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	ioodisj 10201	If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.)
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = (/) )
Theorem	iccshftr 10202	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A + R ) = C   &    |- ( B + R ) = D   =>    |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X + R ) e. ( C [,] D ) ) )
Theorem	iccshftri 10203	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. RR   &    |- B e. RR   &    |- R e. RR   &    |- ( A + R ) = C   &    |- ( B + R ) = D   =>    |- ( X e. ( A [,] B ) -> ( X + R ) e. ( C [,] D ) )
Theorem	iccshftl 10204	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A - R ) = C   &    |- ( B - R ) = D   =>    |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
Theorem	iccshftli 10205	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. RR   &    |- B e. RR   &    |- R e. RR   &    |- ( A - R ) = C   &    |- ( B - R ) = D   =>    |- ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) )
Theorem	iccdil 10206	Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A x. R ) = C   &    |- ( B x. R ) = D   =>    |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) )
Theorem	iccdili 10207	Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. RR   &    |- B e. RR   &    |- R e. RR+   &    |- ( A x. R ) = C   &    |- ( B x. R ) = D   =>    |- ( X e. ( A [,] B ) -> ( X x. R ) e. ( C [,] D ) )
Theorem	icccntr 10208	Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A / R ) = C   &    |- ( B / R ) = D   =>    |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X / R ) e. ( C [,] D ) ) )
Theorem	icccntri 10209	Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. RR   &    |- B e. RR   &    |- R e. RR+   &    |- ( A / R ) = C   &    |- ( B / R ) = D   =>    |- ( X e. ( A [,] B ) -> ( X / R ) e. ( C [,] D ) )
Theorem	divelunit 10210	A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) e. ( 0 [,] 1 ) <-> A <_ B ) )
Theorem	lincmb01cmp 10211	A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.)
|- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) )
Theorem	iccf1o 10212*	Describe a bijection from [ 0 , 1 ] to an arbitrary nontrivial closed interval [ A , B ]. (Contributed by Mario Carneiro, 8-Sep-2015.)
|- F = ( x e. ( 0 [,] 1 ) |-> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) )   =>    |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) /\ `' F = ( y e. ( A [,] B ) |-> ( ( y - A ) / ( B - A ) ) ) ) )
Theorem	unitssre 10213	( 0 [,] 1 ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
|- ( 0 [,] 1 ) C_ RR
Theorem	iccen 10214	Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
|- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( 0 [,] 1 ) ~~ ( A [,] B ) )
Theorem	zltaddlt1le 10215	The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.)
|- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) < N <-> ( M + A ) <_ N ) )
4.5.4  Finite intervals of integers
Syntax	cfz 10216	Extend class notation to include the notation for a contiguous finite set of integers. Read " M ... N " as "the set of integers from M to N inclusive". This symbol is also used informally in some comments to denote an ellipsis, e.g., A + A ^ 2 + ... + A ^ ( N - 1 ).
class ...
Definition	df-fz 10217*	Define an operation that produces a finite set of sequential integers. Read " M ... N " as "the set of integers from M to N inclusive". See fzval 10218 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
|- ... = ( m e. ZZ , n e. ZZ |-> { k e. ZZ | ( m <_ k /\ k <_ n ) } )
Theorem	fzval 10218*	The value of a finite set of sequential integers. E.g., 2 ... 5 means the set { 2 , 3 , 4 , 5 }. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where NNk means our 1 ... k; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
Theorem	fzval2 10219	An alternate way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = ( ( M [,] N ) i^i ZZ ) )
Theorem	fzf 10220	Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- ... : ( ZZ X. ZZ ) --> ~P ZZ
Theorem	elfz1 10221	Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> ( K e. ZZ /\ M <_ K /\ K <_ N ) ) )
Theorem	elfz 10222	Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> ( M <_ K /\ K <_ N ) ) )
Theorem	elfz2 10223	Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show M e. ZZ and N e. ZZ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) <-> ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M <_ K /\ K <_ N ) ) )
Theorem	elfzd 10224	Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K e. ZZ )   &    |- ( ph -> M <_ K )   &    |- ( ph -> K <_ N )   =>    |- ( ph -> K e. ( M ... N ) )
Theorem	elfz5 10225	Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
|- ( ( K e. ( ZZ>= ` M ) /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> K <_ N ) )
Theorem	elfz4 10226	Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M <_ K /\ K <_ N ) ) -> K e. ( M ... N ) )
Theorem	elfzuzb 10227	Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) <-> ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) )
Theorem	eluzfz 10228	Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) -> K e. ( M ... N ) )
Theorem	elfzuz 10229	A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
Theorem	elfzuz3 10230	Membership in a finite set of sequential integers implies membership in an upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
Theorem	elfzel2 10231	Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> N e. ZZ )
Theorem	elfzel1 10232	Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> M e. ZZ )
Theorem	elfzelz 10233	A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> K e. ZZ )
Theorem	elfzelzd 10234	A member of a finite set of sequential integers is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> K e. ( M ... N ) )   =>    |- ( ph -> K e. ZZ )
Theorem	elfzle1 10235	A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> M <_ K )
Theorem	elfzle2 10236	A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> K <_ N )
Theorem	elfzuz2 10237	Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` M ) )
Theorem	elfzle3 10238	Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> M <_ N )
Theorem	eluzfz1 10239	Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
Theorem	eluzfz2 10240	Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
Theorem	eluzfz2b 10241	Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) <-> N e. ( M ... N ) )
Theorem	elfz3 10242	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
|- ( N e. ZZ -> N e. ( N ... N ) )
Theorem	elfz1eq 10243	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
|- ( K e. ( N ... N ) -> K = N )
Theorem	elfzubelfz 10244	If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018.)
