HomeHome Metamath Proof Explorer
Theorem List (p. 334 of 386)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-25952)
  Hilbert Space Explorer  Hilbert Space Explorer
(25953-27476)
  Users' Mathboxes  Users' Mathboxes
(27477-38550)
 

Theorem List for Metamath Proof Explorer - 33301-33400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremzlmodzxzldeplem1 33301 Lemma 1 for zlmodzxzldep 33305. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   &    |-  F  =  { <. A ,  2 >. ,  <. B ,  -u 3 >. }   =>    |-  F  e.  ( ZZ 
 ^m  { A ,  B } )
 
Theoremzlmodzxzldeplem2 33302 Lemma 2 for zlmodzxzldep 33305. (Contributed by AV, 24-May-2019.) (Revised by AV, 30-Jul-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   &    |-  F  =  { <. A ,  2 >. ,  <. B ,  -u 3 >. }   =>    |-  F finSupp  0
 
Theoremzlmodzxzldeplem3 33303 Lemma 3 for zlmodzxzldep 33305. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   &    |-  F  =  { <. A ,  2 >. ,  <. B ,  -u 3 >. }   =>    |-  ( F ( linC  `  Z ) { A ,  B } )  =  ( 0g `  Z )
 
Theoremzlmodzxzldeplem4 33304* Lemma 4 for zlmodzxzldep 33305. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   &    |-  F  =  { <. A ,  2 >. ,  <. B ,  -u 3 >. }   =>    |- 
 E. y  e.  { A ,  B }  ( F `  y )  =/=  0
 
Theoremzlmodzxzldep 33305 { A , B } is a linearly dependent set within the  ZZ-module  ZZ  X.  ZZ (see example in [Roman] p. 112). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   =>    |- 
 { A ,  B } linDepS  Z
 
Theoremldepsnlinclem1 33306 Lemma 1 for ldepsnlinc 33309. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   =>    |-  ( F  e.  (
 ( Base ` ring )  ^m  { B } )  ->  ( F ( linC  `  Z ) { B } )  =/= 
 A )
 
Theoremldepsnlinclem2 33307 Lemma 2 for ldepsnlinc 33309. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   =>    |-  ( F  e.  (
 ( Base ` ring )  ^m  { A } )  ->  ( F ( linC  `  Z ) { A } )  =/= 
 B )
 
21.25.12.5  Differences between (left) modules and (left) vector spaces
 
Theoremlvecpsslmod 33308 The class of all (left) vector spaces is a proper subclass of the class of all (left) modules. Although it is obvious (and proven by lveclmod 17865) that every left vector space is a left module, there is (at least) one left module which is no left vector space, for example the zero module over the zero ring, see lmod1zrnlvec 33295. (Contributed by AV, 29-Apr-2019.)
 |-  LVec  C. 
 LMod
 
Theoremldepsnlinc 33309* The reverse implication of islindeps2 33284 does not hold for arbitrary (left) modules, see note in [Roman] p. 112: "... if a nontrivial linear combination of the elements ... in an R-module M is 0, ... where not all of the coefficients are 0, then we cannot conclude ... that one of the elements ... is a linear combination of the others." This means that there is at least one left module having a linearly dependent subset in which there is at least one element which is not a linear combinantion of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 33297 and zlmodzxznm 33298 (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
 |-  E. m  e.  LMod  E. s  e.  ~P  ( Base `  m )
 ( s linDepS  m  /\  A. v  e.  s  A. f  e.  ( ( Base `  (Scalar `  m ) )  ^m  ( s 
 \  { v }
 ) ) ( f finSupp  ( 0g `  (Scalar `  m ) )  ->  ( f ( linC  `  m )
 ( s  \  {
 v } ) )  =/=  v ) )
 
