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Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1nuz2 11201 1 is not in  ( ZZ>= `  2
). (Contributed by Paul Chapman, 21-Nov-2012.)
 |- 
 -.  1  e.  ( ZZ>=
 `  2 )
 
Theoremelnn1uz2 11202 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 11203 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 11204* 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 step. (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 11205 Extract the lower bound of an upper set of 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 11206 The infimum of the set of positive integers is one. Note that " `'  < " turns sup into inf. (Contributed by NM, 16-Jun-2005.)
 |- 
 sup ( NN ,  RR ,  `'  <  )  =  1
 
Theoremnn0infm 11207 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 11208 The infimum of a subset of an upper set of 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 11209 The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by NM, 11-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  sup ( S ,  RR ,  `'  <  )  e.  S )
 
Theoremublbneg 11210* 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 11211* 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 11212* 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 11213* 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 11214* 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 11215* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 9599.) (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 11216* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 10982 avoids ax-pre-sup 9599.) (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 11217* 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 11218* Any bounded subset of an upper set of 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
 ) ) )
 
Theoremnn01to3 11219 A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
 |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) )
 
Theoremnn0ge2m1nnALT 11220 Alternate proof of nn0ge2m1nn 10901: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 11132, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 10901. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( N  e.  NN0  /\  2  <_  N ) 
 ->  ( N  -  1
 )  e.  NN )
 
5.4.11  Well-ordering principle for bounded-below sets of integers
 
Theoremuzwo3 11221* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 11190 allows the lower bound  B to be any real number. See also nnwo 11191 and nnwos 11193. (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 11222* 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 11223* 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 11224* 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 11225* 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 ) ) )
 
5.4.12  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 11226 Extend class notation to include the class of rationals.
 class  QQ
 
Definitiondf-q 11227 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 11228 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
 |- 
 QQ  =  (  /  " ( ZZ  X.  NN ) )
 
Theoremelq 11228* 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 11229* 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 11230 The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B )  e.  QQ )
 
Theoremqre 11231 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
 |-  ( A  e.  QQ  ->  A  e.  RR )
 
Theoremzq 11232 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
 |-  ( A  e.  ZZ  ->  A  e.  QQ )
 
Theoremzssq 11233 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
 |- 
 ZZ  C_  QQ
 
Theoremnn0ssq 11234 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN0  C_  QQ
 
Theoremnnssq 11235 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN  C_  QQ
 
Theoremqssre 11236 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
 |- 
 QQ  C_  RR
 
Theoremqsscn 11237 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 QQ  C_  CC
 
Theoremqex 11238 The set of rational numbers exists. See also qexALT 11241. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 QQ  e.  _V
 
Theoremnnq 11239 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  A  e.  QQ )
 
Theoremqcn 11240 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
 |-  ( A  e.  QQ  ->  A  e.  CC )
 
TheoremqexALT 11241 Alternate proof of qex 11238. (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 11242 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  +  B )  e.  QQ )
 
Theoremqnegcl 11243 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 11244 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  x.  B )  e.  QQ )
 
Theoremqsubcl 11245 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  -  B )  e.  QQ )
 
Theoremqreccl 11246 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  QQ )
 
Theoremqdivcl 11247 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 ) 
 ->  ( A  /  B )  e.  QQ )
 
Theoremqrevaddcl 11248 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 11249 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  ( 1  /  A )  e.  QQ )
 
Theoremirradd 11250 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 11251 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 ) )
 
5.4.13  Existence of the set of complex numbers
 
Theoremrpnnen1lem1 11252* Lemma for rpnnen1 11257. (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 11253* Lemma for rpnnen1 11257. (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 11254* Lemma for rpnnen1 11257. (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 11255* Lemma for rpnnen1 11257. (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 11256* Lemma for rpnnen1 11257. (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 11257* One half of rpnnen 14167, 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 11258 Alternate proof of reex 9612. (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 11259* 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 9527), 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 11260 The set of complex numbers exists. See also ax-cnex 9577. (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 11261 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
 |-  RR*  e.  _V
 
Theoremaddex 11262 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 +  e.  _V
 
Theoremmulex 11263 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 11264 Extend class notation to include the class of positive reals.
 class  RR+
 
Definitiondf-rp 11265 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 11266 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremelrpii 11267 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A  e.  RR+
 
Theorem1rp 11268 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
 |-  1  e.  RR+
 
Theorem2rp 11269 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  2  e.  RR+
 
Theoremrpre 11270 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  ->  A  e.  RR )
 
Theoremrpxr 11271 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( A  e.  RR+  ->  A  e.  RR* )
 
Theoremrpcn 11272 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  A  e.  CC )
 
Theoremnnrp 11273 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
 |-  ( A  e.  NN  ->  A  e.  RR+ )
 
Theoremrpssre 11274 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
 |-  RR+  C_  RR
 
Theoremrpgt0 11275 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
 |-  ( A  e.  RR+  -> 
 0  <  A )
 
Theoremrpge0 11276 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
 |-  ( A  e.  RR+  -> 
 0  <_  A )
 
Theoremrpregt0 11277 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 11278 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <_  A )
 )
 
Theoremrpne0 11279 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
 |-  ( A  e.  RR+  ->  A  =/=  0 )
 
Theoremrprene0 11280 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A  =/=  0 ) )
 
Theoremrpcnne0 11281 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  CC  /\  A  =/=  0 ) )
 
Theoremrpcndif0 11282 A positive real number is a complex number not being 0. (Contributed by AV, 29-May-2020.)
 |-  ( A  e.  RR+  ->  A  e.  ( CC  \  { 0 } )
 )
 
Theoremralrp 11283 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
 |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  <  x  -> 
 ph ) )
 
Theoremrexrp 11284 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  <  x  /\  ph ) )
 
Theoremrpaddcl 11285 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 11286 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 11287 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  /  B )  e.  RR+ )
 
Theoremrpreccl 11288 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
 
Theoremrphalfcl 11289 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  /  2
 )  e.  RR+ )
 
Theoremrpgecl 11290 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 11291 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 11292 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 11293 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 11294 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+ ) )
 
Theoremnegelrp 11295 Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)
 |-  ( A  e.  RR  ->  ( -u A  e.  RR+  <->  A  <  0 ) )
 
Theorem0nrp 11296 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |- 
 -.  0  e.  RR+
 
Theoremltsubrp 11297 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  -  B )  <  A )
 
Theoremltaddrp 11298 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  <  ( A  +  B )
 )
 
Theoremdifrp 11299 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( B  -  A )  e.  RR+ ) )
 
Theoremelrpd 11300 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  A  e.  RR+ )
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