|- ( K e. ( M ... N ) -> N e. ( M ... N ) )
Theorem	peano2fzr 10245	A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
|- ( ( K e. ( ZZ>= ` M ) /\ ( K + 1 ) e. ( M ... N ) ) -> K e. ( M ... N ) )
Theorem	fzm 10246*	Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
|- ( E. x x e. ( M ... N ) <-> N e. ( ZZ>= ` M ) )
Theorem	fztri3or 10247	Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
Theorem	fzdcel 10248	Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID K e. ( M ... N ) )
Theorem	fznlem 10249	A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M -> ( M ... N ) = (/) ) )
Theorem	fzn 10250	A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> ( M ... N ) = (/) ) )
Theorem	fzen 10251	A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M ... N ) ~~ ( ( M + K ) ... ( N + K ) ) )
Theorem	fz1n 10252	A 1-based finite set of sequential integers is empty iff it ends at index 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. NN0 -> ( ( 1 ... N ) = (/) <-> N = 0 ) )
Theorem	0fz1 10253	Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
|- ( ( N e. NN0 /\ F Fn ( 1 ... N ) ) -> ( F = (/) <-> N = 0 ) )
Theorem	fz10 10254	There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( 1 ... 0 ) = (/)
Theorem	uzsubsubfz 10255	Membership of an integer greater than L decreased by ( L - M ) in an M based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
|- ( ( L e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` L ) ) -> ( N - ( L - M ) ) e. ( M ... N ) )
Theorem	uzsubsubfz1 10256	Membership of an integer greater than L decreased by ( L - 1 ) in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
|- ( ( L e. NN /\ N e. ( ZZ>= ` L ) ) -> ( N - ( L - 1 ) ) e. ( 1 ... N ) )
Theorem	ige3m2fz 10257	Membership of an integer greater than 2 decreased by 2 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
|- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. ( 1 ... N ) )
Theorem	fzsplit2 10258	Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
|- ( ( ( K + 1 ) e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) -> ( M ... N ) = ( ( M ... K ) u. ( ( K + 1 ) ... N ) ) )
Theorem	fzsplit 10259	Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
|- ( K e. ( M ... N ) -> ( M ... N ) = ( ( M ... K ) u. ( ( K + 1 ) ... N ) ) )
Theorem	fzdisj 10260	Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( K < M -> ( ( J ... K ) i^i ( M ... N ) ) = (/) )
Theorem	fz01en 10261	0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ( N e. ZZ -> ( 0 ... ( N - 1 ) ) ~~ ( 1 ... N ) )
Theorem	elfznn 10262	A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.)
