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Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzolb2 11101 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with  M  <  N. This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 
M  e.  ( ZZ>= `  N ). (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ( M..^ N )  <->  M  <  N ) )
 
Theoremelfzole1 11102 A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  M  <_  K )
 
Theoremelfzolt2 11103 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  <  N )
 
Theoremelfzolt3 11104 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  M  <  N )
 
Theoremelfzolt2b 11105 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( K..^ N ) )
 
Theoremelfzolt3b 11106 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  M  e.  ( M..^ N ) )
 
Theoremfzonel 11107 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |- 
 -.  B  e.  ( A..^ B )
 
Theoremelfzouz2 11108 The upper bound of a half-open range is greater or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  N  e.  ( ZZ>= `  K )
 )
 
Theoremelfzofz 11109 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( M ... N ) )
 
Theoremelfzo3 11110 Express membership in a half-open integer interval in terms of the "less than or equal" and "less than" predicates on integers, resp.  K  e.  (
ZZ>= `  M )  <->  M  <_  K,  K  e.  ( K..^ N )  <->  K  <  N. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  <->  ( K  e.  ( ZZ>= `  M )  /\  K  e.  ( K..^ N ) ) )
 
Theoremfzon0 11111 A half-open integer interval is nonempty iff it contains its left endpoint. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( ( M..^ N )  =/=  (/)  <->  M  e.  ( M..^ N ) )
 
Theoremfzossfz 11112 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( A..^ B ) 
 C_  ( A ... B )
 
Theoremfzon 11113 A half-open set of sequential integers is empty if the bounds are equal or reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <_  M  <-> 
 ( M..^ N )  =  (/) ) )
 
Theoremfzo0 11114 Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( A..^ A )  =  (/)
 
Theoremfzonnsub 11115 If  K  <  N then 
N  -  K is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
 |-  ( K  e.  ( M..^ N )  ->  ( N  -  K )  e. 
 NN )
 
Theoremfzonnsub2 11116 If  M  <  N then 
N  -  M is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( K  e.  ( M..^ N )  ->  ( N  -  M )  e. 
 NN )
 
Theoremfzoss1 11117 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( ZZ>=
 `  M )  ->  ( K..^ N )  C_  ( M..^ N ) )
 
Theoremfzoss2 11118 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( N  e.  ( ZZ>=
 `  K )  ->  ( M..^ K )  C_  ( M..^ N ) )
 
Theoremfzossrbm1 11119 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  ( N  e.  NN0  ->  ( 0..^ ( N  -  1 ) )  C_  ( 0..^ N ) )
 
Theoremfzospliti 11120 One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( ( A  e.  ( B..^ C )  /\  D  e.  ZZ )  ->  ( A  e.  ( B..^ D )  \/  A  e.  ( D..^ C ) ) )
 
Theoremfzosplit 11121 Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( D  e.  ( B ... C )  ->  ( B..^ C )  =  ( ( B..^ D )  u.  ( D..^ C ) ) )
 
Theoremfzodisj 11122 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( ( A..^ B )  i^i  ( B..^ C ) )  =  (/)
 
Theoremfzouzsplit 11123 Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( ZZ>= `  A )  =  ( ( A..^ B )  u.  ( ZZ>= `  B ) ) )
 
Theoremfzouzdisj 11124 A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.)
 |-  ( ( A..^ B )  i^i  ( ZZ>= `  B ) )  =  (/)
 
Theoremlbfzo0 11125 An integer is strictly greater than zero iff it is a member of  NN. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( 0  e.  (
 0..^ A )  <->  A  e.  NN )
 
Theoremelfzo0 11126 Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( A  e.  (
 0..^ B )  <->  ( A  e.  NN0  /\  B  e.  NN  /\  A  <  B ) )
 
Theoremfzossnn 11127 Half-opened integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  ( 1..^ N ) 
 C_  NN
 
