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Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssfzo12bi 9601 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 )
 ) )
 
Theoremubmelm1fzo 9602 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 ) )
 
Theoremfzofzp1 9603 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 ) )
 
Theoremfzofzp1b 9604 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 ) ) )
 
Theoremelfzom1b 9605 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 ) ) ) )
 
Theoremelfzonelfzo 9606 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 ) ) )
 
Theoremelfzomelpfzo 9607 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 ) ) )
 
Theorempeano2fzor 9608 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 ) )
 
Theoremfzosplitsn 9609 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 }
 ) )
 
Theoremfzosplitprm1 9610 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 }
 ) )
 
Theoremfzosplitsni 9611 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 ) ) )
 
Theoremfzisfzounsn 9612 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 } ) )
 
Theoremfzostep1 9613 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 ) )
 
Theoremfzoshftral 9614* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 9489. (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 )
 )
 
Theoremfzind2 9615* 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 8831 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 )
 
Theoremexfzdc 9616* 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 )
 
Theoremfvinim0ffz 9617 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 ) ) ) ) )
 
Theoremsubfzo0 9618 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 ) )
 
3.5.7  Rational numbers (cont.)
 
Theoremqtri3or 9619 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
 |-  ( ( M  e.  QQ  /\  N  e.  QQ )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
 
Theoremqletric 9620 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  <_  B  \/  B  <_  A ) )
 
Theoremqlelttric 9621 Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremqltnle 9622 '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 )
 )
 
Theoremqdceq 9623 Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  -> DECID  A  =  B )
 
Theoremexbtwnzlemstep 9624* Lemma for exbtwnzlemex 9626. 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 ) ) )
 
Theoremexbtwnzlemshrink 9625* Lemma for exbtwnzlemex 9626. 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 ) ) )
 
Theoremexbtwnzlemex 9626* 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 )
 ) )
 
Theoremexbtwnz 9627* 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 ) ) )
 
Theoremqbtwnz 9628* 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 ) ) )
 
Theoremrebtwn2zlemstep 9629* Lemma for rebtwn2z 9631. 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 ) ) )
 
Theoremrebtwn2zlemshrink 9630* Lemma for rebtwn2z 9631. 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 ) ) )
 
Theoremrebtwn2z 9631* 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 ) ) )
 
Theoremqbtwnrelemcalc 9632 Lemma for qbtwnre 9633. 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 )
 
Theoremqbtwnre 9633* 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 ) )
 
Theoremqbtwnxr 9634* 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 ) )
 
Theoremqavgle 9635 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 ) )
 
Theoremioo0 9636 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A (,) B )  =  (/)  <->  B  <_  A ) )
 
Theoremioom 9637* 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 ) )
 
Theoremico0 9638 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A [,) B )  =  (/)  <->  B  <_  A ) )
 
Theoremioc0 9639 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A (,] B )  =  (/)  <->  B  <_  A ) )
 
3.6  Elementary integer functions
 
3.6.1  The floor and ceiling functions
 
Syntaxcfl 9640 Extend class notation with floor (greatest integer) function.
 class  |_
 
Syntaxcceil 9641 Extend class notation to include the ceiling function.
 class
 
Definitiondf-fl 9642* Define the floor (greatest integer less than or equal to) function. See flval 9644 for its value, flqlelt 9648 for its basic property, and flqcl 9645 for its closure. For example,  ( |_ `  (
3  /  2 ) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 11309).

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 ) ) ) )
 
Definitiondf-ceil 9643 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 9678 for its value, ceilqge 9682 and ceilqm1lt 9684 for its basic properties, and ceilqcl 9680 for its closure. For example,  ( `  (
3  /  2 ) )  =  2 while  ( `  -u ( 3  /  2
) )  =  -u
1 (ex-ceil 11310).

