Theorem elrpd 8771

Description: Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Hypotheses

Ref	Expression
elrpd.1	|- ( ph -> A e. RR )
elrpd.2	|- ( ph -> 0 < A )

Assertion

Ref	Expression
elrpd	|- ( ph -> A e. RR+ )

Proof of Theorem elrpd

Step	Hyp	Ref	Expression
1		elrpd.1	. 2 |- ( ph -> A e. RR )
2		elrpd.2	. 2 |- ( ph -> 0 < A )
3		elrp 8736	. 2 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
4	1, 2, 3	sylanbrc 408	1 |- ( ph -> A e. RR+ )

Syntax hints:   -> wi 4   e. wcel 1433   class class class wbr 3785   RRcr 6980   0cc0 6981   < clt 7153   RR+crp 8734

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-rp 8735

This theorem is referenced by:  zltaddlt1le  9028  modqval  9326  ltexp2a  9528  leexp2a  9529  expnlbnd2  9598  resqrexlem1arp  9891  resqrexlemp1rp  9892  resqrexlemcalc2  9901  resqrexlemcalc3  9902  resqrexlemgt0  9906  resqrexlemglsq  9908  rpsqrtcl  9927  absrpclap  9947  mulcn2  10151  climge0  10163