Theoremldepslinc 33310* For (left) vector spaces, isldepslvec2 33286 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 33309 indicates that there is not an analogous alternative definition for arbitrary (left) modules. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
 |-  ( A. m  e.  LVec  A. s  e.  ~P  ( Base `  m ) ( s linDepS  m  <->  E. v  e.  s  E. f  e.  ( ( Base `  (Scalar `  m ) )  ^m  ( s 
 \  { v }
 ) ) ( f finSupp  ( 0g `  (Scalar `  m ) )  /\  ( f ( linC  `  m )
 ( s  \  {
 v } ) )  =  v ) ) 
 /\  -.  A. m  e. 
 LMod  A. s  e.  ~P  ( Base `  m )
 ( s linDepS  m  <->  E. v  e.  s  E. f  e.  (
 ( Base `  (Scalar `  m ) )  ^m  ( s 
 \  { v }
 ) ) ( f finSupp  ( 0g `  (Scalar `  m ) )  /\  ( f ( linC  `  m )
 ( s  \  {
 v } ) )  =  v ) ) )
 
21.25.13  Complexity theory
 
21.25.13.1  Auxiliary theorems
 
Theoremoffval0 33311* Value of an operation applied to two functions. (Contributed by AV, 15-May-2020.)
 |-  (
 ( F  e.  V  /\  G  e.  W ) 
 ->  ( F  oF R G )  =  ( x  e.  ( dom 
 F  i^i  dom  G ) 
 |->  ( ( F `  x ) R ( G `  x ) ) ) )
 
Theoremsuppdm 33312 If the range of a function does not contain the zero, the support of the function equals its domain. (Contributed by AV, 20-May-2020.)
 |-  (
 ( ( Fun  F  /\  F  e.  V  /\  Z  e.  W )  /\  Z  e/  ran  F )  ->  ( F supp  Z )  =  dom  F )
 
Theoremeluz2gt1 33313 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
 |-  ( N  e.  ( ZZ>= `  2 )  ->  1  <  N )
 
Theoremeluz2n0 33314 An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
 |-  ( N  e.  ( ZZ>= `  2 )  ->  N  =/=  0 )
 
Theoremeluz2cnn0n1 33315 An integer greater than 1 is a complex number not equal to 0 or 1. (Contributed by AV, 23-May-2020.)
 |-  ( B  e.  ( ZZ>= `  2 )  ->  B  e.  ( CC  \  { 0 ,  1 } )
 )
 
Theoremdivge1b 33316 The ratio of a real number to a positive real number is greater than or equal to 1 iff the divisor (the positive real number) is less than or equal to the dividend (the real number). (Contributed by AV, 26-May-2020.)
 |-  (
 ( A  e.  RR+  /\  B  e.  RR )  ->  ( A  <_  B  <->  1 
 <_  ( B  /  A ) ) )
 
Theoremdivgt1b 33317 The ratio of a real number to a positive real number is greater than 1 iff the divisor (the positive real number) is less than the dividend (the real number). (Contributed by AV, 30-May-2020.)
 |-  (
 ( A  e.  RR+  /\  B  e.  RR )  ->  ( A  <  B  <->  1  <  ( B  /  A ) ) )
 
Theoremdivlt1lt 33318 A real number divided by a positive real number is less than 1 iff the real number is less than the positive real number. (Contributed by AV, 25-May-2020.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  /  B )  <  1  <->  A  <  B ) )
 
Theoremltsubaddb 33319 Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  -  C )  <  ( B  -  D )  <->  ( A  +  D )  <  ( B  +  C ) ) )
 
Theoremltsubsubb 33320 Equivalence for the "less than" relation between differences. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  -  C )  <  ( B  -  D )  <->  ( A  -  B )  <  ( C  -  D ) ) )
 
Theoremltsubadd2b 33321 Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( D  -  C )  <  ( B  -  A )  <->  ( A  +  D )  <  ( B  +  C ) ) )
 
Theoremdivsub1dir 33322 Distribution of division over subtraction by 1. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( ( A  /  B )  -  1
 )  =  ( ( A  -  B ) 
 /  B ) )
 