As described in df-fl 9642 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 ) )
 
Theoremflval 9644* 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 )
 ) ) )
 
Theoremflqcl 9645 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 9647. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  A )  e.  ZZ )
 
Theoremapbtwnz 9646* 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 ) ) )
 
Theoremflapcl 9647* 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 11251) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
 |-  ( ( A  e.  RR  /\  A. n  e. 
 ZZ  A #  n )  ->  ( |_ `  A )  e.  ZZ )
 
Theoremflqlelt 9648 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( ( |_ `  A )  <_  A  /\  A  <  ( ( |_ `  A )  +  1 )
 ) )
 
Theoremflqcld 9649 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 )
 
Theoremflqle 9650 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  A )  <_  A )
 
Theoremflqltp1 9651 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  A  <  ( ( |_ `  A )  +  1 ) )
 
Theoremqfraclt1 9652 The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( A  -  ( |_ `  A ) )  <  1 )
 
Theoremqfracge0 9653 The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  0  <_  ( A  -  ( |_ `  A ) ) )
 
Theoremflqge 9654 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 ) ) )
 
Theoremflqlt 9655 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 ) )
 
Theoremflid 9656 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
 
Theoremflqidm 9657 The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A ) )
 
Theoremflqidz 9658 A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( ( |_ `  A )  =  A  <->  A  e.  ZZ ) )
 
Theoremflqltnz 9659 If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  -.  A  e.  ZZ )  ->  ( |_ `  A )  <  A )
 
Theoremflqwordi 9660 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  A  <_  B )  ->  ( |_ `  A )  <_  ( |_ `  B ) )
 
Theoremflqword2 9661 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  A  <_  B )  ->  ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) ) )
 
Theoremflqbi 9662 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  ZZ )  ->  ( ( |_ `  A )  =  B  <->  ( B  <_  A  /\  A  <  ( B  +  1 ) ) ) )
 
Theoremflqbi2 9663 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( N  e.  ZZ  /\  F  e.  QQ )  ->  ( ( |_ `  ( N  +  F ) )  =  N  <->  ( 0  <_  F  /\  F  <  1 ) ) )
 
Theoremadddivflid 9664 The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN )  ->  ( B  <  C  <->  ( |_ `  ( A  +  ( B  /  C ) ) )  =  A ) )
 
Theoremflqge0nn0 9665 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  0  <_  A )  ->  ( |_ `  A )  e.  NN0 )
 
Theoremflqge1nn 9666 The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  1  <_  A )  ->  ( |_ `  A )  e.  NN )
 
Theoremfldivnn0 9667 The floor function of a division of a nonnegative integer by a positive integer is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  e.  NN0 )
 
Theoremdivfl0 9668 The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  <  B  <->  ( |_ `  ( A 
 /  B ) )  =  0 ) )
 
Theoremflqaddz 9669 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  N  e.  ZZ )  ->  ( |_ `  ( A  +  N )
 )  =  ( ( |_ `  A )  +  N ) )
 
Theoremflqzadd 9670 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( N  e.  ZZ  /\  A  e.  QQ )  ->  ( |_ `  ( N  +  A )
 )  =  ( N  +  ( |_ `  A ) ) )
 
Theoremflqmulnn0 9671 Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( N  e.  NN0  /\  A  e.  QQ )  ->  ( N  x.  ( |_ `  A ) ) 
 <_  ( |_ `  ( N  x.  A ) ) )
 
Theorembtwnzge0 9672 A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (Contributed by NM, 12-Mar-2005.)
 |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  ( 0  <_  A  <->  0 
 <_  N ) )
 
Theorem2tnp1ge0ge0 9673 Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( 0  <_  (
 ( 2  x.  N )  +  1 )  <->  0 
 <_  N ) )
 
Theoremflhalf 9674 Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
 |-  ( N  e.  ZZ  ->  N  <_  ( 2  x.  ( |_ `  (
 ( N  +  1 )  /  2 ) ) ) )
 
Theoremfldivnn0le 9675 The floor function of a division of a nonnegative integer by a positive integer is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) ) 
 <_  ( K  /  L ) )
 
