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Theorem reval 14998
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 6847 . . . 4 (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴))
2 oveq12 7371 . . . 4 ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
31, 2mpdan 686 . . 3 (𝑥 = 𝐴 → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
43oveq1d 7377 . 2 (𝑥 = 𝐴 → ((𝑥 + (∗‘𝑥)) / 2) = ((𝐴 + (∗‘𝐴)) / 2))
5 df-re 14992 . 2 ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
6 ovex 7395 . 2 ((𝐴 + (∗‘𝐴)) / 2) ∈ V
74, 5, 6fvmpt 6953 1 (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6501  (class class class)co 7362  cc 11056   + caddc 11061   / cdiv 11819  2c2 12215  ccj 14988  cre 14989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-re 14992
This theorem is referenced by:  recl  15002  ref  15004  crre  15006  addcj  15040  sqreulem  15251  recosval  16025  dvmptre  25349  cosargd  25979  lnopunilem1  30994  cnre2csqima  32532
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