P. 105 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fz0sn 10401	An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
|- ( 0 ... 0 ) = { 0 }
Theorem	fz0tp 10402	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 }
Theorem	fz0to3un2pr 10403	An integer range from 0 to 3 is the union of two unordered pairs. (Contributed by AV, 7-Feb-2021.)
|- ( 0 ... 3 ) = ( { 0 , 1 } u. { 2 , 3 } )
Theorem	fz0to4untppr 10404	An integer range from 0 to 4 is the union of a triple and a pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
|- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )
Theorem	elfz0ubfz0 10405	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 ) )
Theorem	elfz0fzfz0 10406	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 ) )
Theorem	fz0fzelfz0 10407	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 ) )
Theorem	fznn0sub2 10408	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 ) )
Theorem	uzsubfz0 10409	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 ) )
Theorem	fz0fzdiffz0 10410	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 ) )
Theorem	elfzmlbm 10411	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 ) ) )
Theorem	elfzmlbp 10412	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 ) )
Theorem	fzctr 10413	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 ) ) )
Theorem	difelfzle 10414	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 ) )
Theorem	difelfznle 10415	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 ) )
Theorem	nn0split 10416	Express the set of nonnegative integers as the disjoint (see nn0disj 10418) 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 ) ) ) )
Theorem	nnsplit 10417	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 ) ) ) )
Theorem	nn0disj 10418	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 ) ) ) = (/)
Theorem	1fv 10419	A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
|- ( ( N e. V /\ P = { <. 0 , N >. } ) -> ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) )
Theorem	4fvwrd4 10420*	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 ) ) )
Theorem	2ffzeq 10421*	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 ) ) ) )
4.5.6  Half-open integer ranges
Syntax	cfzo 10422	Syntax for half-open integer ranges.
class ..^
Definition	df-fzo 10423*	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 10289, which includes N. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 10459 with fzsplit 10331, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ..^ = ( m e. ZZ , n e. ZZ |-> ( m ... ( n - 1 ) ) )
Theorem	fzof 10424	Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ..^ : ( ZZ X. ZZ ) --> ~P ZZ
Theorem	elfzoel1 10425	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> B e. ZZ )
Theorem	elfzoel2 10426	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> C e. ZZ )
Theorem	elfzoelz 10427	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> A e. ZZ )
Theorem	fzoval 10428	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 ) ) )
Theorem	elfzo 10429	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 ) ) )
Theorem	elfzo2 10430	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 ) )
Theorem	elfzouz 10431	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( M..^ N ) -> K e. ( ZZ>= ` M ) )
Theorem	nelfzo 10432	An integer not being a member of a half-open finite set of integers. (Contributed by AV, 29-Apr-2020.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e/ ( M..^ N ) <-> ( K < M \/ N <_ K ) ) )
Theorem	fzodcel 10433	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 ) )
Theorem	fzolb 10434	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 ) )
Theorem	fzolb2 10435	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 ) )
Theorem	elfzole1 10436	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 )
Theorem	elfzolt2 10437	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 )
Theorem	elfzolt3 10438	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 )
Theorem	elfzolt2b 10439	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 ) )
Theorem	elfzolt3b 10440	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 ) )
Theorem	fzonel 10441	A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
|- -. B e. ( A..^ B )
Theorem	elfzouz2 10442	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 ) )
Theorem	elfzofz 10443	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 ) )
Theorem	elfzo3 10444	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 ) ) )
Theorem	fzom 10445*	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 ) )
Theorem	fzossfz 10446	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 )
Theorem	fzon 10447	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 ) = (/) ) )
Theorem	fzo0n 10448	A half-open range of nonnegative integers is empty iff the upper bound is not positive. (Contributed by AV, 2-May-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N <_ M <-> ( 0..^ ( N - M ) ) = (/) ) )
Theorem	fzonlt0 10449	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 ) = (/) ) )
Theorem	fzo0 10450	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 ) = (/)
Theorem	fzonnsub 10451	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 )
Theorem	fzonnsub2 10452	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 )
Theorem	fzoss1 10453	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 ) )
Theorem	fzoss2 10454	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 ) )
Theorem	fzossrbm1 10455	Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
|- ( N e. ZZ -> ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) )
Theorem	fzo0ss1 10456	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 )
Theorem	fzossnn0 10457	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 )
Theorem	fzospliti 10458	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 ) ) )
Theorem	fzosplit 10459	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 ) ) )
Theorem	fzodisj 10460	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( ( A..^ B ) i^i ( B..^ C ) ) = (/)
Theorem	fzouzsplit 10461	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 ) ) )
Theorem	fzouzdisj 10462	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 ) ) = (/)
Theorem	fzoun 10463	A half-open integer range as union of two half-open integer ranges. (Contributed by AV, 23-Apr-2022.)
