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Mirrors > Home > ILE Home > Th. List > imre | GIF version |
Description: The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
imre | โข (๐ด โ โ โ (โโ๐ด) = (โโ(-i ยท ๐ด))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imval 10873 | . 2 โข (๐ด โ โ โ (โโ๐ด) = (โโ(๐ด / i))) | |
2 | ax-icn 7920 | . . . . 5 โข i โ โ | |
3 | iap0 9156 | . . . . 5 โข i # 0 | |
4 | divrecap2 8660 | . . . . 5 โข ((๐ด โ โ โง i โ โ โง i # 0) โ (๐ด / i) = ((1 / i) ยท ๐ด)) | |
5 | 2, 3, 4 | mp3an23 1339 | . . . 4 โข (๐ด โ โ โ (๐ด / i) = ((1 / i) ยท ๐ด)) |
6 | irec 10634 | . . . . 5 โข (1 / i) = -i | |
7 | 6 | oveq1i 5898 | . . . 4 โข ((1 / i) ยท ๐ด) = (-i ยท ๐ด) |
8 | 5, 7 | eqtrdi 2236 | . . 3 โข (๐ด โ โ โ (๐ด / i) = (-i ยท ๐ด)) |
9 | 8 | fveq2d 5531 | . 2 โข (๐ด โ โ โ (โโ(๐ด / i)) = (โโ(-i ยท ๐ด))) |
10 | 1, 9 | eqtrd 2220 | 1 โข (๐ด โ โ โ (โโ๐ด) = (โโ(-i ยท ๐ด))) |
Colors of variables: wff set class |
Syntax hints: โ wi 4 = wceq 1363 โ wcel 2158 class class class wbr 4015 โcfv 5228 (class class class)co 5888 โcc 7823 0cc0 7825 1c1 7826 ici 7827 ยท cmul 7830 -cneg 8143 # cap 8552 / cdiv 8643 โcre 10863 โcim 10864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-pre-mulext 7943 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-reap 8546 df-ap 8553 df-div 8644 df-2 8992 df-cj 10865 df-re 10866 df-im 10867 |
This theorem is referenced by: imcl 10877 absimle 11107 recan 11132 |
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