Theorem rpxrd 10026

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 10025	. 2 |- ( ph -> A e. RR )
3	2	rexrd 8319	1 |- ( ph -> A e. RR* )

Syntax hints:   -> wi 4   e. wcel 2203   RR*cxr 8303   RR+crp 9982

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-xr 8308  df-rp 9983

This theorem is referenced by:  ssblex  15283  metequiv2  15348  metss2lem  15349  metcnp  15364  metcnpi3  15369