Theoremexpnegico01 33323 An integer greater than 1 to the power of a negative integer is in the closed-below, open-above interval between 0 and 1. (Contributed by AV, 24-May-2020.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ  /\  N  <  0 )  ->  ( B ^ N )  e.  ( 0 [,) 1
 ) )
 
Theoremnn0rp0 33324 A nonnegative integer is a nonnegative real number. (Contributed by AV, 24-May-2020.)
 |-  ( N  e.  NN0  ->  N  e.  ( 0 [,) +oo ) )
 
Theoremelfzolborelfzop1 33325 An element of a half-open integer interval is either equal to the left bound of the interval or an element of a half-open integer interval with a left bound increased by 1. (Contributed by AV, 2-Jun-2020.)
 |-  ( K  e.  ( M..^ N )  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) )
 
Theorem3halfnz 33326 Three halfs is not an integer. (Contributed by AV, 2-Jun-2020.)
 |-  -.  ( 3  /  2
 )  e.  ZZ
 
Theorempw2m1lepw2m1 33327 2 to the power of a positive integer decreased by 1 is less than or equal to 2 to the power of the integer minus 1. (Contributed by AV, 30-May-2020.)
 |-  ( I  e.  NN  ->  ( 2 ^ ( I  -  1 ) ) 
 <_  ( ( 2 ^ I )  -  1
 ) )
 
Theoremzgeltp1eq 33328 If an integer is between another integer and its successor, the integer is equal to the other integer. (Contributed by AV, 30-May-2020.)
 |-  (
 ( I  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( A  <_  I 
 /\  I  <  ( A  +  1 )
 )  ->  I  =  A ) )
 
Theoremzgtp1leeq 33329 If an integer is between another integer and its predecessor, the integer is equal to the other integer. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( I  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( ( A  -  1 )  < 
 I  /\  I  <_  A )  ->  I  =  A ) )
 
Theoremflsubz 33330 An integer can be moved in and out of the floor of a difference. (Contributed by AV, 29-May-2020.)
 |-  (
 ( A  e.  RR  /\  N  e.  ZZ )  ->  ( |_ `  ( A  -  N ) )  =  ( ( |_ `  A )  -  N ) )
 
21.25.13.2  The modulo (remainder) operation (extension)
 
Theoremfldivmod 33331 Expressing the floor of a division by the modulo operator. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( |_ `  ( A  /  B ) )  =  ( ( A  -  ( A  mod  B ) )  /  B ) )
 
Theoremmod0mul 33332* If an integer is 0 modulo a positive integer, this integer must be the product of another integer and the modulus. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN )  ->  ( ( A  mod  N )  =  0  ->  E. x  e.  ZZ  A  =  ( x  x.  N ) ) )
 
Theoremmodn0mul 33333* If an integer is not 0 modulo a positive integer, this integer must be the sum of the product of another integer and the modulus and a positive integer less than the modulus. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN )  ->  ( ( A  mod  N )  =/=  0  ->  E. x  e.  ZZ  E. y  e.  ( 1..^ N ) A  =  ( ( x  x.  N )  +  y
 ) ) )
 
Theoremm1modmmod 33334 An integer decreased by 1 modulo a positive integer minus the integer modulo the same modulus is either -1 or the modulus minus 1. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN )  ->  ( ( ( A  -  1 )  mod  N )  -  ( A 
 mod  N ) )  =  if ( ( A 
 mod  N )  =  0 ,  ( N  -  1 ) ,  -u 1
 ) )
 
Theoremdifmodm1lt 33335 The difference between an integer modulo a positive integer and the integer decreased by 1 modulo the same modulus is less than the modulus decreased by 1 (if the modulus is greater than 2). This theorem would not be valid for an odd  A and  N  =  2, since  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) ) would be  ( 1  -  0 )  =  1 which is not less than  ( N  -  1 )  =  1. (Contributed by AV, 6-Jun-2012.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( ( A  mod  N )  -  ( ( A  -  1 ) 
 mod  N ) )  < 
 ( N  -  1
 ) )
 
