P. 102 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	0e0iccpnf 10101	0 is a member of ( 0 [,] +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 e. ( 0 [,] +oo )
Theorem	ge0addcl 10102	The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) -> ( A + B ) e. ( 0 [,) +oo ) )
Theorem	ge0mulcl 10103	The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) -> ( A x. B ) e. ( 0 [,) +oo ) )
Theorem	ge0xaddcl 10104	The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) ) -> ( A +e B ) e. ( 0 [,] +oo ) )
Theorem	lbicc2 10105	The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
Theorem	ubicc2 10106	The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
Theorem	0elunit 10107	Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- 0 e. ( 0 [,] 1 )
Theorem	1elunit 10108	One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- 1 e. ( 0 [,] 1 )
Theorem	iooneg 10109	Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A (,) B ) <-> -u C e. ( -u B (,) -u A ) ) )
Theorem	iccneg 10110	Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A [,] B ) <-> -u C e. ( -u B [,] -u A ) ) )
Theorem	icoshft 10111	A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) -> ( X + C ) e. ( ( A + C ) [,) ( B + C ) ) ) )
Theorem	icoshftf1o 10112*	Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
|- F = ( x e. ( A [,) B ) |-> ( x + C ) )   =>    |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) )
Theorem	icodisj 10113	End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )
Theorem	ioodisj 10114	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 10115	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 10116	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 10117	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 10118	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 10119	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 10120	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 10121	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 10122	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 10123	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 10124	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 10125*	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 10126	( 0 [,] 1 ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
|- ( 0 [,] 1 ) C_ RR
Theorem	iccen 10127	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 10128	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 10129	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 10130*	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 10131 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 10131*	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 10132	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 10133	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 10134	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 10135	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 10136	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 10137	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 10138	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 10139	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 10140	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 10141	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 10142	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 10143	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 10144	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 10145	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 10146	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 10147	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 10148	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 10149	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 10150	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 10151	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 10152	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 10153	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 10154	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 10155	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 10156	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 10157	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 10158	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 10159*	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 10160	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 10161	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 10162	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 10163	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 10164	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 10165	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 10166	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 10167	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 10168	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 10169	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 10170	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 10171	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 10172	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 10173	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 10174	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 10175	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 10176	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 10177	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 10178	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 10179	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 10180	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 10181	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 10182	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 10183	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 10184	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 10185	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 10186	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 10187	A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( M e. ZZ -> ( M ... M ) = { M } )
Theorem	fzssp1 10188	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 10189	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 10190	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 10191	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 10192	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 10193	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 10194	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 10195	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 10196	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 10197	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 10198	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 10199	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 10200	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 ) } ) )