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Theorem rpgt0d 9237
Description: A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
Hypothesis
Ref Expression
rpred.1  |-  ( ph  ->  A  e.  RR+ )
Assertion
Ref Expression
rpgt0d  |-  ( ph  ->  0  <  A )

Proof of Theorem rpgt0d
StepHypRef Expression
1 rpred.1 . 2  |-  ( ph  ->  A  e.  RR+ )
2 rpgt0 9206 . 2  |-  ( A  e.  RR+  ->  0  < 
A )
31, 2syl 14 1  |-  ( ph  ->  0  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1439   class class class wbr 3851   0cc0 7411    < clt 7583   RR+crp 9195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-rp 9196
This theorem is referenced by:  rpregt0d  9241  ltmulgt11d  9270  ltmulgt12d  9271  gt0divd  9272  ge0divd  9273  lediv12ad  9294  expgt0  10049  nnesq  10134  bccl2  10237  resqrexlemp1rp  10500  resqrexlemover  10504  resqrexlemnm  10512  resqrexlemgt0  10514  resqrexlemglsq  10516  sqrtgt0d  10653  fsumlt  10919  eirraplem  11125  prmind2  11441  sqrt2irrlem  11479  mulc1cncf  11918
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