21.25.13.3  Even and odd integers
 
Theoremnn0enne 33336 A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.)
 |-  ( N  e.  NN  ->  ( ( N  /  2
 )  e.  NN0  <->  ( N  / 
 2 )  e.  NN ) )
 
Theoremnn0o1gt2 33337 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  ( N  =  1  \/  2  <  N ) )
 
Theoremnno 33338 An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.)
 |-  (
 ( N  e.  ( ZZ>=
 `  2 )  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  ( ( N  -  1 )  /  2
 )  e.  NN )
 
Theoremnn0o 33339 An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  ( ( N  -  1 )  /  2
 )  e.  NN0 )
 
Theoremnn0ob 33340 Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.)
 |-  ( N  e.  NN0  ->  (
 ( ( N  +  1 )  /  2
 )  e.  NN0  <->  ( ( N  -  1 )  / 
 2 )  e.  NN0 ) )
 
Theoremnn0onn0ex 33341* For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.)
 |-  (
 ( N  e.  NN0  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  E. m  e.  NN0  N  =  ( ( 2  x.  m )  +  1 ) )
 
Theoremnn0enn0ex 33342* For each even nonnegative integer there is a nonnegative integer which, multiplied by 2, results in the even nonnegative integer. (Contributed by AV, 30-May-2020.)
 |-  (
 ( N  e.  NN0  /\  ( N  /  2
 )  e.  NN0 )  ->  E. m  e.  NN0  N  =  ( 2  x.  m ) )
 
Theoremnneop 33343 A positive integer is even or odd. (Contributed by AV, 30-May-2020.)
 |-  ( N  e.  NN  ->  ( ( N  /  2
 )  e.  NN  \/  ( ( N  +  1 )  /  2
 )  e.  NN )
 )
 
Theoremnneom 33344 A positive integer is even or odd. (Contributed by AV, 30-May-2020.)
 |-  ( N  e.  NN  ->  ( ( N  /  2
 )  e.  NN  \/  ( ( N  -  1 )  /  2
 )  e.  NN0 )
 )
 
Theoremnn0eo 33345 A nonnegative integer is even or odd. (Contributed by AV, 27-May-2020.)
 |-  ( N  e.  NN0  ->  (
 ( N  /  2
 )  e.  NN0  \/  ( ( N  +  1 )  /  2
 )  e.  NN0 )
 )
 
Theoremnnpw2even 33346 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.)
 |-  ( N  e.  NN  ->  ( ( 2 ^ N )  /  2 )  e. 
 NN )
 
Theoremzefldiv2 33347 The floor of an even integer divided by 2 is equal to the integer divided by 2. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( N  e.  ZZ  /\  ( N  /  2
 )  e.  ZZ )  ->  ( |_ `  ( N  /  2 ) )  =  ( N  / 
 2 ) )
 
Theoremzofldiv2 33348 The floor of an odd integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( N  e.  ZZ  /\  ( ( N  +  1 )  /  2
 )  e.  ZZ )  ->  ( |_ `  ( N  /  2 ) )  =  ( ( N  -  1 )  / 
 2 ) )
 
Theoremnn0ofldiv2 33349 The floor of an odd nonnegative integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.) (Proof shortened by AV, 7-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  ( |_ `  ( N  /  2 ) )  =  ( ( N  -  1 )  / 
 2 ) )
 
Theoremflnn0div2ge 33350 The floor of a positive integer divided by 2 is greater than or equal to the integer decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.)
 |-  ( N  e.  NN0  ->  (
 ( N  -  1
 )  /  2 )  <_  ( |_ `  ( N  /  2 ) ) )
 
Theoremflnn0ohalf 33351 The floor of the half of an odd positive integer is equal to the floor of the half of the integer decreased by 1. (Contributed by AV, 5-Jun-2012.)
 |-  (
 ( N  e.  NN0  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  ( |_ `  ( N  /  2 ) )  =  ( |_ `  (
 ( N  -  1
 )  /  2 )
 ) )
 
