Theorem rpcn 9694

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 9692	. 2 |- ( A e. RR+ -> A e. RR )
2	1	recnd 8017	1 |- ( A e. RR+ -> A e. CC )

Syntax hints:   -> wi 4   e. wcel 2160   CCcc 7840   RR+crp 9685

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-resscn 7934

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-in 3150  df-ss 3157  df-rp 9686

This theorem is referenced by:  rpcnne0  9705  rpcnap0  9706  divge1  9755  sqrtdiv  11086  efgt1p2  11738  efgt1p  11739  pilem1  14677  rpcxp0  14796  rpcxp1  14797  cxprec  14808  rplogbval  14840  rprelogbdiv  14852