P. 105 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ubmelfzo 10401	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 10402	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 10403	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 10404	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 10405	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 10406	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 10407	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 10408	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 10409	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 10410	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 10411	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 10412	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 10413	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 10414	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 10415	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 10416	Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
|- ( 0..^ N ) C_ NN0
Theorem	fzo01 10417	Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( 0..^ 1 ) = { 0 }
Theorem	fzo12sn 10418	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 10419	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 10420	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 10421	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 10422	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 10423	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 10424	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 10425	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 10426	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 10427	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 10428	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 10429	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 10430	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 10431	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 10432	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 10433	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 10434	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	fzosplitprm1 10435	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 10436	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 10437	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 10438	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 10439*	Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10300. (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 10440*	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 9558 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 10441*	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 10442	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 10443	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 10444*	Lemma for zsupcl 10446. 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 10445*	Lemma for zsupcl 10446. 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 10446*	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 10447*	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 10448*	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 10449*	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 10450*	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	suprzubdc 10451*	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 10452*	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 10453*	An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8116.) (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 10454*	The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8116.) (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 10455	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( M e. QQ /\ N e. QQ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	qletric 10456	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B <_ A ) )
Theorem	qlelttric 10457	Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B < A ) )
Theorem	qltnle 10458	'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 10459	Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A = B )
Theorem	qdclt 10460	Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A < B )
Theorem	qdcle 10461	Rational <_ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A <_ B )
Theorem	exbtwnzlemstep 10462*	Lemma for exbtwnzlemex 10464. 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 10463*	Lemma for exbtwnzlemex 10464. 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 10464*	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 10465*	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 10466*	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 10467*	Lemma for rebtwn2z 10469. 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 10468*	Lemma for rebtwn2z 10469. 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 10469*	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 10470	Lemma for qbtwnre 10471. 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 )
Theorem	qbtwnre 10471*	The rational numbers are dense in RR: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.)
|- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
Theorem	qbtwnxr 10472*	The rational numbers are dense in RR*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
Theorem	qavgle 10473	The average of two rational numbers is less than or equal to at least one of them. (Contributed by Jim Kingdon, 3-Nov-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( ( ( A + B ) / 2 ) <_ A \/ ( ( A + B ) / 2 ) <_ B ) )
Theorem	ioo0 10474	An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> B <_ A ) )
Theorem	ioom 10475*	An open interval of extended reals is inhabited iff the lower argument is less than the upper argument. (Contributed by Jim Kingdon, 27-Nov-2021.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) <-> A < B ) )
Theorem	ico0 10476	An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> B <_ A ) )
Theorem	ioc0 10477	An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,] B ) = (/) <-> B <_ A ) )
Theorem	dfrp2 10478	Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
|- RR+ = ( 0 (,) +oo )
Theorem	elicod 10479	Membership in a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> C e. RR* )   &    |- ( ph -> A <_ C )   &    |- ( ph -> C < B )   =>    |- ( ph -> C e. ( A [,) B ) )
Theorem	icogelb 10480	An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,) B ) ) -> A <_ C )
Theorem	elicore 10481	A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C e. RR )
Theorem	xqltnle 10482	"Less than" expressed in terms of "less than or equal to", for extended numbers which are rational or +oo. We have not yet had enough usage of such numbers to warrant fully developing the concept, as in NN0* or RR*, so for now we just have a handful of theorems for what we need. (Contributed by Jim Kingdon, 5-Jun-2025.)
|- ( ( ( A e. QQ \/ A = +oo ) /\ ( B e. QQ \/ B = +oo ) ) -> ( A < B <-> -. B <_ A ) )
4.6  Elementary integer functions
4.6.1  The floor and ceiling functions
Syntax	cfl 10483	Extend class notation with floor (greatest integer) function.
class |_
Syntax	cceil 10484	Extend class notation to include the ceiling function.
class ⌈
Definition	df-fl 10485*	Define the floor (greatest integer less than or equal to) function. See flval 10487 for its value, flqlelt 10491 for its basic property, and flqcl 10488 for its closure. For example, ( |_ ` ( 3 / 2 ) ) = 1 while ( |_ ` -u ( 3 / 2 ) ) = -u 2 (ex-fl 16047). Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision. The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of an L-shaped left bracket and its reflection as a right bracket. In APL, the left-bracket alone is used, and we borrow this idea. (Thanks to Paul Chapman for this information.) (Contributed by NM, 14-Nov-2004.)
|- |_ = ( x e. RR |-> ( iota_ y e. ZZ ( y <_ x /\ x < ( y + 1 ) ) ) )
Definition	df-ceil 10486	The ceiling (least integer greater than or equal to) function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. See ceilqval 10523 for its value, ceilqge 10527 and ceilqm1lt 10529 for its basic properties, and ceilqcl 10525 for its closure. For example, (⌈ ` ( 3 / 2 ) ) = 2 while (⌈ ` -u ( 3 / 2 ) ) = -u 1 (ex-ceil 16048). As described in df-fl 10485 most theorems are only for rationals, not reals. The symbol ⌈ is inspired by the gamma shaped left bracket of the usual notation. (Contributed by David A. Wheeler, 19-May-2015.)
|- ⌈ = ( x e. RR |-> -u ( |_ ` -u x ) )
Theorem	flval 10487*	Value of the floor (greatest integer) function. The floor of A is the (unique) integer less than or equal to A whose successor is strictly greater than A. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
|- ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
Theorem	flqcl 10488	The floor (greatest integer) function yields an integer when applied to a rational (closure law). For a similar closure law for real numbers apart from any integer, see flapcl 10490. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) e. ZZ )
Theorem	apbtwnz 10489*	There is a unique greatest integer less than or equal to a real number which is apart from all integers. (Contributed by Jim Kingdon, 11-May-2022.)
|- ( ( A e. RR /\ A. n e. ZZ A # n ) -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
Theorem	flapcl 10490*	The floor (greatest integer) function yields an integer when applied to a real number apart from any integer. For example, an irrational number (see for example sqrt2irrap 12697) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
|- ( ( A e. RR /\ A. n e. ZZ A # n ) -> ( |_ ` A ) e. ZZ )
Theorem	flqlelt 10491	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )
Theorem	flqcld 10492	The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ph -> A e. QQ )   =>    |- ( ph -> ( |_ ` A ) e. ZZ )
Theorem	flqle 10493	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) <_ A )
Theorem	flqltp1 10494	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> A < ( ( |_ ` A ) + 1 ) )
Theorem	qfraclt1 10495	The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( A - ( |_ ` A ) ) < 1 )
Theorem	qfracge0 10496	The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> 0 <_ ( A - ( |_ ` A ) ) )
Theorem	flqge 10497	The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ ) -> ( B <_ A <-> B <_ ( |_ ` A ) ) )
Theorem	flqlt 10498	The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ ) -> ( A < B <-> ( |_ ` A ) < B ) )
Theorem	flid 10499	An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
|- ( A e. ZZ -> ( |_ ` A ) = A )
Theorem	flqidm 10500	The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` ( |_ ` A ) ) = ( |_ ` A ) )