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Theorem rpregt0 10581
Description: A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
Assertion
Ref Expression
rpregt0  |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <  A ) )

Proof of Theorem rpregt0
StepHypRef Expression
1 elrp 10570 . 2  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
21biimpi 187 1  |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   class class class wbr 4172   RRcr 8945   0cc0 8946    < clt 9076   RR+crp 10568
This theorem is referenced by:  rpne0  10583  modge0  11212  modlt  11213  modid  11225  expnlbnd  11464  o1fsum  12547  isprm6  13064  gexexlem  15422  lmnn  19169  aaliou2b  20211  harmonicbnd4  20802  logfaclbnd  20959  logfacrlim  20961  chto1ub  21123  vmadivsum  21129  dchrmusumlema  21140  dchrvmasumlem2  21145  dchrisum0lem2a  21164  dchrisum0lem2  21165  dchrisum0lem3  21166  mulogsumlem  21178  mulog2sumlem2  21182  selberg2lem  21197  selberg3lem1  21204  pntrmax  21211  pntrsumo1  21212  pntibndlem3  21239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-rp 10569
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