Theoremflltdivnn0lt 9676 The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( K  <  N  ->  ( |_ `  ( K  /  L ) )  < 
 ( N  /  L ) ) )
 
Theoremfldiv4p1lem1div2 9677 The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
 |-  ( ( N  =  3  \/  N  e.  ( ZZ>=
 `  5 ) ) 
 ->  ( ( |_ `  ( N  /  4 ) )  +  1 )  <_  ( ( N  -  1 )  /  2
 ) )
 
Theoremceilqval 9678 The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( `  A )  =  -u ( |_ `  -u A ) )
 
Theoremceiqcl 9679 The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  -> 
 -u ( |_ `  -u A )  e.  ZZ )
 
Theoremceilqcl 9680 Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( `  A )  e.  ZZ )
 
Theoremceiqge 9681 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  A  <_  -u ( |_ `  -u A ) )
 
Theoremceilqge 9682 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  A  <_  ( `  A ) )
 
Theoremceiqm1l 9683 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( -u ( |_ `  -u A )  -  1 )  <  A )
 
Theoremceilqm1lt 9684 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( ( `  A )  -  1 )  <  A )
 
Theoremceiqle 9685 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  A  <_  B )  -> 
 -u ( |_ `  -u A )  <_  B )
 
Theoremceilqle 9686 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  A  <_  B )  ->  ( `  A )  <_  B )
 
Theoremceilid 9687 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
 |-  ( A  e.  ZZ  ->  ( `  A )  =  A )
 
Theoremceilqidz 9688 A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  ( `  A )  =  A ) )
 
Theoremflqleceil 9689 The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  A )  <_  ( `  A )
 )
 
Theoremflqeqceilz 9690 A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  ( |_ `  A )  =  ( `  A )
 ) )
 
Theoremintqfrac2 9691 Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  Z  =  ( |_ `  A )   &    |-  F  =  ( A  -  Z )   =>    |-  ( A  e.  QQ  ->  ( 0  <_  F  /\  F  <  1  /\  A  =  ( Z  +  F ) ) )
 
Theoremintfracq 9692 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 9691. (Contributed by NM, 16-Aug-2008.)
 |-  Z  =  ( |_ `  ( M  /  N ) )   &    |-  F  =  ( ( M  /  N )  -  Z )   =>    |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  (
 0  <_  F  /\  F  <_  ( ( N  -  1 )  /  N )  /\  ( M 
 /  N )  =  ( Z  +  F ) ) )
 
Theoremflqdiv 9693 Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  N  e.  NN )  ->  ( |_ `  (
 ( |_ `  A )  /  N ) )  =  ( |_ `  ( A  /  N ) ) )
 
3.6.2  The modulo (remainder) operation
 
Syntaxcmo 9694 Extend class notation with the modulo operation.
 class  mod
 
Definitiondf-mod 9695* Define the modulo (remainder) operation. See modqval 9696 for its value. For example,  ( 5  mod  3 )  =  2 and  ( -u 7  mod  2 )  =  1. As with df-fl 9642 we define this for first and second arguments which are real and positive real, respectively, even though many theorems will need to be more restricted (for example, specify rational arguments). (Contributed by NM, 10-Nov-2008.)
 |- 
 mod  =  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  ( x  /  y ) ) ) ) )
 
Theoremmodqval 9696 The value of the modulo operation. The modulo congruence notation of number theory,  J  ==  K (modulo  N), can be expressed in our notation as  ( J  mod  N )  =  ( K  mod  N ). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive numbers to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) As with flqcl 9645 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
 
Theoremmodqvalr 9697 The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( A  mod  B )  =  ( A  -  ( ( |_ `  ( A  /  B ) )  x.  B ) ) )
 
Theoremmodqcl 9698 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( A  mod  B )  e.  QQ )
 
Theoremflqpmodeq 9699 Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( ( |_ `  ( A  /  B ) )  x.  B )  +  ( A  mod  B ) )  =  A )
 
Theoremmodqcld 9700 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  ( A  mod  B )  e. 
 QQ )
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