HomeHome Intuitionistic Logic Explorer
Theorem List (p. 103 of 144)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 10201-10300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzosplitsn 10201 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 10202 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 10203 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 10204 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 10205 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 10206* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10076. (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 10207* 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 9339 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 10208* 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 10209 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 10210 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 ) )
 
4.5.7  Rational numbers (cont.)
 
Theoremqtri3or 10211 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
 |-  ( ( M  e.  QQ  /\  N  e.  QQ )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
 
Theoremqletric 10212 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  <_  B  \/  B  <_  A ) )
 
Theoremqlelttric 10213 Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremqltnle 10214 '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 10215 Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  -> DECID  A  =  B )
 
Theoremexbtwnzlemstep 10216* Lemma for exbtwnzlemex 10218. 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 10217* Lemma for exbtwnzlemex 10218. 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 10218* 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 10219* 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 10220* 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 10221* Lemma for rebtwn2z 10223. 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 10222* Lemma for rebtwn2z 10223. 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 10223* 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 10224 Lemma for qbtwnre 10225. 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 10225* 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 10226* 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 10227 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 10228 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A (,) B )  =  (/)  <->  B  <_  A ) )
 
Theoremioom 10229* 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 10230 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A [,) B )  =  (/)  <->  B  <_  A ) )
 
Theoremioc0 10231 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A (,] B )  =  (/)  <->  B  <_  A ) )
 
Theoremdfrp2 10232 Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
 |-  RR+  =  ( 0 (,) +oo )
 
Theoremelicod 10233 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 ) )
 
Theoremicogelb 10234 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 )
 
Theoremelicore 10235 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 )
 
4.6  Elementary integer functions
 
4.6.1  The floor and ceiling functions
 
Syntaxcfl 10236 Extend class notation with floor (greatest integer) function.
 class  |_
 
Syntaxcceil 10237 Extend class notation to include the ceiling function.
 class
 
Definitiondf-fl 10238* Define the floor (greatest integer less than or equal to) function. See flval 10240 for its value, flqlelt 10244 for its basic property, and flqcl 10241 for its closure. For example,  ( |_ `  (
3  /  2 ) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 14017).

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 10239 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 10274 for its value, ceilqge 10278 and ceilqm1lt 10280 for its basic properties, and ceilqcl 10276 for its closure. For example,  ( `  (
3  /  2 ) )  =  2 while  ( `  -u ( 3  /  2
) )  =  -u
1 (ex-ceil 14018).

As described in df-fl 10238 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 10240* 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 10241 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 10243. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  A )  e.  ZZ )
 
Theoremapbtwnz 10242* 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 10243* 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 12145) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
 |-  ( ( A  e.  RR  /\  A. n  e. 
 ZZ  A #  n )  ->  ( |_ `  A )  e.  ZZ )
 
Theoremflqlelt 10244 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 10245 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 10246 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  A )  <_  A )
 
Theoremflqltp1 10247 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  A  <  ( ( |_ `  A )  +  1 ) )
 
Theoremqfraclt1 10248 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 10249 The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  0  <_  ( A  -  ( |_ `  A ) ) )
 
Theoremflqge 10250 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 10251 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 10252 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
 
Theoremflqidm 10253 The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A ) )
 
Theoremflqidz 10254 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 10255 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 10256 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 10257 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 10258 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 10259 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 10260 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 10261 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 10262 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 10263 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 10264 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 10265 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 10266 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 10267 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 10268 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 10269 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 10270 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 10271 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 10272 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 10273 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 10274 The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( `  A )  =  -u ( |_ `  -u A ) )
 
Theoremceiqcl 10275 The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  -> 
 -u ( |_ `  -u A )  e.  ZZ )
 
Theoremceilqcl 10276 Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( `  A )  e.  ZZ )
 
Theoremceiqge 10277 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 10278 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 10279 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 10280 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 10281 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 10282 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 10283 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
 |-  ( A  e.  ZZ  ->  ( `  A )  =  A )
 
Theoremceilqidz 10284 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 10285 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 10286 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 10287 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 10288 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10287. (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 10289 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 ) ) )
 
4.6.2  The modulo (remainder) operation
 
Syntaxcmo 10290 Extend class notation with the modulo operation.
 class  mod
 
Definitiondf-mod 10291* Define the modulo (remainder) operation. See modqval 10292 for its value. For example,  ( 5  mod  3 )  =  2 and  ( -u 7  mod  2 )  =  1. As with df-fl 10238 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 10292 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 10241 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 10293 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 10294 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 10295 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 10296 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 )
 
Theoremmodq0 10297  A  mod  B is zero iff  A is evenly divisible by  B. (Contributed by Jim Kingdon, 17-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  mod  B )  =  0  <->  ( A  /  B )  e.  ZZ ) )
 
Theoremmulqmod0 10298 The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M ) 
 ->  ( ( A  x.  M )  mod  M )  =  0 )
 
Theoremnegqmod0 10299  A is divisible by  B iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  mod  B )  =  0  <->  ( -u A  mod  B )  =  0 ) )
 
Theoremmodqge0 10300 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  0  <_  ( A 
 mod  B ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14370
  Copyright terms: Public domain < Previous  Next >