21.25.13.4  The natural logarithm on complex numbers (extension)
 
Theoremlogge0b 33352 The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.)
 |-  ( A  e.  RR+  ->  (
 0  <_  ( log `  A )  <->  1  <_  A ) )
 
Theoremloggt0b 33353 The logarithm of a number is positive iff the number is greater than 1. (Contributed by AV, 30-May-2020.)
 |-  ( A  e.  RR+  ->  (
 0  <  ( log `  A )  <->  1  <  A ) )
 
Theoremlogle1b 33354 The logarithm of a number is less than or equal to 1 iff the number is less than or equal to Euler's constant. (Contributed by AV, 30-May-2020.)
 |-  ( A  e.  RR+  ->  (
 ( log `  A )  <_  1  <->  A  <_  _e ) )
 
Theoremloglt1b 33355 The logarithm of a number is less than 1 iff the number is less than Euler's constant. (Contributed by AV, 30-May-2020.)
 |-  ( A  e.  RR+  ->  (
 ( log `  A )  <  1  <->  A  <  _e ) )
 
Theoremlogcxp0 33356 Logarithm of a complex power. Generalisation of logcxp 23137. (Contributed by AV, 22-May-2020.)
 |-  (
 ( A  e.  ( CC  \  { 0 } )  /\  B  e.  CC  /\  ( B  x.  ( log `  A )
 )  e.  ran  log )  ->  ( log `  ( A  ^c  B ) )  =  ( B  x.  ( log `  A ) ) )
 
Theoremregt1loggt0 33357 The natural logarithm for a real number greater than 1 is greater than 0. (Contributed by AV, 25-May-2020.)
 |-  ( B  e.  ( 1 (,) +oo )  ->  0  <  ( log `  B ) )
 
21.25.13.5  Division of functions
 
Syntaxcfdiv 33358 Extend class notation with the division operator of two functions.
 class /_f
 
Definitiondf-fdiv 33359* Define the division of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)
 |- /_f  =  ( f  e.  _V ,  g  e.  _V  |->  ( ( f  oF  /  g )  |`  ( g supp  0 ) ) )
 
Theoremfdivval 33360 The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)
 |-  (
 ( F  e.  V  /\  G  e.  W ) 
 ->  ( F /_f  G )  =  ( ( F  oF  /  G )  |`  ( G supp  0
 ) ) )
 
Theoremfdivmpt 33361* The quotient of two functions into the complex numbers as mapping. (Contributed by AV, 16-May-2020.)
 |-  (
 ( F : A --> CC  /\  G : A --> CC  /\  A  e.  V )  ->  ( F /_f  G )  =  ( x  e.  ( G supp  0 ) 
 |->  ( ( F `  x )  /  ( G `  x ) ) ) )
 
Theoremfdivmptf 33362 The quotient of two functions into the complex numbers is a function into the complex numbers. (Contributed by AV, 16-May-2020.)
 |-  (
 ( F : A --> CC  /\  G : A --> CC  /\  A  e.  V )  ->  ( F /_f  G ) : ( G supp  0
 ) --> CC )
 
Theoremrefdivmptf 33363 The quotient of two functions into the real numbers is a function into the real numbers. (Contributed by AV, 16-May-2020.)
 |-  (
 ( F : A --> RR  /\  G : A --> RR  /\  A  e.  V )  ->  ( F /_f  G ) : ( G supp  0
 ) --> RR )
 
Theoremfdivpm 33364 The quotient of two functions into the complex numbers is a partial function. (Contributed by AV, 16-May-2020.)
 |-  (
 ( F : A --> CC  /\  G : A --> CC  /\  A  e.  V )  ->  ( F /_f  G )  e.  ( CC  ^pm 
 A ) )
 
Theoremrefdivpm 33365 The quotient of two functions into the real numbers is a partial function. (Contributed by AV, 16-May-2020.)
 |-  (
 ( F : A --> RR  /\  G : A --> RR  /\  A  e.  V )  ->  ( F /_f  G )  e.  ( RR  ^pm 
 A ) )
 
