HomeHome Metamath Proof Explorer
Theorem List (p. 104 of 328)
< 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
(1-21514)
  Hilbert Space Explorer  Hilbert Space Explorer
(21515-23037)
  Users' Mathboxes  Users' Mathboxes
(23038-32776)
 

Theorem List for Metamath Proof Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnwof 10301* Well-ordering principle: any non-empty set of natural numbers has a least element. This version allows  x and  y to be present in  A as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   =>    |-  ( ( A 
 C_  NN  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  x  <_  y
 )
 
Theoremnnwos 10302* Well-ordering principle: any non-empty set of natural numbers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  NN  ph  ->  E. x  e.  NN  ( ph  /\  A. y  e.  NN  ( ps  ->  x  <_  y
 ) ) )
 
Theoremindstr 10303* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  e.  NN  ->  (
 A. y  e.  NN  ( y  <  x  ->  ps )  ->  ph )
 )   =>    |-  ( x  e.  NN  -> 
 ph )
 
Theoremeluznn0 10304 Membership in a nonegative set of upper integers implies membership in  NN0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN0 )
 
Theoremeluz2b1 10305 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  <->  ( N  e.  ZZ  /\  1  <  N ) )
 
Theoremeluz2b2 10306 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  <->  ( N  e.  NN  /\  1  <  N ) )
 
Theoremeluz2b3 10307 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  <->  ( N  e.  NN  /\  N  =/=  1
 ) )
 
Theoremuz2m1nn 10308 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( N  -  1
 )  e.  NN )
 
Theorem1nuz2 10309 1 is not in  ( ZZ>= `  2
). (Contributed by Paul Chapman, 21-Nov-2012.)
 |- 
 -.  1  e.  ( ZZ>=
 `  2 )
 
Theoremelnn1uz2 10310 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( N  e.  NN  <->  ( N  =  1  \/  N  e.  ( ZZ>= `  2 ) ) )
 
Theoremuz2mulcl 10311 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( M  e.  ( ZZ>= `  2 )  /\  N  e.  ( ZZ>= `  2 ) )  ->  ( M  x.  N )  e.  ( ZZ>= `  2 ) )
 
Theoremindstr2 10312* Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by Paul Chapman, 21-Nov-2012.)
 |-  ( x  =  1 
 ->  ( ph  <->  ch ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ch   &    |-  ( x  e.  ( ZZ>= `  2 )  ->  ( A. y  e.  NN  (
 y  <  x  ->  ps )  ->  ph ) )   =>    |-  ( x  e.  NN  -> 
 ph )
 
Theoremuzinfmi 10313 Extract the lower bound of a set of upper integers as its infimum. Note that the " `'  < " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 7-Oct-2005.)
 |-  M  e.  ZZ   =>    |-  sup ( (
 ZZ>= `  M ) ,  RR ,  `'  <  )  =  M
 
Theoremnninfm 10314 The infimum of the set of natural numbers is one. Note that " `'  < " turns sup into inf. (Contributed by NM, 16-Jun-2005.)
 |- 
 sup ( NN ,  RR ,  `'  <  )  =  1
 
Theoremnn0infm 10315 The infimum of the set of nonnegative integers is zero. Note that " `'  < " turns sup into inf. (Contributed by NM, 16-Jun-2005.)
 |- 
 sup ( NN0 ,  RR ,  `'  <  )  =  0
 
Theoreminfmssuzle 10316 The infimum of a subset of a set of upper integers is less than or equal to all members of the subset. Note that the " `'  < " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 11-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  A  e.  S ) 
 ->  sup ( S ,  RR ,  `'  <  )  <_  A )
 
Theoreminfmssuzcl 10317 The infimum of a subset of a set of upper integers belongs to the subset. (Contributed by NM, 11-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  sup ( S ,  RR ,  `'  <  )  e.  S )
 
Theoremublbneg 10318* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( E. x  e. 
 RR  A. y  e.  A  y  <_  x  ->  E. x  e.  RR  A. y  e. 
 { z  e.  RR  |  -u z  e.  A } x  <_  y )
 
Theoremeqreznegel 10319* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( A  C_  ZZ  ->  { z  e.  RR  |  -u z  e.  A }  =  { z  e.  ZZ  |  -u z  e.  A } )
 
Theoremnegn0 10320* The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
 
Theoremsupminf 10321* The supremum of a bounded-above set of reals is the negation of the supremum of that set's image under negation. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  sup ( A ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  ) )
 
Theoremlbzbi 10322* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( A  C_  RR  ->  ( E. x  e. 
 RR  A. y  e.  A  x  <_  y  <->  E. x  e.  ZZ  A. y  e.  A  x  <_  y ) )
 
Theoremzsupss 10323* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 8831.) (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  A  y  <_  x )  ->  E. x  e.  A  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  B  ( y  <  x  ->  E. z  e.  A  y  <  z ) ) )
 
Theoremsuprzcl2 10324* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 10107 avoids ax-pre-sup 8831.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  A  y  <_  x )  ->  sup ( A ,  RR ,  <  )  e.  A )
 
