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Theorem reval 13694
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 6103 . . . 4 (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴))
2 oveq12 6558 . . . 4 ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
31, 2mpdan 699 . . 3 (𝑥 = 𝐴 → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
43oveq1d 6564 . 2 (𝑥 = 𝐴 → ((𝑥 + (∗‘𝑥)) / 2) = ((𝐴 + (∗‘𝐴)) / 2))
5 df-re 13688 . 2 ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
6 ovex 6577 . 2 ((𝐴 + (∗‘𝐴)) / 2) ∈ V
74, 5, 6fvmpt 6191 1 (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  cfv 5804  (class class class)co 6549  cc 9813   + caddc 9818   / cdiv 10563  2c2 10947  ccj 13684  cre 13685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-re 13688
This theorem is referenced by:  recl  13698  ref  13700  crre  13702  addcj  13736  sqreulem  13947  recosval  14705  dvmptre  23538  cosargd  24158  lnopunilem1  28253  cnre2csqima  29285
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