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Theorem List for Intuitionistic Logic Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzosubel 10001 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 ) ) )
 
Theoremfzosubel2 10002 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 ) )
 
Theoremfzosubel3 10003 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 ) )
 
Theoremeluzgtdifelfzo 10004 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 ) ) ) )
 
Theoremige2m2fzo 10005 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
 ) ) )
 
Theoremfzocatel 10006 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 ) )
 
Theoremubmelfzo 10007 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 ) )
 
Theoremelfzodifsumelfzo 10008 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 ) ) )
 
Theoremelfzom1elp1fzo 10009 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 ) )
 
Theoremelfzom1elfzo 10010 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 ) )
 
Theoremfzval3 10011 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
 ) ) )
 
Theoremfzosn 10012 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ZZ  ->  ( A..^ ( A  +  1 ) )  =  { A }
 )
 
Theoremelfzomin 10013 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 ) ) )
 
Theoremzpnn0elfzo 10014 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
 ) ) )
 
Theoremzpnn0elfzo1 10015 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 )
 ) ) )
 
Theoremfzosplitsnm1 10016 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
 ) } ) )
 
Theoremelfzonlteqm1 10017 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 ) )
 
Theoremfzonn0p1 10018 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
 ) ) )
 
Theoremfzossfzop1 10019 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 ) ) )
 
Theoremfzonn0p1p1 10020 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 ) ) )
 
Theoremelfzom1p1elfzo 10021 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 ) )
 
Theoremfzo0ssnn0 10022 Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
 |-  ( 0..^ N ) 
 C_  NN0
 
Theoremfzo01 10023 Expressing the singleton of  0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( 0..^ 1 )  =  { 0 }
 
Theoremfzo12sn 10024 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 }
 
Theoremfzo0to2pr 10025 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 }
 
Theoremfzo0to3tp 10026 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 }
 
Theoremfzo0to42pr 10027 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 } )
 
Theoremfzo0sn0fzo1 10028 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 ) ) )
 
Theoremfzoend 10029 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( A  e.  ( A..^ B )  ->  ( B  -  1 )  e.  ( A..^ B ) )
 
Theoremfzo0end 10030 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 ) )
 
Theoremssfzo12 10031 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 ) ) )
 
Theoremssfzo12bi 10032 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 10033 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 10034 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 10035 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 10036 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 10037 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 10038 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 10039 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 10040 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 10041 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 10042 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 10043 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 10044 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 10045* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 9918. (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 10046* 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 9189 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 10047* 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 10048 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 10049 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 10050 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
 |-  ( ( M  e.  QQ  /\  N  e.  QQ )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
 
Theoremqletric 10051 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  <_  B  \/  B  <_  A ) )
 
Theoremqlelttric 10052 Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremqltnle 10053 '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 10054 Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  -> DECID  A  =  B )
 
Theoremexbtwnzlemstep 10055* Lemma for exbtwnzlemex 10057. 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 10056* Lemma for exbtwnzlemex 10057. 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 10057* 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 10058* 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 10059* 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 10060* Lemma for rebtwn2z 10062. 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 10061* Lemma for rebtwn2z 10062. 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 10062* 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 10063 Lemma for qbtwnre 10064. 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 10064* 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 10065* 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 10066 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 10067 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A (,) B )  =  (/)  <->  B  <_  A ) )
 
Theoremioom 10068* 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 10069 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A [,) B )  =  (/)  <->  B  <_  A ) )
 
Theoremioc0 10070 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A (,] B )  =  (/)  <->  B  <_  A ) )
 
4.6  Elementary integer functions
 
4.6.1  The floor and ceiling functions
 
Syntaxcfl 10071 Extend class notation with floor (greatest integer) function.
 class  |_
 
Syntaxcceil 10072 Extend class notation to include the ceiling function.
 class
 
Definitiondf-fl 10073* Define the floor (greatest integer less than or equal to) function. See flval 10075 for its value, flqlelt 10079 for its basic property, and flqcl 10076 for its closure. For example,  ( |_ `  (
3  /  2 ) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 13106).

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 10074 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 10109 for its value, ceilqge 10113 and ceilqm1lt 10115 for its basic properties, and ceilqcl 10111 for its closure. For example,  ( `  (
3  /  2 ) )  =  2 while  ( `  -u ( 3  /  2
) )  =  -u
1 (ex-ceil 13107).

As described in df-fl 10073 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 10075* 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 10076 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 10078. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  A )  e.  ZZ )
 
Theoremapbtwnz 10077* 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 10078* 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 11892) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
 |-  ( ( A  e.  RR  /\  A. n  e. 
 ZZ  A #  n )  ->  ( |_ `  A )  e.  ZZ )
 
Theoremflqlelt 10079 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 10080 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 10081 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  A )  <_  A )
 
Theoremflqltp1 10082 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  A  <  ( ( |_ `  A )  +  1 ) )
 
Theoremqfraclt1 10083 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 10084 The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  0  <_  ( A  -  ( |_ `  A ) ) )
 
Theoremflqge 10085 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 10086 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 10087 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
 
Theoremflqidm 10088 The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A ) )
 
Theoremflqidz 10089 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 10090 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 10091 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 10092 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 10093 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 10094 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 10095 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 10096 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 10097 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 10098 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 10099 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 10100 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 ) )
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