Theoremfdivmptfv 33366 The function value of a quotient of two functions into the complex numbers. (Contributed by AV, 19-May-2020.)
 |-  (
 ( ( F : A
 --> CC  /\  G : A
 --> CC  /\  A  e.  V )  /\  X  e.  ( G supp  0 )
 )  ->  ( ( F /_f  G ) `  X )  =  ( ( F `  X )  /  ( G `  X ) ) )
 
Theoremrefdivmptfv 33367 The function value of a quotient of two functions into the real numbers. (Contributed by AV, 19-May-2020.)
 |-  (
 ( ( F : A
 --> RR  /\  G : A
 --> RR  /\  A  e.  V )  /\  X  e.  ( G supp  0 )
 )  ->  ( ( F /_f  G ) `  X )  =  ( ( F `  X )  /  ( G `  X ) ) )
 
21.25.13.6  Upper bounds
 
Syntaxcbigo 33368 Extend class notation with the class of the "big-O" function.
 class _O
 
Definitiondf-bigo 33369* Define the function "big-O", mapping a real function g to the set of real functions "of order g(x)". Definition in section 1.1 of [AhoHopUll] p. 2. This is a generalisation of "big-O of one", see df-o1 13315 and df-lo1 13316. As explained in the comment of df-o1 , any big-O can be represented in terms of  O(1) and division, see elbigolo1 33378. (Contributed by AV, 15-May-2020.)
 |- _O  =  ( g  e.  ( RR  ^pm  RR )  |->  { f  e.  ( RR 
 ^pm  RR )  |  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( f `
  y )  <_  ( m  x.  (
 g `  y )
 ) } )
 
Theorembigoval 33370* Set of functions of order G(x). (Contributed by AV, 15-May-2020.)
 |-  ( G  e.  ( RR  ^pm 
 RR )  ->  (_O `  G )  =  {
 f  e.  ( RR 
 ^pm  RR )  |  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( f `
  y )  <_  ( m  x.  ( G `  y ) ) } )
 
Theoremelbigofrcl 33371 Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.)
 |-  ( F  e.  (_O `  G )  ->  G  e.  ( RR  ^pm  RR ) )
 
Theoremelbigo 33372* Properties of a function of order G(x). (Contributed by AV, 16-May-2020.)
 |-  ( F  e.  (_O `  G ) 
 <->  ( F  e.  ( RR  ^pm  RR )  /\  G  e.  ( RR  ^pm 
 RR )  /\  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  F  i^i  ( x [,) +oo )
 ) ( F `  y )  <_  ( m  x.  ( G `  y ) ) ) )
 
Theoremelbigo2 33373* Properties of a function of order G(x) under certain assumptions. (Contributed by AV, 17-May-2020.)
 |-  (
 ( ( G : A
 --> RR  /\  A  C_  RR )  /\  ( F : B --> RR  /\  B  C_  A ) ) 
 ->  ( F  e.  (_O `  G )  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  B  ( x  <_  y  ->  ( F `  y )  <_  ( m  x.  ( G `  y ) ) ) ) )
 
Theoremelbigo2r 33374* Sufficient condition for a function to be of order G(x). (Contributed by AV, 18-May-2020.)
 |-  (
 ( ( G : A
 --> RR  /\  A  C_  RR )  /\  ( F : B --> RR  /\  B  C_  A )  /\  ( C  e.  RR  /\  M  e.  RR  /\  A. x  e.  B  ( C  <_  x  ->  ( F `  x ) 
 <_  ( M  x.  ( G `  x ) ) ) ) )  ->  F  e.  (_O `  G ) )
 
Theoremelbigof 33375 A function of order G(x) is a function. (Contributed by AV, 18-May-2020.)
 |-  ( F  e.  (_O `  G )  ->  F : dom  F --> RR )
 
Theoremelbigodm 33376 The domain of a function of order G(x) is a subset of the reals. (Contributed by AV, 18-May-2020.)
 |-  ( F  e.  (_O `  G )  ->  dom  F  C_  RR )
 
