Theorem rpcn 9784

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 9782	. 2 |- ( A e. RR+ -> A e. RR )
2	1	recnd 8101	1 |- ( A e. RR+ -> A e. CC )

Syntax hints:   -> wi 4   e. wcel 2176   CCcc 7923   RR+crp 9775

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-resscn 8017

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-in 3172  df-ss 3179  df-rp 9776

This theorem is referenced by:  rpcnne0  9795  rpcnap0  9796  divge1  9845  sqrtdiv  11353  efgt1p2  12006  efgt1p  12007  pilem1  15251  rpcxp0  15370  rpcxp1  15371  cxprec  15382  rplogbval  15417  rprelogbdiv  15429