Theorem rpcn 10013

Description: A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)

Assertion

Ref	Expression
rpcn	|- ( A e. RR+ -> A e. CC )

Proof of Theorem rpcn

Step	Hyp	Ref	Expression
1		rpre 10011	. 2 |- ( A e. RR+ -> A e. RR )
2	1	recnd 8318	1 |- ( A e. RR+ -> A e. CC )

Syntax hints:   -> wi 4   e. wcel 2205   CCcc 8141   RR+crp 10004

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 2216  ax-resscn 8235

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-in 3220  df-ss 3227  df-rp 10005

This theorem is referenced by:  rpcnne0  10024  rpcnap0  10025  divge1  10074  sqrtdiv  11752  efgt1p2  12406  efgt1p  12407  pilem1  15770  rpcxp0  15889  rpcxp1  15890  cxprec  15901  rplogbval  15936  rprelogbdiv  15948