|- ( K e. ( 1 ... N ) -> K e. NN )
Theorem	elfz1end 10263	A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( A e. NN <-> A e. ( 1 ... A ) )
Theorem	fz1ssnn 10264	A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
|- ( 1 ... A ) C_ NN
Theorem	fznn0sub 10265	Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> ( N - K ) e. NN0 )
Theorem	fzmmmeqm 10266	Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
|- ( M e. ( L ... N ) -> ( ( N - L ) - ( M - L ) ) = ( N - M ) )
Theorem	fzaddel 10267	Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( J e. ( M ... N ) <-> ( J + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
Theorem	fzsubel 10268	Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( J e. ( M ... N ) <-> ( J - K ) e. ( ( M - K ) ... ( N - K ) ) ) )
Theorem	fzopth 10269	A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) <-> ( M = J /\ N = K ) ) )
Theorem	fzass4 10270	Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( ( B e. ( A ... D ) /\ C e. ( B ... D ) ) <-> ( B e. ( A ... C ) /\ C e. ( A ... D ) ) )
Theorem	fzss1 10271	Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( ZZ>= ` M ) -> ( K ... N ) C_ ( M ... N ) )
Theorem	fzss2 10272	Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- ( N e. ( ZZ>= ` K ) -> ( M ... K ) C_ ( M ... N ) )
Theorem	fzssuz 10273	A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
|- ( M ... N ) C_ ( ZZ>= ` M )
Theorem	fzsn 10274	A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( M e. ZZ -> ( M ... M ) = { M } )
Theorem	fzssp1 10275	Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( M ... N ) C_ ( M ... ( N + 1 ) )
Theorem	fzssnn 10276	Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
|- ( M e. NN -> ( M ... N ) C_ NN )
Theorem	fzsuc 10277	Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
Theorem	fzpred 10278	Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
|- ( N e. ( ZZ>= ` M ) -> ( M ... N ) = ( { M } u. ( ( M + 1 ) ... N ) ) )
Theorem	fzpreddisj 10279	A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
|- ( N e. ( ZZ>= ` M ) -> ( { M } i^i ( ( M + 1 ) ... N ) ) = (/) )
Theorem	elfzp1 10280	Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... ( N + 1 ) ) <-> ( K e. ( M ... N ) \/ K = ( N + 1 ) ) ) )
Theorem	fzp1ss 10281	Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( M e. ZZ -> ( ( M + 1 ) ... N ) C_ ( M ... N ) )
Theorem	fzelp1 10282	Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> K e. ( M ... ( N + 1 ) ) )
Theorem	fzp1elp1 10283	Add one to an element of a finite set of integers. (Contributed by Jeff Madsen, 6-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> ( K + 1 ) e. ( M ... ( N + 1 ) ) )
Theorem	fznatpl1 10284	Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
|- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ( I + 1 ) e. ( 1 ... N ) )
Theorem	fzpr 10285	A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( M e. ZZ -> ( M ... ( M + 1 ) ) = { M , ( M + 1 ) } )
Theorem	fztp 10286	A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)
|- ( M e. ZZ -> ( M ... ( M + 2 ) ) = { M , ( M + 1 ) , ( M + 2 ) } )
Theorem	fzsuc2 10287	Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M - 1 ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
Theorem	fzp1disj 10288	( M ... ( N + 1 ) ) is the disjoint union of ( M ... N ) with { ( N + 1 ) }. (Contributed by Mario Carneiro, 7-Mar-2014.)
|- ( ( M ... N ) i^i { ( N + 1 ) } ) = (/)
Theorem	fzdifsuc 10289	Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)
|- ( N e. ( ZZ>= ` M ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
Theorem	fzprval 10290*	Two ways of defining the first two values of a sequence on NN. (Contributed by NM, 5-Sep-2011.)
|- ( A. x e. ( 1 ... 2 ) ( F ` x ) = if ( x = 1 , A , B ) <-> ( ( F ` 1 ) = A /\ ( F ` 2 ) = B ) )
Theorem	fztpval 10291*	Two ways of defining the first three values of a sequence on NN. (Contributed by NM, 13-Sep-2011.)
|- ( A. x e. ( 1 ... 3 ) ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( ( F ` 1 ) = A /\ ( F ` 2 ) = B /\ ( F ` 3 ) = C ) )
Theorem	fzrev 10292	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( K e. ( ( J - N ) ... ( J - M ) ) <-> ( J - K ) e. ( M ... N ) ) )
Theorem	fzrev2 10293	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( K e. ( M ... N ) <-> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) ) )
Theorem	fzrev2i 10294	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
|- ( ( J e. ZZ /\ K e. ( M ... N ) ) -> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) )
Theorem	fzrev3 10295	The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
|- ( K e. ZZ -> ( K e. ( M ... N ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )
Theorem	fzrev3i 10296	The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
|- ( K e. ( M ... N ) -> ( ( M + N ) - K ) e. ( M ... N ) )
Theorem	fznn 10297	Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)
|- ( N e. ZZ -> ( K e. ( 1 ... N ) <-> ( K e. NN /\ K <_ N ) ) )
Theorem	elfz1b 10298	Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
|- ( N e. ( 1 ... M ) <-> ( N e. NN /\ M e. NN /\ N <_ M ) )
Theorem	elfzm11 10299	Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... ( N - 1 ) ) <-> ( K e. ZZ /\ M <_ K /\ K < N ) ) )
Theorem	uzsplit 10300	Express an upper integer set as the disjoint (see uzdisj 10301) union of the first N values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
|- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` M ) = ( ( M ... ( N - 1 ) ) u. ( ZZ>= ` N ) ) )