P. 104 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ige2m1fz1 10301	Membership of an integer greater than 1 decreased by 1 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
|- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. ( 1 ... N ) )
Theorem	ige2m1fz 10302	Membership in a 0 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.)
|- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. ( 0 ... N ) )
Theorem	fz01or 10303	An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.)
|- ( A e. ( 0 ... 1 ) <-> ( A = 0 \/ A = 1 ) )
4.5.5  Finite intervals of nonnegative integers Finite intervals of nonnegative integers (or "finite sets of sequential nonnegative integers") are finite intervals of integers with 0 as lower bound: ( 0 ... N ), usually abbreviated by "fz0".
Theorem	elfz2nn0 10304	Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( 0 ... N ) <-> ( K e. NN0 /\ N e. NN0 /\ K <_ N ) )
Theorem	fznn0 10305	Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
|- ( N e. NN0 -> ( K e. ( 0 ... N ) <-> ( K e. NN0 /\ K <_ N ) ) )
Theorem	elfznn0 10306	A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( 0 ... N ) -> K e. NN0 )
Theorem	elfz3nn0 10307	The upper bound of a nonempty finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( 0 ... N ) -> N e. NN0 )
Theorem	fz0ssnn0 10308	Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
|- ( 0 ... N ) C_ NN0
Theorem	fz1ssfz0 10309	Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( 1 ... N ) C_ ( 0 ... N )
Theorem	0elfz 10310	0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.)
|- ( N e. NN0 -> 0 e. ( 0 ... N ) )
Theorem	nn0fz0 10311	A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.)
|- ( N e. NN0 <-> N e. ( 0 ... N ) )
Theorem	elfz0add 10312	An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( N e. ( 0 ... A ) -> N e. ( 0 ... ( A + B ) ) ) )
Theorem	fz0sn 10313	An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
|- ( 0 ... 0 ) = { 0 }
Theorem	fz0tp 10314	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 10315	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 10316	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 10317	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 10318	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 10319	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 10320	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 10321	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 10322	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 10323	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 10324	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 10325	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 10326	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 10327	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 10328	Express the set of nonnegative integers as the disjoint (see nn0disj 10330) 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 10329	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 10330	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 10331	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 10332*	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 10333*	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 10334	Syntax for half-open integer ranges.
class ..^
Definition	df-fzo 10335*	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 10201, which includes N. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 10371 with fzsplit 10243, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ..^ = ( m e. ZZ , n e. ZZ |-> ( m ... ( n - 1 ) ) )
Theorem	fzof 10336	Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ..^ : ( ZZ X. ZZ ) --> ~P ZZ
Theorem	elfzoel1 10337	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> B e. ZZ )
Theorem	elfzoel2 10338	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> C e. ZZ )
Theorem	elfzoelz 10339	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> A e. ZZ )
Theorem	fzoval 10340	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 10341	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 10342	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 10343	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( M..^ N ) -> K e. ( ZZ>= ` M ) )
Theorem	nelfzo 10344	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 10345	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 10346	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 10347	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 10348	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 10349	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 10350	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 10351	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 10352	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 10353	A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
|- -. B e. ( A..^ B )
Theorem	elfzouz2 10354	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 10355	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 10356	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 10357*	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 10358	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 10359	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 10360	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 10361	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 10362	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 10363	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 10364	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 10365	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 10366	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 10367	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 10368	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 10369	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 10370	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 10371	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 10372	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( ( A..^ B ) i^i ( B..^ C ) ) = (/)
Theorem	fzouzsplit 10373	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 10374	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 10375	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 10376	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 10377	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 10378	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	fzo1fzo0n0 10379	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 10380	Membership in a half-open range of nonnegative integers, generalization of elfzo0 10378 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 10381	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 10382	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 10383	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 10384	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	fzossnn 10385	Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
|- ( 1..^ N ) C_ NN
Theorem	elfzo1 10386	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 10387*	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 10388	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 10389	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 10390	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 10391	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 10392	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 10393	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 10394	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 10395	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 10396	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 10397	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 10398	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 10399	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 10400	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 ) )