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Mirrors > Home > MPE Home > Th. List > reval | Structured version Visualization version GIF version |
Description: The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
reval | ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6695 | . . . 4 ⊢ (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴)) | |
2 | oveq12 7200 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴))) | |
3 | 1, 2 | mpdan 687 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴))) |
4 | 3 | oveq1d 7206 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + (∗‘𝑥)) / 2) = ((𝐴 + (∗‘𝐴)) / 2)) |
5 | df-re 14628 | . 2 ⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) | |
6 | ovex 7224 | . 2 ⊢ ((𝐴 + (∗‘𝐴)) / 2) ∈ V | |
7 | 4, 5, 6 | fvmpt 6796 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 + caddc 10697 / cdiv 11454 2c2 11850 ∗ccj 14624 ℜcre 14625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-re 14628 |
This theorem is referenced by: recl 14638 ref 14640 crre 14642 addcj 14676 sqreulem 14888 recosval 15660 dvmptre 24820 cosargd 25450 lnopunilem1 30045 cnre2csqima 31529 |
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