ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rpcnd Unicode version

Theorem rpcnd 9701
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 9699 . 2  |-  ( ph  ->  A  e.  RR )
32recnd 7989 1  |-  ( ph  ->  A  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148   CCcc 7812   RR+crp 9656
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-resscn 7906
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-in 3137  df-ss 3144  df-rp 9657
This theorem is referenced by:  rpcnne0d  9709  ltaddrp2d  9734  iccf1o  10007  bcp1nk  10745  bcpasc  10749  cvg1nlemcxze  10994  cvg1nlemres  10997  resqrexlemdec  11023  resqrexlemlo  11025  resqrexlemcalc2  11027  resqrexlemcalc3  11028  resqrexlemnm  11030  resqrexlemcvg  11031  resqrexlemoverl  11033  sqrtdiv  11054  absdivap  11082  bdtrilem  11250  isumrpcl  11505  expcnvap0  11513  absgtap  11521  cvgratz  11543  mertenslemi1  11546  effsumlt  11703  pythagtriplem12  12278  pythagtriplem14  12280  pythagtriplem16  12282  limcimolemlt  14273  rpdivcxp  14472  rpcxple2  14478  rpcxplt2  14479  rpcxpsqrt  14482  rpabscxpbnd  14499  logbgcd1irr  14525  iooref1o  14923  trilpolemclim  14925  trilpolemisumle  14927  trilpolemeq1  14929  trilpolemlt1  14930
  Copyright terms: Public domain W3C validator