Theoremelfzo1 11128 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( N  e.  (
 1..^ M )  <->  ( N  e.  NN  /\  M  e.  NN  /\  N  <  M ) )
 
Theoremfzo0n0 11129 A half-open integer range based at 0 is nonempty precisely if the upper bound is a positive integer. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( ( 0..^ A )  =/=  (/)  <->  A  e.  NN )
 
Theoremfzoaddel 11130 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 ) ) )
 
Theoremfzoaddel2 11131 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( 0..^ ( B  -  C ) )  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  +  C )  e.  ( C..^ B ) )
 
Theoremfzosubel 11132 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 11133 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 11134 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 ) )
 
Theoremfzval3 11135 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 11136 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ZZ  ->  ( A..^ ( A  +  1 ) )  =  { A }
 )
 
Theoremfzo01 11137 Expressing the singleton of  0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( 0..^ 1 )  =  { 0 }
 
Theoremfzo12sn 11138 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 11139 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 11140 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 11141 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 } )
 
Theoremfzoend 11142 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 11143 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 ) )
 
Theoremfzofzp1 11144 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 11145 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 11146 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 ) ) ) )
 
Theoremelfznelfzo 11147 A value in a finite set of sequential integers is a border value if it is not contained in the half-open integer range contained in the finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( ( y  e.  ( 0 ... K )  /\  -.  y  e.  ( 1..^ K ) )  ->  ( y  =  0  \/  y  =  K ) )
 
Theoremelfznelfzob 11148 A value in a finite set of sequential integers is a border value if and only if it is not contained in the half-open integer range contained in the finite set of sequential integerss. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
 |-  ( y  e.  (
 0 ... K )  ->  ( -.  y  e.  (
 1..^ K )  <->  ( y  =  0  \/  y  =  K ) ) )
 
Theorempeano2fzor 11149 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  ( K  +  1 )  e.  ( M..^ N ) )  ->  K  e.  ( M..^ N ) )
 
Theoremfzosplitsn 11150 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 }
 ) )
 
Theoremfzosplitsni 11151 Membership in a half-open range extende 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 ) ) )
 
Theoremfzostep1 11152 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 ) )
 
Theoremfzind2 11153* 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 hypothesis. Version of fzind 10324 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 )
 
Theoreminjresinjlem 11154 Lemma for injresinj 11155. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( -.  y  e.  ( 1..^ K ) 
 ->  ( ( F `  0 )  =/=  ( F `  K )  ->  ( ( F :
 ( 0 ... K )
 --> V  /\  K  e.  NN0 )  ->  ( (
 ( F " {
 0 ,  K }
 )  i^i  ( F " ( 1..^ K ) ) )  =  (/)  ->  ( ( x  e.  ( 0 ... K )  /\  y  e.  (
 0 ... K ) ) 
 ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
 ) ) ) ) )
 
Theoreminjresinj 11155 A function whose restriction is injective and the values of the remaining arguments are different from all other values is injective itself. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( K  e.  NN0  ->  ( ( F :
 ( 0 ... K )
 --> V  /\  Fun  `' ( F  |`  ( 1..^ K ) )  /\  ( F `  0 )  =/=  ( F `  K ) )  ->  ( ( ( F
 " { 0 ,  K } )  i^i  ( F " (
 1..^ K ) ) )  =  (/)  ->  Fun  `' F ) ) )
 
5.6  Elementary integer functions
 
5.6.1  The floor (greatest integer) function
 
Syntaxcfl 11156 Extend class notation with floor (greatest integer) function.
 class  |_
 
Definitiondf-fl 11157* Define the floor (greatest integer) function. See flval 11158 for its value, fllelt 11161 for its basic property, and flcl 11159 for its closure. For example,  ( |_ `  ( 3  /  2
) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 21708).