Theoremelbigoimp 33377* The defining property of a function of order G(x). (Contributed by AV, 18-May-2020.)
 |-  (
 ( F  e.  (_O `  G )  /\  F : A --> RR  /\  A  C_  dom 
 G )  ->  E. x  e.  RR  E. m  e. 
 RR  A. y  e.  A  ( x  <_  y  ->  ( F `  y ) 
 <_  ( m  x.  ( G `  y ) ) ) )
 
Theoremelbigolo1 33378 A function (into the positive reals) is of order G(x) iff the quotient of the function and G(x) (also a function into the positive reals) is an eventually upper bounded function. (Contributed by AV, 20-May-2020.)
 |-  (
 ( A  C_  RR  /\  G : A --> RR+  /\  F : A --> RR+ )  ->  ( F  e.  (_O `  G ) 
 <->  ( F /_f  G )  e.  <_O(1) ) )
 
21.25.13.7  Logarithm to an arbitrary base (extension)
 
Theoremrege1logbrege0 33379 The general logarithm, with a real base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.)
 |-  (
 ( B  e.  (
 1 (,) +oo )  /\  X  e.  ( 1 [,) +oo ) )  -> 
 0  <_  ( B logb  X ) )
 
Theoremrege1logbzge0 33380 The general logarithm, with an integer base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  ( 1 [,) +oo ) )  -> 
 0  <_  ( B logb  X ) )
 
Theoremfllogbd 33381 A real number is between the base of a logarithm to the power of the floor of the logarithm of the number and the base of the logarithm to the power of the floor of the logarithm of the number plus one. (Contributed by AV, 23-May-2020.)
 |-  ( ph  ->  B  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  X  e.  RR+ )   &    |-  E  =  ( |_ `  ( B logb  X ) )   =>    |-  ( ph  ->  (
 ( B ^ E )  <_  X  /\  X  <  ( B ^ ( E  +  1 )
 ) ) )
 
Theoremrelogbmulbexp 33382 The logarithm of the product of a positive real number and the base to the power of a real number is the logarithm of the positive real number plus the real number. (Contributed by AV, 29-May-2020.)
 |-  (
 ( B  e.  ( RR+  \  { 1 } )  /\  ( A  e.  RR+  /\  C  e.  RR )
 )  ->  ( B logb  ( A  x.  ( B  ^c  C ) ) )  =  ( ( B logb  A )  +  C ) )
 
Theoremrelogbdivb 33383 The logarithm of the quotient of a positive real number and the base is the logarithm of the number minus 1. (Contributed by AV, 29-May-2020.)
 |-  (
 ( B  e.  ( RR+  \  { 1 } )  /\  A  e.  RR+ )  ->  ( B logb  ( A  /  B ) )  =  ( ( B logb  A )  -  1 ) )
 
Theoremlogbge0b 33384 The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+ )  ->  ( 0  <_  ( B logb  X )  <->  1  <_  X ) )
 
Theoremlogblt1b 33385 The logarithm of a number is less than 1 iff the number is less than the base of the logarithm. (Contributed by AV, 30-May-2020.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+ )  ->  ( ( B logb  X )  <  1  <->  X  <  B ) )
 
21.25.13.8  The binary logarithm

If the binary logarithm is used more often, a separate symbol/definition could be provided for it, e.g. log2 =  ( x  e.  ( CC  \  { 0 } )  |->  ( 2 logb  X ) ). Then we can write "( log2 ` x )" (analogous to  ( log x ) for the natural logarithm) instead of  ( 2 logb  x ).

