Theorem rpregt0d 9735

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

Step	Hyp	Ref	Expression
1		rpred.1	. . 3 |- ( ph -> A e. RR+ )
2	1	rpred 9728	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9731	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 306	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   class class class wbr 4018   RRcr 7841   0cc0 7842   < clt 8023   RR+crp 9685

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-rp 9686

This theorem is referenced by:  reclt1d  9742  recgt1d  9743  ltrecd  9747  lerecd  9748  ltrec1d  9749  lerec2d  9750  lediv2ad  9751  ltdiv2d  9752  lediv2d  9753  ledivdivd  9754  divge0d  9769  ltmul1d  9770  ltmul2d  9771  lemul1d  9772  lemul2d  9773  ltdiv1d  9774  lediv1d  9775  ltmuldivd  9776  ltmuldiv2d  9777  lemuldivd  9778  lemuldiv2d  9779  ltdivmuld  9780  ltdivmul2d  9781  ledivmuld  9782  ledivmul2d  9783  ltdiv23d  9789  lediv23d  9790  lt2mul2divd  9797  mertenslemi1  11578  isprm6  12182