P. 103 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10201-10300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ige2m1fz1 10201	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 10202	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 10203	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 10204	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 10205	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 10206	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 10207	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 10208	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 10209	Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( 1 ... N ) C_ ( 0 ... N )
Theorem	0elfz 10210	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 10211	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 10212	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 10213	An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
|- ( 0 ... 0 ) = { 0 }
Theorem	fz0tp 10214	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 10215	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 10216	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 10217	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 10218	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 10219	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 10220	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 10221	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 10222	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 10223	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 10224	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 10225	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 10226	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 10227	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 10228	Express the set of nonnegative integers as the disjoint (see nn0disj 10230) 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 10229	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 10230	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 10231	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 10232*	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 10233*	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 10234	Syntax for half-open integer ranges.
class ..^
Definition	df-fzo 10235*	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 10101, which includes N. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 10270 with fzsplit 10143, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ..^ = ( m e. ZZ , n e. ZZ |-> ( m ... ( n - 1 ) ) )
Theorem	fzof 10236	Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ..^ : ( ZZ X. ZZ ) --> ~P ZZ
Theorem	elfzoel1 10237	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> B e. ZZ )
Theorem	elfzoel2 10238	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> C e. ZZ )
Theorem	elfzoelz 10239	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> A e. ZZ )
Theorem	fzoval 10240	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 10241	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 10242	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 10243	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( M..^ N ) -> K e. ( ZZ>= ` M ) )
Theorem	nelfzo 10244	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 10245	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 10246	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 10247	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 10248	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 10249	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 10250	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 10251	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 10252	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 10253	A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
|- -. B e. ( A..^ B )
Theorem	elfzouz2 10254	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 10255	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 10256	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 10257*	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 10258	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 10259	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	fzonlt0 10260	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 10261	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 10262	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 10263	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 10264	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 10265	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 10266	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 10267	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 10268	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 10269	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 10270	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 10271	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( ( A..^ B ) i^i ( B..^ C ) ) = (/)
Theorem	fzouzsplit 10272	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 10273	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	lbfzo0 10274	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 10275	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 10276	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 10277	Membership in a half-open range of nonnegative integers, generalization of elfzo0 10275 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 10278	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 10279	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 10280	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 10281	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 10282	Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
|- ( 1..^ N ) C_ NN
Theorem	elfzo1 10283	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 10284*	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 10285	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	fzoaddel2 10286	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	fzosubel 10287	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 10288	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 10289	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 10290	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 10291	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 10292	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 10293	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 10294	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 10295	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 10296	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 10297	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 10298	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 10299	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 10300	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 ) ) )