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Theorem List for Intuitionistic Logic Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfz0ubfz0 9501 An element of a finite set of sequential nonnegative integers is an element of a finite set of sequential nonnegative integers with the upper bound being an element of the finite set of sequential nonnegative 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 ) )
 
Theoremelfz0fzfz0 9502 A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.)
 |-  ( ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... X ) ) 
 ->  M  e.  ( 0
 ... N ) )
 
Theoremfz0fzelfz0 9503 If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.)
 |-  ( ( N  e.  ( 0 ... R )  /\  M  e.  ( N ... R ) ) 
 ->  M  e.  ( 0
 ... R ) )
 
Theoremfznn0sub2 9504 Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  -  K )  e.  ( 0 ... N ) )
 
Theoremuzsubfz0 9505 Membership of an integer greater than L decreased by L in a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
 |-  ( ( L  e.  NN0  /\  N  e.  ( ZZ>= `  L ) )  ->  ( N  -  L )  e.  ( 0 ... N ) )
 
Theoremfz0fzdiffz0 9506 The difference of an integer in a finite set of sequential nonnegative integers and and an integer 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 nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.)
 |-  ( ( M  e.  ( 0 ... N )  /\  K  e.  ( M ... N ) ) 
 ->  ( K  -  M )  e.  ( 0 ... N ) )
 
Theoremelfzmlbm 9507 Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
 |-  ( K  e.  ( M ... N )  ->  ( K  -  M )  e.  ( 0 ... ( N  -  M ) ) )
 
Theoremelfzmlbp 9508 Subtracting the lower bound of a finite set 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 ) )
 
Theoremfzctr 9509 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 ) ) )
 
Theoremdifelfzle 9510 The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
 |-  ( ( K  e.  ( 0 ... N )  /\  M  e.  (
 0 ... N )  /\  K  <_  M )  ->  ( M  -  K )  e.  ( 0 ... N ) )
 
Theoremdifelfznle 9511 The difference of two integers from a finite set of sequential nonnegative integers increased by the upper bound is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
 |-  ( ( K  e.  ( 0 ... N )  /\  M  e.  (
 0 ... N )  /\  -.  K  <_  M )  ->  ( ( M  +  N )  -  K )  e.  ( 0 ... N ) )
 
Theoremnn0split 9512 Express the set of nonnegative integers as the disjoint (see nn0disj 9514) union of the first  N  +  1 values and the rest. (Contributed by AV, 8-Nov-2019.)
 |-  ( N  e.  NN0  ->  NN0  =  ( ( 0
 ... N )  u.  ( ZZ>= `  ( N  +  1 ) ) ) )
 
Theoremnnsplit 9513 Express the set of positive integers as the disjoint union of the first  N values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  ( N  e.  NN  ->  NN  =  ( ( 1 ... N )  u.  ( ZZ>= `  ( N  +  1 )
 ) ) )
 
Theoremnn0disj 9514 The first  N  +  1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)
 |-  ( ( 0 ...
 N )  i^i  ( ZZ>=
 `  ( N  +  1 ) ) )  =  (/)
 
Theorem1fv 9515 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 9516* 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 ) ) )
 
Theorem2ffzeq 9517* Two functions over 0 based finite set of sequential 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 ) ) ) )
 
3.5.6  Half-open integer ranges
 
Syntaxcfzo 9518 Syntax for half-open integer ranges.
 class ..^
 
Definitiondf-fzo 9519* 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 9394, which includes  N. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 9553 with fzsplit 9434, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ 
 =  ( m  e. 
 ZZ ,  n  e. 
 ZZ  |->  ( m ... ( n  -  1
 ) ) )
 
Theoremfzof 9520 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ : ( ZZ  X.  ZZ ) --> ~P ZZ
 
Theoremelfzoel1 9521 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  B  e.  ZZ )
 
Theoremelfzoel2 9522 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  C  e.  ZZ )
 
Theoremelfzoelz 9523 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  A  e.  ZZ )
 
Theoremfzoval 9524 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 9525 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 9526 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 9527 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( ZZ>= `  M )
 )
 
Theoremfzodcel 9528 Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  K  e.  ( M..^ N ) )
 
Theoremfzolb 9529 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 alternate 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 9530 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 alternate 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 9531 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 9532 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 9533 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 9534 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 9535 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 9536 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |- 
 -.  B  e.  ( A..^ B )
 
Theoremelfzouz2 9537 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 9538 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 9539 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 9540* 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 9541 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 9542 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 9543 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 9544 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 9545 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 9546 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 9547 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 9548 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 9549 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 9550 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 9551 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 9552 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 9553 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 9554 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( ( A..^ B )  i^i  ( B..^ C ) )  =  (/)
 
Theoremfzouzsplit 9555 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 9556 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 9557 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 9558 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 9559 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 9560 Membership in a half-open range of nonnegative integers, generalization of elfzo0 9558 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 9561 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 9562 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 9563 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 9564 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 9565 Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  ( 1..^ N ) 
 C_  NN
 
Theoremelfzo1 9566 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 9567* 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 9568 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 9569 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 9570 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 9571 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 9572 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 9573 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 9574 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 9575 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 9576 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 9577 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 9578 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 9579 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 9580 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 9581 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ZZ  ->  ( A..^ ( A  +  1 ) )  =  { A }
 )
 
Theoremelfzomin 9582 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 9583 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 9584 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 9585 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 9586 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 9587 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 9588 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 9589 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 9590 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 9591 Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
 |-  ( 0..^ N ) 
 C_  NN0
 
Theoremfzo01 9592 Expressing the singleton of  0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( 0..^ 1 )  =  { 0 }
 
Theoremfzo12sn 9593 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 9594 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 9595 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 9596 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 9597 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 9598 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 9599 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 9600 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 ) ) )
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