P. 107 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10601-10700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fzossnn0 10601	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 10602	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 10603	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 10604	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( ( A..^ B ) i^i ( B..^ C ) ) = (/)
Theorem	fzouzsplit 10605	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 10606	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 10607	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 10608	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 10609	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 10610	Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|- ( A e. ( 0..^ B ) <-> ( A e. NN0 /\ B e. NN /\ A < B ) )
Theorem	nn0p1elfzo 10611	A nonnegative integer increased by 1 which is less than or equal to another integer is an element of a half-open range of integers. (Contributed by AV, 27-Feb-2021.)
|- ( ( K e. NN0 /\ N e. NN0 /\ ( K + 1 ) <_ N ) -> K e. ( 0..^ N ) )
Theorem	fzo1fzo0n0 10612	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 10613	Membership in a half-open range of nonnegative integers, generalization of elfzo0 10610 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 10614	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 10615	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 10616	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 10617	If a nonnegative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
|- ( ( K =/= M /\ K e. ( 0 ... M ) ) -> K e. ( 0..^ M ) )
Theorem	fz1fzo0m1 10618	Translation of one between closed and open integer ranges. (Contributed by Thierry Arnoux, 28-Jul-2020.)
|- ( M e. ( 1 ... N ) -> ( M - 1 ) e. ( 0..^ N ) )
Theorem	fzossnn 10619	Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
|- ( 1..^ N ) C_ NN
Theorem	elfzo1 10620	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 10621*	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 10622	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 10623	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 10624	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 10625	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 10626	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 10627	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 10628	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 10629	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 10630	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 10631	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 10632	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 10633	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 10634	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 10635	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 10636	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 10637	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 10638	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 10639	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 10640	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 10641	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 10642	Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
|- ( ( Z e. ZZ /\ N e. NN0 ) -> ( Z + N ) e. ( Z..^ ( ( Z + N ) + 1 ) ) )
Theorem	zpnn0elfzo1 10643	Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
|- ( ( Z e. ZZ /\ N e. NN0 ) -> ( Z + N ) e. ( Z..^ ( Z + ( N + 1 ) ) ) )
Theorem	fzosplitsnm1 10644	Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
|- ( ( A e. ZZ /\ B e. ( ZZ>= ` ( A + 1 ) ) ) -> ( A..^ B ) = ( ( A..^ ( B - 1 ) ) u. { ( B - 1 ) } ) )
Theorem	elfzonlteqm1 10645	If an element of a half-open integer range is not less than the upper bound of the range decreased by 1, it must be equal to the upper bound of the range decreased by 1. (Contributed by AV, 3-Nov-2018.)
|- ( ( A e. ( 0..^ B ) /\ -. A < ( B - 1 ) ) -> A = ( B - 1 ) )
Theorem	fzonn0p1 10646	A nonnegative integer is element of the half-open range of nonnegative integers with the element increased by one as an upper bound. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
|- ( N e. NN0 -> N e. ( 0..^ ( N + 1 ) ) )
Theorem	fzossfzop1 10647	A half-open range of nonnegative integers is a subset of a half-open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
|- ( N e. NN0 -> ( 0..^ N ) C_ ( 0..^ ( N + 1 ) ) )
Theorem	fzonn0p1p1 10648	If a nonnegative integer is element of a half-open range of nonnegative integers, increasing this integer by one results in an element of a half- open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
|- ( I e. ( 0..^ N ) -> ( I + 1 ) e. ( 0..^ ( N + 1 ) ) )
Theorem	elfzom1p1elfzo 10649	Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
|- ( ( N e. NN /\ X e. ( 0..^ ( N - 1 ) ) ) -> ( X + 1 ) e. ( 0..^ N ) )
Theorem	fzo0ssnn0 10650	Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
|- ( 0..^ N ) C_ NN0
Theorem	fzo01 10651	Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( 0..^ 1 ) = { 0 }
Theorem	fzo12sn 10652	A 1-based half-open integer interval up to, but not including, 2 is a singleton. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
|- ( 1..^ 2 ) = { 1 }
Theorem	fzo0to2pr 10653	A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
|- ( 0..^ 2 ) = { 0 , 1 }
Theorem	fzo0to3tp 10654	A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
|- ( 0..^ 3 ) = { 0 , 1 , 2 }
Theorem	fzo0to42pr 10655	A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
|- ( 0..^ 4 ) = ( { 0 , 1 } u. { 2 , 3 } )
Theorem	fzo0sn0fzo1 10656	A half-open range of nonnegative integers is the union of the singleton set containing 0 and a half-open range of positive integers. (Contributed by Alexander van der Vekens, 18-May-2018.)
|- ( N e. NN -> ( 0..^ N ) = ( { 0 } u. ( 1..^ N ) ) )
Theorem	fzoend 10657	The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( A e. ( A..^ B ) -> ( B - 1 ) e. ( A..^ B ) )
Theorem	fzo0end 10658	The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|- ( B e. NN -> ( B - 1 ) e. ( 0..^ B ) )
Theorem	ssfzo12 10659	Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
|- ( ( K e. ZZ /\ L e. ZZ /\ K < L ) -> ( ( K..^ L ) C_ ( M..^ N ) -> ( M <_ K /\ L <_ N ) ) )
Theorem	ssfzo12bi 10660	Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.)