Theoremsuprzub 10325* The supremum of a bounded-above set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  ( ( A  C_  ZZ  /\  E. x  e. 
 ZZ  A. y  e.  A  y  <_  x  /\  B  e.  A )  ->  B  <_  sup ( A ,  RR ,  <  ) )
 
Theoremuzsupss 10326* Any bounded subset of upper integers has a supremum. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 21-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  A  C_  Z  /\  E. x  e.  ZZ  A. y  e.  A  y 
 <_  x )  ->  E. x  e.  Z  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  Z  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )
 
5.4.10  Well-ordering principle for bounded-below sets of integers
 
Theoremuzwo3 10327* Well-ordering principle: any non-empty subset of upper integers has a unique least element. This generalization of uzwo2 10299 allows the lower bound  B to be any real number. See also nnwo 10300 and nnwos 10302. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.)
 |-  ( ( B  e.  RR  /\  ( A  C_  { z  e.  ZZ  |  B  <_  z }  /\  A  =/=  (/) ) )  ->  E! x  e.  A  A. y  e.  A  x  <_  y )
 
Theoremzmin 10328* There is a unique smallest integer greater than or equal to a given real number. (Contributed by NM, 12-Nov-2004.) (Revised by Mario Carneiro, 13-Jun-2014.)
 |-  ( A  e.  RR  ->  E! x  e.  ZZ  ( A  <_  x  /\  A. y  e.  ZZ  ( A  <_  y  ->  x  <_  y ) ) )
 
Theoremzmax 10329* There is a unique largest integer less than or equal to a given real number. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  RR  ->  E! x  e.  ZZ  ( x  <_  A  /\  A. y  e.  ZZ  (
 y  <_  A  ->  y 
 <_  x ) ) )
 
Theoremzbtwnre 10330* There is a unique integer between a real number and the number plus one. Exercise 5 of [Apostol] p. 28. (Contributed by NM, 13-Nov-2004.)
 |-  ( A  e.  RR  ->  E! x  e.  ZZ  ( A  <_  x  /\  x  <  ( A  +  1 ) ) )
 
Theoremrebtwnz 10331* There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  RR  ->  E! x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  +  1 ) ) )
 
5.4.11  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 10332 Extend class notation to include the class of rationals.
 class  QQ
 
Definitiondf-q 10333 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 10334 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
 |- 
 QQ  =  (  /  " ( ZZ  X.  NN ) )
 
Theoremelq 10334* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)
 |-  ( A  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  /  y ) )
 
Theoremqmulz 10335* If  A is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)
 |-  ( A  e.  QQ  ->  E. x  e.  NN  ( A  x.  x )  e.  ZZ )
 
Theoremznq 10336 The ratio of an integer and a natural number is a rational number. (Contributed by NM, 12-Jan-2002.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B )  e.  QQ )
 
Theoremqre 10337 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
 |-  ( A  e.  QQ  ->  A  e.  RR )
 
Theoremzq 10338 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
 |-  ( A  e.  ZZ  ->  A  e.  QQ )
 
Theoremzssq 10339 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
 |- 
 ZZ  C_  QQ
 
Theoremnn0ssq 10340 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN0  C_  QQ
 
Theoremnnssq 10341 The natural numbers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN  C_  QQ
 
Theoremqssre 10342 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
 |- 
 QQ  C_  RR
 
Theoremqsscn 10343 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 QQ  C_  CC
 
Theoremqex 10344 The set of rational numbers exists. See also qexALT 10347. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 QQ  e.  _V
 
Theoremnnq 10345 A natural number is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  A  e.  QQ )
 
Theoremqcn 10346 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
 |-  ( A  e.  QQ  ->  A  e.  CC )
 
TheoremqexALT 10347 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 QQ  e.  _V
 
Theoremqaddcl 10348 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  +  B )  e.  QQ )
 
Theoremqnegcl 10349 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  QQ  -> 
 -u A  e.  QQ )
 
Theoremqmulcl 10350 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  x.  B )  e.  QQ )
 
Theoremqsubcl 10351 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  -  B )  e.  QQ )
 
Theoremqreccl 10352 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  QQ )
 
Theoremqdivcl 10353 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 ) 
 ->  ( A  /  B )  e.  QQ )
 
Theoremqrevaddcl 10354 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
 |-  ( B  e.  QQ  ->  ( ( A  e.  CC  /\  ( A  +  B )  e.  QQ ) 
 <->  A  e.  QQ )
 )
 
Theoremnnrecq 10355 The reciprocal of a natural number is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  ( 1  /  A )  e.  QQ )
 
Theoremirradd 10356 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)
 |-  ( ( A  e.  ( RR  \  QQ )  /\  B  e.  QQ )  ->  ( A  +  B )  e.  ( RR  \  QQ ) )
 
Theoremirrmul 10357 The product of an irrational with a nonzero rational is irrational. (Contributed by NM, 7-Nov-2008.)
 |-  ( ( A  e.  ( RR  \  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  x.  B )  e.  ( RR  \  QQ ) )
 
5.4.12  Existence of the set of complex numbers
 
Theoremrpnnen1lem1 10358* Lemma for rpnnen1 10363. (Contributed by Mario Carneiro, 12-May-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m 
 NN ) )
 
