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	fzodisjsn 10601	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 10602	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 10603	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 10604	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 10605	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 10606	Membership in a half-open range of nonnegative integers, generalization of elfzo0 10603 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 10607	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 10608	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 10609	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 10610	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 10611	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 10612	Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
|- ( 1..^ N ) C_ NN
Theorem	elfzo1 10613	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 10614*	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 10615	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 10616	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 10617	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 10618	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 10619	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 10620	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 10621	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 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	fzosubel2 10623	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 10624	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 10625	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 10626	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 10627	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 10628	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 10629	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 10630	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 10631	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 10632	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 10633	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 10634	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 10635	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 10636	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 10637	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 10638	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 10639	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 10640	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 10641	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 10642	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 10643	Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
|- ( 0..^ N ) C_ NN0
Theorem	fzo01 10644	Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( 0..^ 1 ) = { 0 }
Theorem	fzo12sn 10645	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 10646	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 10647	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 10648	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 10649	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 10650	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 10651	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 10652	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 10653	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 10654	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 10655	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 10656	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 10657	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 10658	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 10659	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 10660	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 10661	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 10662	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 10663	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 10664	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 10665	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 10666	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 10667*	Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10525. (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 10668*	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 9765 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 10669*	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 10670	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 10671	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 10672*	Lemma for zsupcl 10674. 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 10673*	Lemma for zsupcl 10674. 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 10674*	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 10675*	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 10676*	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 10677*	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 10678*	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 10679*	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 10680*	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 10681*	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 10682*	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 10683*	An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8300.) (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 10684*	The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8300.) (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 10685	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( M e. QQ /\ N e. QQ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	qletric 10686	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B <_ A ) )
Theorem	qlelttric 10687	Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B < A ) )
Theorem	qltnle 10688	'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 10689	Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A = B )
Theorem	qdclt 10690	Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A < B )
Theorem	qdcle 10691	Rational <_ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A <_ B )
Theorem	exbtwnzlemstep 10692*	Lemma for exbtwnzlemex 10694. 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 10693*	Lemma for exbtwnzlemex 10694. 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 ) ) )
Theorem	exbtwnzlemex 10694*	Existence of an integer so that a given real number is between the integer and its successor. The real number must satisfy the n <_ A \/ A < n hypothesis. For example either a rational number or a number which is irrational (in the sense of being apart from any rational number) will meet this condition. The proof starts by finding two integers which are less than and greater than A. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on the n <_ A \/ A < n hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ph -> A e. RR )   &    |- ( ( ph /\ n e. ZZ ) -> ( n <_ A \/ A < n ) )   =>    |- ( ph -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
Theorem	exbtwnz 10695*	If a real number is between an integer and its successor, there is a unique greatest integer less than or equal to the real number. (Contributed by Jim Kingdon, 10-May-2022.)
|- ( ph -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )   &    |- ( ph -> A e. RR )   =>    |- ( ph -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
Theorem	qbtwnz 10696*	There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
Theorem	rebtwn2zlemstep 10697*	Lemma for rebtwn2z 10699. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
|- ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR /\ E. m e. ZZ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> E. m e. ZZ ( m < A /\ A < ( m + K ) ) )
Theorem	rebtwn2zlemshrink 10698*	Lemma for rebtwn2z 10699. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
|- ( ( A e. RR /\ J e. ( ZZ>= ` 2 ) /\ E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
Theorem	rebtwn2z 10699*	A real number can be bounded by integers above and below which are two apart. The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)
|- ( A e. RR -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
Theorem	qbtwnrelemcalc 10700	Lemma for qbtwnre 10701. Calculations involved in showing the constructed rational number is less than B. (Contributed by Jim Kingdon, 14-Oct-2021.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> M < ( A x. ( 2 x. N ) ) )   &    |- ( ph -> ( 1 / N ) < ( B - A ) )   =>    |- ( ph -> ( ( M + 2 ) / ( 2 x. N ) ) < B )