Theorem rpcn 10042

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 10040	. 2 |- ( A e. RR+ -> A e. RR )
2	1	recnd 8344	1 |- ( A e. RR+ -> A e. CC )

Syntax hints:   -> wi 4   e. wcel 2209   CCcc 8167   RR+crp 10033

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-in 3226  df-ss 3233  df-rp 10034

This theorem is referenced by:  rpcnne0  10053  rpcnap0  10054  divge1  10103  sqrtdiv  11786  efgt1p2  12440  efgt1p  12441  pilem1  15803  rpcxp0  15923  rpcxp1  15924  cxprec  15935  rplogbval  15970  rprelogbdiv  15982