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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 9453 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 7054 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
3 | ax-icn 6979 | . . . . 5 ⊢ i ∈ ℂ | |
4 | imcl 9454 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
5 | 4 | recnd 7054 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
6 | mulcl 7008 | . . . . 5 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
7 | 3, 5, 6 | sylancr 393 | . . . 4 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
8 | 2, 7 | neg2subd 7339 | . . 3 ⊢ (𝐴 ∈ ℂ → (-(ℜ‘𝐴) − -(i · (ℑ‘𝐴))) = ((i · (ℑ‘𝐴)) − (ℜ‘𝐴))) |
9 | reneg 9468 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴)) | |
10 | imneg 9476 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘-𝐴) = -(ℑ‘𝐴)) | |
11 | 10 | oveq2d 5528 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘-𝐴)) = (i · -(ℑ‘𝐴))) |
12 | mulneg2 7393 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) | |
13 | 3, 5, 12 | sylancr 393 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) |
14 | 11, 13 | eqtrd 2072 | . . . 4 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘-𝐴)) = -(i · (ℑ‘𝐴))) |
15 | 9, 14 | oveq12d 5530 | . . 3 ⊢ (𝐴 ∈ ℂ → ((ℜ‘-𝐴) − (i · (ℑ‘-𝐴))) = (-(ℜ‘𝐴) − -(i · (ℑ‘𝐴)))) |
16 | 2, 7 | negsubdi2d 7338 | . . 3 ⊢ (𝐴 ∈ ℂ → -((ℜ‘𝐴) − (i · (ℑ‘𝐴))) = ((i · (ℑ‘𝐴)) − (ℜ‘𝐴))) |
17 | 8, 15, 16 | 3eqtr4d 2082 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℜ‘-𝐴) − (i · (ℑ‘-𝐴))) = -((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) |
18 | negcl 7211 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
19 | remim 9460 | . . 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 9460 | . . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = ((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) | |
22 | 21 | negeqd 7206 | . 2 ⊢ (𝐴 ∈ ℂ → -(∗‘𝐴) = -((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) |
23 | 17, 20, 22 | 3eqtr4d 2082 | 1 ⊢ (𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 ici 6891 · cmul 6894 − cmin 7182 -cneg 7183 ∗ccj 9439 ℜcre 9440 ℑcim 9441 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-2 7973 df-cj 9442 df-re 9443 df-im 9444 |
This theorem is referenced by: cjsub 9492 cjnegi 9526 cjnegd 9556 absneg 9648 |
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