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Theorem rpcnd 8722
 Description: A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
Hypothesis
Ref Expression
rpred.1 (𝜑𝐴 ∈ ℝ+)
Assertion
Ref Expression
rpcnd (𝜑𝐴 ∈ ℂ)

Proof of Theorem rpcnd
StepHypRef Expression
1 rpred.1 . . 3 (𝜑𝐴 ∈ ℝ+)
21rpred 8720 . 2 (𝜑𝐴 ∈ ℝ)
32recnd 7113 1 (𝜑𝐴 ∈ ℂ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1409  ℂcc 6945  ℝ+crp 8681 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-resscn 7034 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-in 2952  df-ss 2959  df-rp 8682 This theorem is referenced by:  rpcnne0d  8730  ltaddrp2d  8755  iccf1o  8973  bcp1nk  9630  bcpasc  9634  cvg1nlemcxze  9809  cvg1nlemres  9812  resqrexlemdec  9838  resqrexlemlo  9840  resqrexlemcalc2  9842  resqrexlemcalc3  9843  resqrexlemnm  9845  resqrexlemcvg  9846  resqrexlemoverl  9848  sqrtdiv  9869  absdivap  9897
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