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Theorem reval 14461
Description: The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
Assertion
Ref Expression
reval (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2))

Proof of Theorem reval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6649 . . . 4 (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴))
2 oveq12 7148 . . . 4 ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
31, 2mpdan 686 . . 3 (𝑥 = 𝐴 → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
43oveq1d 7154 . 2 (𝑥 = 𝐴 → ((𝑥 + (∗‘𝑥)) / 2) = ((𝐴 + (∗‘𝐴)) / 2))
5 df-re 14455 . 2 ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
6 ovex 7172 . 2 ((𝐴 + (∗‘𝐴)) / 2) ∈ V
74, 5, 6fvmpt 6749 1 (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  cfv 6328  (class class class)co 7139  cc 10528   + caddc 10533   / cdiv 11290  2c2 11684  ccj 14451  cre 14452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-re 14455
This theorem is referenced by:  recl  14465  ref  14467  crre  14469  addcj  14503  sqreulem  14715  recosval  15485  dvmptre  24576  cosargd  25203  lnopunilem1  29797  cnre2csqima  31268
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