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Theorem rpregt0d 9490
Description: A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
Hypothesis
Ref Expression
rpred.1  |-  ( ph  ->  A  e.  RR+ )
Assertion
Ref Expression
rpregt0d  |-  ( ph  ->  ( A  e.  RR  /\  0  <  A ) )

Proof of Theorem rpregt0d
StepHypRef Expression
1 rpred.1 . . 3  |-  ( ph  ->  A  e.  RR+ )
21rpred 9483 . 2  |-  ( ph  ->  A  e.  RR )
31rpgt0d 9486 . 2  |-  ( ph  ->  0  <  A )
42, 3jca 304 1  |-  ( ph  ->  ( A  e.  RR  /\  0  <  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480   class class class wbr 3929   RRcr 7619   0cc0 7620    < clt 7800   RR+crp 9441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-rp 9442
This theorem is referenced by:  reclt1d  9497  recgt1d  9498  ltrecd  9502  lerecd  9503  ltrec1d  9504  lerec2d  9505  lediv2ad  9506  ltdiv2d  9507  lediv2d  9508  ledivdivd  9509  divge0d  9524  ltmul1d  9525  ltmul2d  9526  lemul1d  9527  lemul2d  9528  ltdiv1d  9529  lediv1d  9530  ltmuldivd  9531  ltmuldiv2d  9532  lemuldivd  9533  lemuldiv2d  9534  ltdivmuld  9535  ltdivmul2d  9536  ledivmuld  9537  ledivmul2d  9538  ltdiv23d  9544  lediv23d  9545  lt2mul2divd  9552  mertenslemi1  11304  isprm6  11825
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