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Theorem rpcnd 9655
Description: A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
Hypothesis
Ref Expression
rpred.1  |-  ( ph  ->  A  e.  RR+ )
Assertion
Ref Expression
rpcnd  |-  ( ph  ->  A  e.  CC )

Proof of Theorem rpcnd
StepHypRef Expression
1 rpred.1 . . 3  |-  ( ph  ->  A  e.  RR+ )
21rpred 9653 . 2  |-  ( ph  ->  A  e.  RR )
32recnd 7948 1  |-  ( ph  ->  A  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141   CCcc 7772   RR+crp 9610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7866
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-in 3127  df-ss 3134  df-rp 9611
This theorem is referenced by:  rpcnne0d  9663  ltaddrp2d  9688  iccf1o  9961  bcp1nk  10696  bcpasc  10700  cvg1nlemcxze  10946  cvg1nlemres  10949  resqrexlemdec  10975  resqrexlemlo  10977  resqrexlemcalc2  10979  resqrexlemcalc3  10980  resqrexlemnm  10982  resqrexlemcvg  10983  resqrexlemoverl  10985  sqrtdiv  11006  absdivap  11034  bdtrilem  11202  isumrpcl  11457  expcnvap0  11465  absgtap  11473  cvgratz  11495  mertenslemi1  11498  effsumlt  11655  pythagtriplem12  12229  pythagtriplem14  12231  pythagtriplem16  12233  limcimolemlt  13427  rpdivcxp  13626  rpcxple2  13632  rpcxplt2  13633  rpcxpsqrt  13636  rpabscxpbnd  13653  logbgcd1irr  13679  iooref1o  14066  trilpolemclim  14068  trilpolemisumle  14070  trilpolemeq1  14072  trilpolemlt1  14073
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