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Theorem reval 15040
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 6881 . . . 4 (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴))
2 oveq12 7405 . . . 4 ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
31, 2mpdan 686 . . 3 (𝑥 = 𝐴 → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
43oveq1d 7411 . 2 (𝑥 = 𝐴 → ((𝑥 + (∗‘𝑥)) / 2) = ((𝐴 + (∗‘𝐴)) / 2))
5 df-re 15034 . 2 ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
6 ovex 7429 . 2 ((𝐴 + (∗‘𝐴)) / 2) ∈ V
74, 5, 6fvmpt 6987 1 (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6535  (class class class)co 7396  cc 11095   + caddc 11100   / cdiv 11858  2c2 12254  ccj 15030  cre 15031
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 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399  df-re 15034
This theorem is referenced by:  recl  15044  ref  15046  crre  15048  addcj  15082  sqreulem  15293  recosval  16066  dvmptre  25455  cosargd  26085  lnopunilem1  31228  cnre2csqima  32822
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