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Mirrors > Home > ILE Home > Th. List > cjth | GIF version |
Description: The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
cjth | ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cju 8712 | . . . 4 ⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) | |
2 | riotasbc 5738 | . . . 4 ⊢ (∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ) → [(℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) / 𝑥]((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℂ → [(℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) / 𝑥]((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) |
4 | cjval 10610 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ))) | |
5 | 4 | sbceq1d 2909 | . . 3 ⊢ (𝐴 ∈ ℂ → ([(∗‘𝐴) / 𝑥]((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ) ↔ [(℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) / 𝑥]((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ))) |
6 | 3, 5 | mpbird 166 | . 2 ⊢ (𝐴 ∈ ℂ → [(∗‘𝐴) / 𝑥]((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) |
7 | riotacl 5737 | . . . . 5 ⊢ (∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ) → (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) ∈ ℂ) | |
8 | 1, 7 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) ∈ ℂ) |
9 | 4, 8 | eqeltrd 2214 | . . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) |
10 | oveq2 5775 | . . . . . 6 ⊢ (𝑥 = (∗‘𝐴) → (𝐴 + 𝑥) = (𝐴 + (∗‘𝐴))) | |
11 | 10 | eleq1d 2206 | . . . . 5 ⊢ (𝑥 = (∗‘𝐴) → ((𝐴 + 𝑥) ∈ ℝ ↔ (𝐴 + (∗‘𝐴)) ∈ ℝ)) |
12 | oveq2 5775 | . . . . . . 7 ⊢ (𝑥 = (∗‘𝐴) → (𝐴 − 𝑥) = (𝐴 − (∗‘𝐴))) | |
13 | 12 | oveq2d 5783 | . . . . . 6 ⊢ (𝑥 = (∗‘𝐴) → (i · (𝐴 − 𝑥)) = (i · (𝐴 − (∗‘𝐴)))) |
14 | 13 | eleq1d 2206 | . . . . 5 ⊢ (𝑥 = (∗‘𝐴) → ((i · (𝐴 − 𝑥)) ∈ ℝ ↔ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ)) |
15 | 11, 14 | anbi12d 464 | . . . 4 ⊢ (𝑥 = (∗‘𝐴) → (((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ) ↔ ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ))) |
16 | 15 | sbcieg 2936 | . . 3 ⊢ ((∗‘𝐴) ∈ ℂ → ([(∗‘𝐴) / 𝑥]((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ) ↔ ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ))) |
17 | 9, 16 | syl 14 | . 2 ⊢ (𝐴 ∈ ℂ → ([(∗‘𝐴) / 𝑥]((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ) ↔ ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ))) |
18 | 6, 17 | mpbid 146 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃!wreu 2416 [wsbc 2904 ‘cfv 5118 ℩crio 5722 (class class class)co 5767 ℂcc 7611 ℝcr 7612 ici 7615 + caddc 7616 · cmul 7618 − cmin 7926 ∗ccj 10604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 df-reap 8330 df-cj 10607 |
This theorem is referenced by: recl 10618 crre 10622 |
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