Theorem rpxrd 9901

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 9900	. 2 |- ( ph -> A e. RR )
3	2	rexrd 8204	1 |- ( ph -> A e. RR* )

Syntax hints:   -> wi 4   e. wcel 2200   RR*cxr 8188   RR+crp 9857

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-xr 8193  df-rp 9858

This theorem is referenced by:  ssblex  15113  metequiv2  15178  metss2lem  15179  metcnp  15194  metcnpi3  15199