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Mirrors > Home > MPE Home > Th. List > cjval | Structured version Visualization version GIF version |
Description: The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
cjval | ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7152 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑥) = (𝐴 + 𝑥)) | |
2 | 1 | eleq1d 2894 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 + 𝑥) ∈ ℝ ↔ (𝐴 + 𝑥) ∈ ℝ)) |
3 | oveq1 7152 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 − 𝑥) = (𝐴 − 𝑥)) | |
4 | 3 | oveq2d 7161 | . . . . 5 ⊢ (𝑦 = 𝐴 → (i · (𝑦 − 𝑥)) = (i · (𝐴 − 𝑥))) |
5 | 4 | eleq1d 2894 | . . . 4 ⊢ (𝑦 = 𝐴 → ((i · (𝑦 − 𝑥)) ∈ ℝ ↔ (i · (𝐴 − 𝑥)) ∈ ℝ)) |
6 | 2, 5 | anbi12d 630 | . . 3 ⊢ (𝑦 = 𝐴 → (((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦 − 𝑥)) ∈ ℝ) ↔ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ))) |
7 | 6 | riotabidv 7105 | . 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦 − 𝑥)) ∈ ℝ)) = (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ))) |
8 | df-cj 14446 | . 2 ⊢ ∗ = (𝑦 ∈ ℂ ↦ (℩𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦 − 𝑥)) ∈ ℝ))) | |
9 | riotaex 7107 | . 2 ⊢ (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) ∈ V | |
10 | 7, 8, 9 | fvmpt 6761 | 1 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 ℂcc 10523 ℝcr 10524 ici 10527 + caddc 10528 · cmul 10530 − cmin 10858 ∗ccj 14443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-riota 7103 df-ov 7148 df-cj 14446 |
This theorem is referenced by: cjth 14450 remim 14464 |
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