Theorem rpxrd 9861

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 9860	. 2 |- ( ph -> A e. RR )
3	2	rexrd 8164	1 |- ( ph -> A e. RR* )

Syntax hints:   -> wi 4   e. wcel 2180   RR*cxr 8148   RR+crp 9817

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rab 2497  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-xr 8153  df-rp 9818

This theorem is referenced by:  ssblex  15070  metequiv2  15135  metss2lem  15136  metcnp  15151  metcnpi3  15156