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 ) ) ) )
 
Theoremflval 11158* 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 )
 ) ) )
 
Theoremflcl 11159 The floor (greatest integer) function is an integer (closure law). (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
 
Theoremreflcl 11160 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
 
Theoremfllelt 11161 A basic property of the floor (greatest integer) function. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( A  e.  RR  ->  ( ( |_ `  A )  <_  A  /\  A  <  ( ( |_ `  A )  +  1 )
 ) )
 
Theoremflcld 11162 The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( |_ `  A )  e. 
 ZZ )
 
Theoremflle 11163 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
 
Theoremflltp1 11164 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  A  <  ( ( |_ `  A )  +  1 ) )
 
Theoremfllep1 11165 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( A  e.  RR  ->  A  <_  ( ( |_ `  A )  +  1 ) )
 
Theoremfraclt1 11166 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( A  -  ( |_ `  A ) )  <  1 )
 
Theoremfracle1 11167 The fractional part of a real number is less than or equal to one. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( A  e.  RR  ->  ( A  -  ( |_ `  A ) ) 
 <_  1 )
 
Theoremfracge0 11168 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
 |-  ( A  e.  RR  ->  0  <_  ( A  -  ( |_ `  A ) ) )
 
Theoremflge 11169 The floor function value is the greatest integer less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( B  <_  A  <->  B  <_  ( |_ `  A ) ) )
 
Theoremfllt 11170 The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( A  <  B  <-> 
 ( |_ `  A )  <  B ) )
 
Theoremflid 11171 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
 
Theoremflidm 11172 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A ) )
 
Theoremflidz 11173 A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR  ->  ( ( |_ `  A )  =  A  <->  A  e.  ZZ ) )
 
Theoremflwordi 11174 Ordering relationship for the greatest integer function. (Contributed by NM, 31-Dec-2005.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( |_ `  A )  <_  ( |_ `  B ) )
 
Theoremflword2 11175 Ordering relationship for the greatest integer function. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) ) )
 
Theoremflval2 11176* An alternate way to define the floor (greatest integer) function. (Contributed by NM, 16-Nov-2004.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  =  ( iota_ x  e. 
 ZZ ( x  <_  A  /\  A. y  e. 
 ZZ  ( y  <_  A  ->  y  <_  x ) ) ) )
 
Theoremflval3 11177* An alternate way to define the floor (greatest integer) function, as the supremum of all integers less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 6-Sep-2014.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  =  sup ( { x  e.  ZZ  |  x  <_  A } ,  RR ,  <  ) )
 
Theoremflbi 11178 A condition equivalent to floor. (Contributed by NM, 11-Mar-2005.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( |_ `  A )  =  B  <->  ( B  <_  A  /\  A  <  ( B  +  1 ) ) ) )
 
Theoremflbi2 11179 A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)
 |-  ( ( N  e.  ZZ  /\  F  e.  RR )  ->  ( ( |_ `  ( N  +  F ) )  =  N  <->  ( 0  <_  F  /\  F  <  1 ) ) )
 
Theoremflge0nn0 11180 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by NM, 26-Apr-2005.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( |_ `  A )  e.  NN0 )
 
Theoremflge1nn 11181 The floor of a number greater than or equal to 1 is a natural number. (Contributed by NM, 26-Apr-2005.)
 |-  ( ( A  e.  RR  /\  1  <_  A )  ->  ( |_ `  A )  e.  NN )
 
Theoremfladdz 11182 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 27-Apr-2005.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
 |-  ( ( A  e.  RR  /\  N  e.  ZZ )  ->  ( |_ `  ( A  +  N )
 )  =  ( ( |_ `  A )  +  N ) )
 
Theoremflzadd 11183 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 2-Jan-2009.)
 |-  ( ( N  e.  ZZ  /\  A  e.  RR )  ->  ( |_ `  ( N  +  A )
 )  =  ( N  +  ( |_ `  A ) ) )
 
