Theorem rpcn 9728

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 9726	. 2 |- ( A e. RR+ -> A e. RR )
2	1	recnd 8048	1 |- ( A e. RR+ -> A e. CC )

Syntax hints:   -> wi 4   e. wcel 2164   CCcc 7870   RR+crp 9719

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 2175  ax-resscn 7964

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-in 3159  df-ss 3166  df-rp 9720

This theorem is referenced by:  rpcnne0  9739  rpcnap0  9740  divge1  9789  sqrtdiv  11186  efgt1p2  11838  efgt1p  11839  pilem1  14914  rpcxp0  15033  rpcxp1  15034  cxprec  15045  rplogbval  15077  rprelogbdiv  15089