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Mirrors > Home > ILE Home > Th. List > cjneg | GIF version |
Description: Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
cjneg | ⊢ (𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recl 10625 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 7794 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
3 | ax-icn 7715 | . . . . 5 ⊢ i ∈ ℂ | |
4 | imcl 10626 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
5 | 4 | recnd 7794 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
6 | mulcl 7747 | . . . . 5 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
7 | 3, 5, 6 | sylancr 410 | . . . 4 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
8 | 2, 7 | neg2subd 8090 | . . 3 ⊢ (𝐴 ∈ ℂ → (-(ℜ‘𝐴) − -(i · (ℑ‘𝐴))) = ((i · (ℑ‘𝐴)) − (ℜ‘𝐴))) |
9 | reneg 10640 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴)) | |
10 | imneg 10648 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘-𝐴) = -(ℑ‘𝐴)) | |
11 | 10 | oveq2d 5790 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘-𝐴)) = (i · -(ℑ‘𝐴))) |
12 | mulneg2 8158 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) | |
13 | 3, 5, 12 | sylancr 410 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) |
14 | 11, 13 | eqtrd 2172 | . . . 4 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘-𝐴)) = -(i · (ℑ‘𝐴))) |
15 | 9, 14 | oveq12d 5792 | . . 3 ⊢ (𝐴 ∈ ℂ → ((ℜ‘-𝐴) − (i · (ℑ‘-𝐴))) = (-(ℜ‘𝐴) − -(i · (ℑ‘𝐴)))) |
16 | 2, 7 | negsubdi2d 8089 | . . 3 ⊢ (𝐴 ∈ ℂ → -((ℜ‘𝐴) − (i · (ℑ‘𝐴))) = ((i · (ℑ‘𝐴)) − (ℜ‘𝐴))) |
17 | 8, 15, 16 | 3eqtr4d 2182 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℜ‘-𝐴) − (i · (ℑ‘-𝐴))) = -((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) |
18 | negcl 7962 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
19 | remim 10632 | . . 3 ⊢ (-𝐴 ∈ ℂ → (∗‘-𝐴) = ((ℜ‘-𝐴) − (i · (ℑ‘-𝐴)))) | |
20 | 18, 19 | syl 14 | . 2 ⊢ (𝐴 ∈ ℂ → (∗‘-𝐴) = ((ℜ‘-𝐴) − (i · (ℑ‘-𝐴)))) |
21 | remim 10632 | . . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = ((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) | |
22 | 21 | negeqd 7957 | . 2 ⊢ (𝐴 ∈ ℂ → -(∗‘𝐴) = -((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) |
23 | 17, 20, 22 | 3eqtr4d 2182 | 1 ⊢ (𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 ici 7622 · cmul 7625 − cmin 7933 -cneg 7934 ∗ccj 10611 ℜcre 10612 ℑcim 10613 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-2 8779 df-cj 10614 df-re 10615 df-im 10616 |
This theorem is referenced by: cjsub 10664 cjnegi 10698 cjnegd 10728 absneg 10822 |
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