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Mirrors > Home > ILE Home > Th. List > imval2 | Unicode version |
Description: The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
imval2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imcl 10594 | . . . 4 | |
2 | 1 | recnd 7762 | . . 3 |
3 | 2mulicn 8910 | . . . 4 | |
4 | 2muliap0 8912 | . . . 4 # | |
5 | divcanap4 8427 | . . . 4 # | |
6 | 3, 4, 5 | mp3an23 1292 | . . 3 |
7 | 2, 6 | syl 14 | . 2 |
8 | recl 10593 | . . . . . . 7 | |
9 | 8 | recnd 7762 | . . . . . 6 |
10 | ax-icn 7683 | . . . . . . 7 | |
11 | mulcl 7715 | . . . . . . 7 | |
12 | 10, 2, 11 | sylancr 410 | . . . . . 6 |
13 | 9, 12 | addcld 7753 | . . . . 5 |
14 | 13, 9, 12 | subsubd 8069 | . . . 4 |
15 | replim 10599 | . . . . 5 | |
16 | remim 10600 | . . . . 5 | |
17 | 15, 16 | oveq12d 5760 | . . . 4 |
18 | 12 | 2timesd 8930 | . . . . 5 |
19 | mulcom 7717 | . . . . . . . 8 | |
20 | 3, 19 | mpan2 421 | . . . . . . 7 |
21 | 2cn 8759 | . . . . . . . 8 | |
22 | mulass 7719 | . . . . . . . 8 | |
23 | 21, 10, 22 | mp3an12 1290 | . . . . . . 7 |
24 | 20, 23 | eqtrd 2150 | . . . . . 6 |
25 | 2, 24 | syl 14 | . . . . 5 |
26 | 9, 12 | pncan2d 8043 | . . . . . 6 |
27 | 26 | oveq1d 5757 | . . . . 5 |
28 | 18, 25, 27 | 3eqtr4d 2160 | . . . 4 |
29 | 14, 17, 28 | 3eqtr4rd 2161 | . . 3 |
30 | 29 | oveq1d 5757 | . 2 |
31 | 7, 30 | eqtr3d 2152 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 wcel 1465 class class class wbr 3899 cfv 5093 (class class class)co 5742 cc 7586 cc0 7588 ci 7590 caddc 7591 cmul 7593 cmin 7901 # cap 8311 cdiv 8400 c2 8739 ccj 10579 cre 10580 cim 10581 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-2 8747 df-cj 10582 df-re 10583 df-im 10584 |
This theorem is referenced by: resinval 11349 |
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