|- ( ( ( K e. ZZ /\ L e. ZZ ) /\ ( M e. ZZ /\ N e. ZZ ) /\ K < L ) -> ( ( K..^ L ) C_ ( M..^ N ) <-> ( M <_ K /\ L <_ N ) ) )
Theorem	ubmelm1fzo 10661	The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.)
|- ( K e. ( 0..^ N ) -> ( ( N - K ) - 1 ) e. ( 0..^ N ) )
Theorem	fzofzp1 10662	If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- ( C e. ( A..^ B ) -> ( C + 1 ) e. ( A ... B ) )
Theorem	fzofzp1b 10663	If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- ( C e. ( ZZ>= ` A ) -> ( C e. ( A..^ B ) <-> ( C + 1 ) e. ( A ... B ) ) )
Theorem	elfzom1b 10664	An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1..^ N ) <-> ( K - 1 ) e. ( 0..^ ( N - 1 ) ) ) )
Theorem	elfzonelfzo 10665	If an element of a half-open integer range is not contained in the lower subrange, it must be in the upper subrange. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
|- ( N e. ZZ -> ( ( K e. ( M..^ R ) /\ -. K e. ( M..^ N ) ) -> K e. ( N..^ R ) ) )
Theorem	elfzomelpfzo 10666	An integer increased by another integer is an element of a half-open integer range if and only if the integer is contained in the half-open integer range with bounds decreased by the other integer. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ L e. ZZ ) ) -> ( K e. ( ( M - L )..^ ( N - L ) ) <-> ( K + L ) e. ( M..^ N ) ) )
Theorem	peano2fzor 10667	A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
|- ( ( K e. ( ZZ>= ` M ) /\ ( K + 1 ) e. ( M..^ N ) ) -> K e. ( M..^ N ) )
Theorem	fzosplitsn 10668	Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- ( B e. ( ZZ>= ` A ) -> ( A..^ ( B + 1 ) ) = ( ( A..^ B ) u. { B } ) )
Theorem	fzosplitpr 10669	Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
|- ( B e. ( ZZ>= ` A ) -> ( A..^ ( B + 2 ) ) = ( ( A..^ B ) u. { B , ( B + 1 ) } ) )
Theorem	fzosplitprm1 10670	Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
|- ( ( A e. ZZ /\ B e. ZZ /\ A < B ) -> ( A..^ ( B + 1 ) ) = ( ( A..^ ( B - 1 ) ) u. { ( B - 1 ) , B } ) )
Theorem	fzosplitsni 10671	Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- ( B e. ( ZZ>= ` A ) -> ( C e. ( A..^ ( B + 1 ) ) <-> ( C e. ( A..^ B ) \/ C = B ) ) )
Theorem	fzisfzounsn 10672	A finite interval of integers as union of a half-open integer range and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
|- ( B e. ( ZZ>= ` A ) -> ( A ... B ) = ( ( A..^ B ) u. { B } ) )
Theorem	fzostep1 10673	Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- ( A e. ( B..^ C ) -> ( ( A + 1 ) e. ( B..^ C ) \/ ( A + 1 ) = C ) )
Theorem	fzoshftral 10674*	Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10532. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( A. j e. ( M..^ N ) ph <-> A. k e. ( ( M + K )..^ ( N + K ) ) [. ( k - K ) / j ]. ph ) )
Theorem	fzind2 10675*	Induction on the integers from M to N inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Version of fzind 9772 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
|- ( x = M -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = K -> ( ph <-> ta ) )   &    |- ( N e. ( ZZ>= ` M ) -> ps )   &    |- ( y e. ( M..^ N ) -> ( ch -> th ) )   =>    |- ( K e. ( M ... N ) -> ta )
Theorem	exfzdc 10676*	Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ n e. ( M ... N ) ) -> DECID ps )   =>    |- ( ph -> DECID E. n e. ( M ... N ) ps )
Theorem	fvinim0ffz 10677	The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
|- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( ( ( F " { 0 , K } ) i^i ( F " ( 1..^ K ) ) ) = (/) <-> ( ( F ` 0 ) e/ ( F " ( 1..^ K ) ) /\ ( F ` K ) e/ ( F " ( 1..^ K ) ) ) ) )
Theorem	subfzo0 10678	The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)
|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) )
Theorem	zsupcllemstep 10679*	Lemma for zsupcl 10681. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.)
|- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps )   =>    |- ( K e. ( ZZ>= ` M ) -> ( ( ( ph /\ A. n e. ( ZZ>= ` K ) -. ps ) -> E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) ) -> ( ( ph /\ A. n e. ( ZZ>= ` ( K + 1 ) ) -. ps ) -> E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) ) ) )
Theorem	zsupcllemex 10680*	Lemma for zsupcl 10681. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.)
