ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rpgt0d Unicode version

Theorem rpgt0d 9701
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 9667 . 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 2148   class class class wbr 4005   0cc0 7813    < clt 7994   RR+crp 9655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-rp 9656
This theorem is referenced by:  rpregt0d  9705  ltmulgt11d  9734  ltmulgt12d  9735  gt0divd  9736  ge0divd  9737  lediv12ad  9758  expgt0  10555  nnesq  10642  bccl2  10750  resqrexlemp1rp  11017  resqrexlemover  11021  resqrexlemnm  11029  resqrexlemgt0  11031  resqrexlemglsq  11033  sqrtgt0d  11170  reccn2ap  11323  fsumlt  11474  eirraplem  11786  dvdsmodexp  11804  prmind2  12122  sqrt2irrlem  12163  modprmn0modprm0  12258  ssblex  14016  mulc1cncf  14161  cncfmptc  14167  mulcncflem  14175  cnplimclemle  14222  pilem3  14289  iooref1o  14867  trilpolemeq1  14873  nconstwlpolemgt0  14897  taupi  14906
  Copyright terms: Public domain W3C validator