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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 6446 | . . . 4 ⊢ (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴)) | |
2 | oveq12 6931 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴))) | |
3 | 1, 2 | mpdan 677 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (∗‘𝑥)) = (𝐴 + (∗‘𝐴))) |
4 | 3 | oveq1d 6937 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + (∗‘𝑥)) / 2) = ((𝐴 + (∗‘𝐴)) / 2)) |
5 | df-re 14247 | . 2 ⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) | |
6 | ovex 6954 | . 2 ⊢ ((𝐴 + (∗‘𝐴)) / 2) ∈ V | |
7 | 4, 5, 6 | fvmpt 6542 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 + caddc 10275 / cdiv 11032 2c2 11430 ∗ccj 14243 ℜcre 14244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-re 14247 |
This theorem is referenced by: recl 14257 ref 14259 crre 14261 addcj 14295 sqreulem 14506 recosval 15268 dvmptre 24169 cosargd 24791 lnopunilem1 29441 cnre2csqima 30555 |
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