Theoremrpnnen1lem2 10359* Lemma for rpnnen1 10363. (Contributed by Mario Carneiro, 12-May-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  ( ( x  e. 
 RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
 
Theoremrpnnen1lem3 10360* Lemma for rpnnen1 10363. (Contributed by Mario Carneiro, 12-May-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  ( x  e.  RR  ->  A. n  e.  ran  ( F `  x ) n  <_  x )
 
Theoremrpnnen1lem4 10361* Lemma for rpnnen1 10363. (Contributed by Mario Carneiro, 12-May-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
 
Theoremrpnnen1lem5 10362* Lemma for rpnnen1 10363. (Contributed by Mario Carneiro, 12-May-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  x )
 
Theoremrpnnen1 10363* One half of rpnnen 12521, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number  x to the sequence  ( F `  x ) : NN --> QQ such that  ( ( F `  x ) `  k ) is the largest rational number with denominator  k that is strictly less than  x. In this manner, we get a monotonically increasing sequence that converges to  x, and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  RR 
 ~<_  ( QQ  ^m  NN )
 
TheoremreexALT 10364 The set of real numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 RR  e.  _V
 
Theoremcnref1o 10365* There is a natural one-to-one mapping from  ( RR  X.  RR ) to  CC, where we map  <. x ,  y
>. to  ( x  +  ( _i  x.  y ) ). In our construction of the complex numbers, this is in fact our definition of  CC (see df-c 8759), but in the axiomatic treatment we can only show that there is the expected mapping between these two sets. (Contributed by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro, 17-Feb-2014.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y
 ) ) )   =>    |-  F : ( RR  X.  RR ) -1-1-onto-> CC
 
TheoremcnexALT 10366 The set of complex numbers exists. See also ax-cnex 8809. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 CC  e.  _V
 
Theoremxrex 10367 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
 |-  RR*  e.  _V
 
Theoremaddex 10368 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 +  e.  _V
 
Theoremmulex 10369 The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 x.  e.  _V
 
5.5  Order sets
 
5.5.1  Positive reals (as a subset of complex numbers)
 
Syntaxcrp 10370 Extend class notation to include the class of positive reals.
 class  RR+
 
Definitiondf-rp 10371 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |-  RR+  =  { x  e. 
 RR  |  0  < 
 x }
 
Theoremelrp 10372 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremelrpii 10373 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A  e.  RR+
 
Theorem1rp 10374 1 is a positive real. (Contributed by Jeffrey Hankins, 23-Nov-2008.)
 |-  1  e.  RR+
 
Theorem2rp 10375 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  2  e.  RR+
 
Theoremrpre 10376 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  ->  A  e.  RR )
 
Theoremrpxr 10377 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( A  e.  RR+  ->  A  e.  RR* )
 
Theoremrpcn 10378 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  A  e.  CC )
 
Theoremnnrp 10379 A natural number is a positive real. (Contributed by NM, 28-Nov-2008.)
 |-  ( A  e.  NN  ->  A  e.  RR+ )
 
Theoremrpssre 10380 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
 |-  RR+  C_  RR
 
Theoremrpgt0 10381 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
 |-  ( A  e.  RR+  -> 
 0  <  A )
 
Theoremrpge0 10382 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
 |-  ( A  e.  RR+  -> 
 0  <_  A )
 
Theoremrpregt0 10383 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremrprege0 10384 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <_  A )
 )
 
Theoremrpne0 10385 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
 |-  ( A  e.  RR+  ->  A  =/=  0 )
 
Theoremrprene0 10386 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A  =/=  0 ) )
 
Theoremrpcnne0 10387 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  CC  /\  A  =/=  0 ) )
 
Theoremralrp 10388 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
 |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  <  x  -> 
 ph ) )
 
Theoremrexrp 10389 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  <  x  /\  ph ) )
 
Theoremrpaddcl 10390 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  +  B )  e.  RR+ )
 
Theoremrpmulcl 10391 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  x.  B )  e.  RR+ )
 
Theoremrpdivcl 10392 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  /  B )  e.  RR+ )
 
Theoremrpreccl 10393 Closure law for reciprocation of positive reals. (Contributed by Jeffrey Hankins, 23-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
 
Theoremrphalfcl 10394 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  /  2
 )  e.  RR+ )
 
Theoremrpgecl 10395 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  A  <_  B )  ->  B  e.  RR+ )
 
Theoremrphalflt 10396 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( A  e.  RR+  ->  ( A  /  2
 )  <  A )
 
Theoremrerpdivcl 10397 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  /  B )  e.  RR )
 
Theoremge0p1rp 10398 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( A  +  1 )  e.  RR+ )
 
Theoremrpneg 10399 Either a nonzero real or its negation is a positive real, but not both. Axiom 8 of [Apostol] p. 20. (Contributed by NM, 7-Nov-2008.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  ( A  e.  RR+  <->  -.  -u A  e.  RR+ ) )
 
Theorem0nrp 10400 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |- 
 -.  0  e.  RR+
    < 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 32701-32776
  Copyright terms: Public domain < Previous  Next >