Theorem rpxrd 9514

Description: A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Hypothesis

Ref	Expression
rpred.1	|- ( ph -> A e. RR+ )

Assertion

Ref	Expression
rpxrd	|- ( ph -> A e. RR* )

Proof of Theorem rpxrd

Step	Hyp	Ref	Expression
1		rpred.1	. . 3 |- ( ph -> A e. RR+ )
2	1	rpred 9513	. 2 |- ( ph -> A e. RR )
3	2	rexrd 7839	1 |- ( ph -> A e. RR* )

Syntax hints:   -> wi 4   e. wcel 1481   RR*cxr 7823   RR+crp 9470

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-xr 7828  df-rp 9471

This theorem is referenced by:  ssblex  12639  metequiv2  12704  metss2lem  12705  metcnp  12720  metcnpi3  12725