 
Theoremfldivexpfllog2 33386 The floor of a positive real number divided by 2 to the power of the floor of the logarithm to base 2 of the number is 1. (Contributed by AV, 26-May-2020.)
 |-  ( X  e.  RR+  ->  ( |_ `  ( X  /  ( 2 ^ ( |_ `  ( 2 logb  X ) ) ) ) )  =  1 )
 
Theoremnnlog2ge0lt1 33387 A positive integer is 1 iff its binary logarithm is between 0 and 1. (Contributed by AV, 30-May-2020.)
 |-  ( N  e.  NN  ->  ( N  =  1  <->  ( 0  <_  ( 2 logb  N )  /\  ( 2 logb  N )  < 
 1 ) ) )
 
Theoremlogbpw2m1 33388 The floor of the binary logarithm of 2 to the power of a positive integer minus 1 is equal to the integer minus 1. (Contributed by AV, 31-May-2020.)
 |-  ( I  e.  NN  ->  ( |_ `  ( 2 logb  ( ( 2 ^ I
 )  -  1 ) ) )  =  ( I  -  1 ) )
 
Theoremfllog2 33389 The floor of the binary logarithm of 2 to the power of an element of a half-open integer interval bounded by powers of 2 is equal to the integer. (Contributed by AV, 31-May-2020.)
 |-  (
 ( I  e.  NN0  /\  N  e.  ( ( 2 ^ I )..^ ( 2 ^ ( I  +  1 )
 ) ) )  ->  ( |_ `  ( 2 logb  N ) )  =  I )
 
21.25.13.9  Binary length
 
Syntaxcblen 33390 Extend class notation with the class of the binary length function.
 class #b
 
Definitiondf-blen 33391 Define the binary length of an integer. Definition in section 1.3 of [AhoHopUll] p. 12. Although not restricted to integers, this definition is only meaningful for  n  e.  ZZ or even for  n  e.  CC. (Contributed by AV, 16-May-2020.)
 |- #b  =  ( n  e.  _V  |->  if ( n  =  0 ,  1 ,  (
 ( |_ `  (
 2 logb  ( abs `  n ) ) )  +  1 ) ) )
 
Theoremblenval 33392 The binary length of an integer. (Contributed by AV, 20-May-2020.)
 |-  ( N  e.  V  ->  (#b
 `  N )  =  if ( N  =  0 ,  1 ,  ( ( |_ `  (
 2 logb  ( abs `  N ) ) )  +  1 ) ) )
 
Theoremblen0 33393 The binary length of 0. (Contributed by AV, 20-May-2020.)
 |-  (#b `  0 )  =  1
 
Theoremblenn0 33394 The binary length of a "number" not being 0. (Contributed by AV, 20-May-2020.)
 |-  (
 ( N  e.  V  /\  N  =/=  0 ) 
 ->  (#b `  N )  =  ( ( |_ `  (
 2 logb  ( abs `  N ) ) )  +  1 ) )
 
Theoremblenre 33395 The binary length of a positive real number. (Contributed by AV, 20-May-2020.)
 |-  ( N  e.  RR+  ->  (#b `  N )  =  ( ( |_ `  (
 2 logb  N ) )  +  1 ) )
 
Theoremblennn 33396 The binary length of a positive integer. (Contributed by AV, 21-May-2020.)
 |-  ( N  e.  NN  ->  (#b
 `  N )  =  ( ( |_ `  (
 2 logb  N ) )  +  1 ) )
 
Theoremblennnelnn 33397 The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.)
 |-  ( N  e.  NN  ->  (#b
 `  N )  e. 
 NN )
 
Theoremblennn0elnn 33398 The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.)
 |-  ( N  e.  NN0  ->  (#b `  N )  e.  NN )
 
Theoremblenpw2 33399 The binary length of a power of 2 is the exponent plus 1. (Contributed by AV, 30-May-2020.)
 |-  ( I  e.  NN0  ->  (#b `  ( 2 ^ I
 ) )  =  ( I  +  1 ) )
 
Theoremblenpw2m1 33400 The binary length of a power of 2 minus 1 is the exponent. (Contributed by AV, 31-May-2020.)
 |-  ( I  e.  NN  ->  (#b
 `  ( ( 2 ^ I )  -  1 ) )  =  I )
    < 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 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38550
  Copyright terms: Public domain < Previous  Next >