Theorem reval 15085

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.

Step	Hyp	Ref	Expression
1		fveq2 6897	. . . 4 ⊢ (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴))
2		oveq12 7429	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
3	1, 2	mpdan 686	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴)))
4	3	oveq1d 7435	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + (∗‘𝑥)) / 2) = ((𝐴 + (∗‘𝐴)) / 2))
5		df-re 15079	. 2 ⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
6		ovex 7453	. 2 ⊢ ((𝐴 + (∗‘𝐴)) / 2) ∈ V
7	4, 5, 6	fvmpt 7005	1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ‘cfv 6548  (class class class)co 7420  ℂcc 11136   + caddc 11141   / cdiv 11901  2c2 12297  ∗ccj 15075  ℜcre 15076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-re 15079

This theorem is referenced by:  recl  15089  ref  15091  crre  15093  addcj  15127  sqreulem  15338  recosval  16112  dvmptre  25900  cosargd  26541  lnopunilem1  31819  cnre2csqima  33512