|- ( ( B e. ( ZZ>= ` A ) /\ C e. NN0 ) -> ( A..^ ( B + C ) ) = ( ( A..^ B ) u. ( B..^ ( B + C ) ) ) )
Theorem	fzodisjsn 10464	A half-open integer range and the singleton of its upper bound are disjoint. (Contributed by AV, 7-Mar-2021.)
|- ( ( A..^ B ) i^i { B } ) = (/)
Theorem	lbfzo0 10465	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 )
Theorem	elfzo0 10466	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 ) )
Theorem	nn0p1elfzo 10467	A nonnegative integer increased by 1 which is less than or equal to another integer is an element of a half-open range of integers. (Contributed by AV, 27-Feb-2021.)
|- ( ( K e. NN0 /\ N e. NN0 /\ ( K + 1 ) <_ N ) -> K e. ( 0..^ N ) )
Theorem	fzo1fzo0n0 10468	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 ) )
Theorem	elfzo0z 10469	Membership in a half-open range of nonnegative integers, generalization of elfzo0 10466 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 ) )
Theorem	elfzo0le 10470	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 )
Theorem	elfzonn0 10471	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 )
Theorem	fzonmapblen 10472	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 )
Theorem	fzofzim 10473	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 ) )
Theorem	fz1fzo0m1 10474	Translation of one between closed and open integer ranges. (Contributed by Thierry Arnoux, 28-Jul-2020.)
|- ( M e. ( 1 ... N ) -> ( M - 1 ) e. ( 0..^ N ) )
Theorem	fzossnn 10475	Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
|- ( 1..^ N ) C_ NN
Theorem	elfzo1 10476	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 ) )
Theorem	fzo0m 10477*	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 )
Theorem	fzoaddel 10478	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 ) ) )
Theorem	fzo0addel 10479	Translate membership in a 0-based half-open integer range. (Contributed by AV, 30-Apr-2020.)
|- ( ( A e. ( 0..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D..^ ( C + D ) ) )
Theorem	fzo0addelr 10480	Translate membership in a 0-based half-open integer range. (Contributed by AV, 30-Apr-2020.)
|- ( ( A e. ( 0..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D..^ ( D + C ) ) )
Theorem	fzoaddel2 10481	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 ) )
Theorem	elfzoextl 10482	Membership of an integer in an extended open range of integers, extension added to the left. (Contributed by AV, 31-Aug-2025.) Generalized by replacing the left border of the ranges. (Revised by SN, 18-Sep-2025.)
|- ( ( Z e. ( M..^ N ) /\ I e. NN0 ) -> Z e. ( M..^ ( I + N ) ) )
Theorem	elfzoext 10483	Membership of an integer in an extended open range of integers, extension added to the right. (Contributed by AV, 30-Apr-2020.) (Proof shortened by AV, 23-Sep-2025.)
|- ( ( Z e. ( M..^ N ) /\ I e. NN0 ) -> Z e. ( M..^ ( N + I ) ) )
Theorem	elincfzoext 10484	Membership of an increased integer in a correspondingly extended half-open range of integers. (Contributed by AV, 30-Apr-2020.)
|- ( ( Z e. ( M..^ N ) /\ I e. NN0 ) -> ( Z + I ) e. ( M..^ ( N + I ) ) )
Theorem	fzosubel 10485	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 ) ) )
Theorem	fzosubel2 10486	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 ) )
Theorem	fzosubel3 10487	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 ) )
Theorem	eluzgtdifelfzo 10488	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 ) ) ) )
Theorem	ige2m2fzo 10489	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 ) ) )
Theorem	fzocatel 10490	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 ) )
Theorem	ubmelfzo 10491	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 ) )
Theorem	elfzodifsumelfzo 10492	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 ) ) )
Theorem	elfzom1elp1fzo 10493	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 ) )
Theorem	elfzom1elfzo 10494	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 ) )
Theorem	fzval3 10495	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 ) ) )
Theorem	fzosn 10496	Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- ( A e. ZZ -> ( A..^ ( A + 1 ) ) = { A } )
Theorem	elfzomin 10497	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 ) ) )
Theorem	zpnn0elfzo 10498	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 ) ) )
Theorem	zpnn0elfzo1 10499	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 ) ) ) )
Theorem	fzosplitsnm1 10500	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 ) } ) )