Theorem rpxrd 9633

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 9632	. 2 |- ( ph -> A e. RR )
3	2	rexrd 7948	1 |- ( ph -> A e. RR* )

Syntax hints:   -> wi 4   e. wcel 2136   RR*cxr 7932   RR+crp 9589

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-xr 7937  df-rp 9590

This theorem is referenced by:  ssblex  13071  metequiv2  13136  metss2lem  13137  metcnp  13152  metcnpi3  13157