Theoremflmulnn0 11184 Move a nonnegative integer in and out of a floor. (Contributed by NM, 2-Jan-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)
 |-  ( ( N  e.  NN0  /\  A  e.  RR )  ->  ( N  x.  ( |_ `  A ) ) 
 <_  ( |_ `  ( N  x.  A ) ) )
 
Theorembtwnzge0 11185 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. (For the first half see rebtwnz 10529.) (Contributed by NM, 12-Mar-2005.)
 |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  ( 0  <_  A  <->  0 
 <_  N ) )
 
Theoremflhalf 11186 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 ) ) ) )
 
Theoremceicl 11187 The ceiling function returns an integer (closure law). (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( A  e.  RR  -> 
 -u ( |_ `  -u A )  e.  ZZ )
 
Theoremceige 11188 The ceiling of a real number is greater than or equal to that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( A  e.  RR  ->  A  <_  -u ( |_ `  -u A ) )
 
Theoremceim1l 11189 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A )  -  1 )  <  A )
 
Theoremceile 11190 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  ZZ  /\  A  <_  B )  -> 
 -u ( |_ `  -u A )  <_  B )
 
Theoremquoremz 11191 Quotient and remainder of an integer divided by a natural number. TO DO - is this really needed for anything? Should we use  mod to simplify it? (Contributed by NM, 14-Aug-2008.)
 |-  Q  =  ( |_ `  ( A  /  B ) )   &    |-  R  =  ( A  -  ( B  x.  Q ) )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( Q  e.  ZZ  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
 
Theoremquoremnn0 11192 Quotient and remainder of a nonnegative integer divided by a natural number. (Contributed by NM, 14-Aug-2008.)
 |-  Q  =  ( |_ `  ( A  /  B ) )   &    |-  R  =  ( A  -  ( B  x.  Q ) )   =>    |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e.  NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
 
Theoremquoremnn0ALT 11193 Quotient and remainder of a nonnegative integer divided by a natural number. TO DO - Keep either quoremnn0ALT 11193 ((if we don't keep quoremz 11191) or quoremnn0 11192 (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  Q  =  ( |_ `  ( A  /  B ) )   &    |-  R  =  ( A  -  ( B  x.  Q ) )   =>    |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e.  NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
 
Theoremintfrac2 11194 Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 11218? (Contributed by NM, 16-Aug-2008.)
 |-  Z  =  ( |_ `  A )   &    |-  F  =  ( A  -  Z )   =>    |-  ( A  e.  RR  ->  ( 0  <_  F  /\  F  <  1  /\  A  =  ( Z  +  F ) ) )
 
Theoremintfracq 11195 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 11194. (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 ) ) )
 
Theoremfldiv 11196 Cancellation of the embedded floor of a real divided by an integer. (Contributed by NM, 16-Aug-2008.)
 |-  ( ( A  e.  RR  /\  N  e.  NN )  ->  ( |_ `  (
 ( |_ `  A )  /  N ) )  =  ( |_ `  ( A  /  N ) ) )
 
Theoremfldiv2 11197 Cancellation of an embedded floor of a ratio. Generalization of Equation 2.4 in [CormenLeisersonRivest] p. 33 (where  A must be an integer). (Contributed by NM, 9-Nov-2008.)
 |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  (
 ( |_ `  ( A  /  M ) ) 
 /  N ) )  =  ( |_ `  ( A  /  ( M  x.  N ) ) ) )
 
Theoremfznnfl 11198 Finite set of sequential integers starting at 1 and ending at a real number. (Contributed by Mario Carneiro, 3-May-2016.)
 |-  ( N  e.  RR  ->  ( K  e.  (
 1 ... ( |_ `  N ) )  <->  ( K  e.  NN  /\  K  <_  N ) ) )
 
Theoremuzsup 11199 A set of upper integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  sup ( Z ,  RR*
 ,  <  )  =  +oo )
 
Theoremioopnfsup 11200 A set of upper reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  sup ( ( A (,)  +oo ) ,  RR* ,  <  )  =  +oo )
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