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Theorem List for Metamath Proof Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqmulcl 10301 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  x.  B )  e.  QQ )
 
Theoremqsubcl 10302 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  -  B )  e.  QQ )
 
Theoremqreccl 10303 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  QQ )
 
Theoremqdivcl 10304 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 ) 
 ->  ( A  /  B )  e.  QQ )
 
Theoremqrevaddcl 10305 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 10306 The reciprocal of a natural number is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  ( 1  /  A )  e.  QQ )
 
Theoremirradd 10307 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 10308 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 10309* Lemma for rpnnen1 10314. (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 10310* Lemma for rpnnen1 10314. (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 10311* Lemma for rpnnen1 10314. (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 10312* Lemma for rpnnen1 10314. (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 10313* Lemma for rpnnen1 10314. (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 10314* One half of rpnnen 12467, 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 10315 The set of real numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.)
 |- 
 RR  e.  _V
 
Theoremcnref1o 10316* 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 8711), 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 10317 The set of complex numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.)
 |- 
 CC  e.  _V
 
Theoremxrex 10318 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
 |-  RR*  e.  _V
 
Theoremaddex 10319 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 +  e.  _V
 
Theoremmulex 10320 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 10321 Extend class notation to include the class of positive reals.
 class  RR+
 
Definitiondf-rp 10322 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 10323 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremelrpii 10324 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A  e.  RR+
 
Theorem1rp 10325 1 is a positive real. (Contributed by Jeffrey Hankins, 23-Nov-2008.)
 |-  1  e.  RR+
 
Theorem2rp 10326 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  2  e.  RR+
 
Theoremrpre 10327 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  ->  A  e.  RR )
 
Theoremrpxr 10328 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( A  e.  RR+  ->  A  e.  RR* )
 
Theoremrpcn 10329 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  A  e.  CC )
 
Theoremnnrp 10330 A natural number is a positive real. (Contributed by NM, 28-Nov-2008.)
 |-  ( A  e.  NN  ->  A  e.  RR+ )
 
Theoremrpssre 10331 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
 |-  RR+  C_  RR
 
Theoremrpgt0 10332 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
 |-  ( A  e.  RR+  -> 
 0  <  A )
 
Theoremrpge0 10333 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
 |-  ( A  e.  RR+  -> 
 0  <_  A )
 
Theoremrpregt0 10334 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 10335 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <_  A )
 )
 
Theoremrpne0 10336 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
 |-  ( A  e.  RR+  ->  A  =/=  0 )
 
Theoremrprene0 10337 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A  =/=  0 ) )
 
Theoremrpcnne0 10338 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  CC  /\  A  =/=  0 ) )
 
Theoremralrp 10339 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
 |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  <  x  -> 
 ph ) )
 
Theoremrexrp 10340 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  <  x  /\  ph ) )
 
Theoremrpaddcl 10341 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 10342 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 10343 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  /  B )  e.  RR+ )
 
Theoremrpreccl 10344 Closure law for reciprocation of positive reals. (Contributed by Jeffrey Hankins, 23-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
 
Theoremrphalfcl 10345 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  /  2
 )  e.  RR+ )
 
Theoremrpgecl 10346 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 10347 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 10348 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 10349 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 10350 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 10351 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |- 
 -.  0  e.  RR+
 
Theoremltsubrp 10352 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 10353 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 10354 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 10355 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+ )
 
Theoremnnrpd 10356 A natural number is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR+ )
 
Theoremrpred 10357 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremrpxrd 10358 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  RR* )
 
Theoremrpcnd 10359 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremrpgt0d 10360 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  0  <  A )
 
Theoremrpge0d 10361 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremrpne0d 10362 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremrpregt0d 10363 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremrprege0d 10364 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A ) )
 
Theoremrprene0d 10365 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  RR  /\  A  =/=  0 ) )
 
Theoremrpcnne0d 10366 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  CC  /\  A  =/=  0 ) )
 
Theoremrpreccld 10367 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR+ )
 
Theoremrprecred 10368 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremrphalfcld 10369 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  /  2 )  e.  RR+ )
 
Theoremreclt1d 10370 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  1  <->  1  <  (
 1  /  A )
 ) )
 
Theoremrecgt1d 10371 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  A  <->  ( 1  /  A )  <  1 ) )
 
Theoremrpaddcld 10372 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  +  B )  e.  RR+ )
 
Theoremrpmulcld 10373 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  x.  B )  e.  RR+ )
 
Theoremrpdivcld 10374 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  /  B )  e.  RR+ )
 
Theoremltrecd 10375 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( 1  /  B )  <  ( 1 
 /  A ) ) )
 
Theoremlerecd 10376 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( 1  /  B )  <_  ( 1 
 /  A ) ) )
 
Theoremltrec1d 10377 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  ( 1  /  A )  <  B )   =>    |-  ( ph  ->  ( 1  /  B )  <  A )
 
Theoremlerec2d 10378 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  A 
 <_  ( 1  /  B ) )   =>    |-  ( ph  ->  B  <_  ( 1  /  A ) )
 
Theoremlediv2ad 10379 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( C  /  B )  <_  ( C  /  A ) )
 
Theoremltdiv2d 10380 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  /  B )  <  ( C 
 /  A ) ) )
 
Theoremlediv2d 10381 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  /  B )  <_  ( C 
 /  A ) ) )
 
Theoremledivdivd 10382 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  ( A  /  B ) 
 <_  ( C  /  D ) )   =>    |-  ( ph  ->  ( D  /  C )  <_  ( B  /  A ) )
 
Theoremge0p1rpd 10383 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( A  +  1 )  e.  RR+ )
 
Theoremrerpdivcld 10384 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  /  B )  e. 
 RR )
 
Theoremltsubrpd 10385 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  -  B )  <  A )
 
Theoremltaddrpd 10386 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  A  <  ( A  +  B ) )
 
Theoremltaddrp2d 10387 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  A  <  ( B  +  A ) )
 
Theoremltmulgt11d 10388 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  A  <->  B  <  ( B  x.  A ) ) )
 
Theoremltmulgt12d 10389 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  A  <->  B  <  ( A  x.  B ) ) )
 
Theoremgt0divd 10390 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 0  <  A  <->  0  <  ( A  /  B ) ) )
 
Theoremge0divd 10391 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 0  <_  A  <->  0  <_  ( A  /  B ) ) )
 
Theoremrpgecld 10392 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  B 
 <_  A )   =>    |-  ( ph  ->  A  e.  RR+ )
 
Theoremdivge0d 10393 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  0  <_  ( A  /  B ) )
 
Theoremltmul1d 10394 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
 
Theoremltmul2d 10395 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
 
Theoremlemul1d 10396 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
 
Theoremlemul2d 10397 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  x.  A )  <_  ( C  x.  B ) ) )
 
Theoremltdiv1d 10398 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  /  C )  <  ( B 
 /  C ) ) )
 
Theoremlediv1d 10399 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  /  C )  <_  ( B 
 /  C ) ) )
 
Theoremltmuldivd 10400 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  C )  <  B  <->  A  <  ( B 
 /  C ) ) )
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