
 |-  ( N  e.  NN  ->  ( 0..^ N )  =  ( { 0 }  u.  ( 1..^ N ) ) )
 
Theoremfzoend 9179 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 9180 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 ) )
 
Theoremssfzo12 9181 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 ) ) )
 
Theoremssfzo12bi 9182 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.)
 |-  ( ( ( K  e.  ZZ  /\  L  e.  ZZ )  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  <  L )  ->  (
 ( K..^ L ) 
 C_  ( M..^ N ) 
 <->  ( M  <_  K  /\  L  <_  N )
 ) )
 
Theoremubmelm1fzo 9183 The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.)
 |-  ( K  e.  (
 0..^ N )  ->  ( ( N  -  K )  -  1
 )  e.  ( 0..^ N ) )
 
Theoremfzofzp1 9184 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 9185 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 9186 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 ) ) ) )
 
Theoremelfzonelfzo 9187 If an element of a half-open integer range 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 ) ) )
 
Theoremelfzomelpfzo 9188 An integer increased by another integer is an element of a half-open integer range if and only if the integer is contained in the half-open integer range 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 ) ) )
 
Theorempeano2fzor 9189 A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  ( K  +  1 )  e.  ( M..^ N ) )  ->  K  e.  ( M..^ N ) )
 
Theoremfzosplitsn 9190 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 }
 ) )
 
Theoremfzosplitprm1 9191 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  <  B ) 
 ->  ( A..^ ( B  +  1 ) )  =  ( ( A..^ ( B  -  1
 ) )  u.  {
 ( B  -  1
 ) ,  B }
 ) )
 
Theoremfzosplitsni 9192 Membership in a half-open range extended 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 ) ) )
 
Theoremfzisfzounsn 9193 A finite interval of integers as union of a half-open integer range and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( A ... B )  =  ( ( A..^ B )  u.  { B } ) )
 
Theoremfzostep1 9194 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 ) )
 
Theoremfzoshftral 9195* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 9071. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
 |-  ( ( 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 )
 )
 
Theoremfzind2 9196* 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 step. Version of fzind 8411 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 )
 
Theoremfvinim0ffz 9197 The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
 |-  ( ( F :
 ( 0 ... K )
 --> V  /\  K  e.  NN0 )  ->  ( (
 ( F " {
 0 ,  K }
 )  i^i  ( F " ( 1..^ K ) ) )  =  (/)  <->  (
 ( F `  0
 )  e/  ( F " ( 1..^ K ) )  /\  ( F `
  K )  e/  ( F " ( 1..^ K ) ) ) ) )
 
Theoremsubfzo0 9198 The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)
 |-  ( ( I  e.  ( 0..^ N ) 
 /\  J  e.  (
 0..^ N ) ) 
 ->  ( -u N  <  ( I  -  J )  /\  ( I  -  J )  <  N ) )
 
3.5.7  Rational numbers (cont.)
 
Theoremqtri3or 9199 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
 |-  ( ( M  e.  QQ  /\  N  e.  QQ )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
 
Theoremqletric 9200 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  <_  B  \/  B  <_  A ) )
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