|- ( ph -> M e. ZZ )   &    |- ( n = M -> ( ps <-> ch ) )   &    |- ( ph -> ch )   &    |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps )   &    |- ( ph -> E. j e. ( ZZ>= ` M ) A. n e. ( ZZ>= ` j ) -. ps )   =>    |- ( ph -> E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) )
Theorem	zsupcl 10681*	Closure of supremum for decidable integer properties. The property which defines the set we are taking the supremum of must (a) be true at M (which corresponds to the nonempty condition of classical supremum theorems), (b) decidable at each value after M, and (c) be false after j (which corresponds to the upper bound condition found in classical supremum theorems). (Contributed by Jim Kingdon, 7-Dec-2021.)
|- ( ph -> M e. ZZ )   &    |- ( n = M -> ( ps <-> ch ) )   &    |- ( ph -> ch )   &    |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps )   &    |- ( ph -> E. j e. ( ZZ>= ` M ) A. n e. ( ZZ>= ` j ) -. ps )   =>    |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) )
Theorem	zssinfcl 10682*	The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. B -. y < x /\ A. y e. RR ( x < y -> E. z e. B z < y ) ) )   &    |- ( ph -> B C_ ZZ )   &    |- ( ph -> inf ( B , RR , < ) e. ZZ )   =>    |- ( ph -> inf ( B , RR , < ) e. B )
Theorem	infssuzex 10683*	Existence of the infimum of a subset of an upper set of integers. (Contributed by Jim Kingdon, 13-Jan-2022.)
|- ( ph -> M e. ZZ )   &    |- S = { n e. ( ZZ>= ` M ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> E. x e. RR ( A. y e. S -. y < x /\ A. y e. RR ( x < y -> E. z e. S z < y ) ) )
Theorem	infssuzledc 10684*	The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-Jan-2022.)
|- ( ph -> M e. ZZ )   &    |- S = { n e. ( ZZ>= ` M ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) <_ A )
Theorem	infssuzcldc 10685*	The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.)
|- ( ph -> M e. ZZ )   &    |- S = { n e. ( ZZ>= ` M ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) e. S )
Theorem	infssfzcldc 10686*	The infimum of a decidable inhabited subset of an integer range is a member of the set. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- S = { n e. ( M ... N ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) e. S )
Theorem	infssfzledc 10687*	The infimum of a decidable inhabited subset of an integer range is a lower bound for that set. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- S = { n e. ( M ... N ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) <_ A )
Theorem	suprzubdc 10688*	The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
|- ( ph -> A C_ ZZ )   &    |- ( ph -> A. x e. ZZ DECID x e. A )   &    |- ( ph -> E. x e. ZZ A. y e. A y <_ x )   &    |- ( ph -> B e. A )   =>    |- ( ph -> B <_ sup ( A , RR , < ) )
Theorem	nninfdcex 10689*	A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> E. y y e. A )   =>    |- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )
Theorem	zsupssdc 10690*	An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8306.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
|- ( ph -> A C_ ZZ )   &    |- ( ph -> E. x x e. A )   &    |- ( ph -> A. x e. ZZ DECID x e. A )   &    |- ( ph -> E. x e. ZZ A. y e. A y <_ x )   =>    |- ( ph -> E. x e. A ( A. y e. A -. x < y /\ A. y e. B ( y < x -> E. z e. A y < z ) ) )
Theorem	suprzcl2dc 10691*	The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8306.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.)
|- ( ph -> A C_ ZZ )   &    |- ( ph -> A. x e. ZZ DECID x e. A )   &    |- ( ph -> E. x e. ZZ A. y e. A y <_ x )   &    |- ( ph -> E. x x e. A )   =>    |- ( ph -> sup ( A , RR , < ) e. A )
4.5.7  Rational numbers (cont.)
Theorem	qtri3or 10692	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( M e. QQ /\ N e. QQ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	qletric 10693	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B <_ A ) )
Theorem	qlelttric 10694	Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B < A ) )
Theorem	qltnle 10695	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A < B <-> -. B <_ A ) )
Theorem	qdceq 10696	Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A = B )
Theorem	qdclt 10697	Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A < B )
Theorem	qdcle 10698	Rational <_ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A <_ B )
Theorem	exbtwnzlemstep 10699*	Lemma for exbtwnzlemex 10701. Induction step. (Contributed by Jim Kingdon, 10-May-2022.)
|- ( ph -> K e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ( ph /\ n e. ZZ ) -> ( n <_ A \/ A < n ) )   =>    |- ( ( ph /\ E. m e. ZZ ( m <_ A /\ A < ( m + ( K + 1 ) ) ) ) -> E. m e. ZZ ( m <_ A /\ A < ( m + K ) ) )
Theorem	exbtwnzlemshrink 10700*	Lemma for exbtwnzlemex 10701. Shrinking the range around A. (Contributed by Jim Kingdon, 10-May-2022.)
|- ( ph -> J e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ( ph /\ n e. ZZ ) -> ( n <_ A \/ A < n ) )   =>    |- ( ( ph /\ E. m e. ZZ ( m <_ A /\ A < ( m + J ) ) ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )