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Theorem rpcnd 8622
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 8620 . 2 (𝜑𝐴 ∈ ℝ)
32recnd 7052 1 (𝜑𝐴 ∈ ℂ)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1393  cc 6885  +crp 8581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-resscn 6974
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-in 2924  df-ss 2931  df-rp 8582
This theorem is referenced by:  rpcnne0d  8630  ltaddrp2d  8655  iccf1o  8870  cvg1nlemcxze  9555  cvg1nlemres  9558  resqrexlemdec  9583  resqrexlemlo  9585  resqrexlemcalc2  9587  resqrexlemcalc3  9588  resqrexlemnm  9590  resqrexlemcvg  9591  resqrexlemoverl  9593  sqrtdiv  9614  absdivap  9642
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