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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 10858 | . 2 โข (๐ด โ โ โ (โโ๐ด) = (โโ(๐ด / i))) | |
2 | ax-icn 7905 | . . . . 5 โข i โ โ | |
3 | iap0 9141 | . . . . 5 โข i # 0 | |
4 | divrecap2 8645 | . . . . 5 โข ((๐ด โ โ โง i โ โ โง i # 0) โ (๐ด / i) = ((1 / i) ยท ๐ด)) | |
5 | 2, 3, 4 | mp3an23 1329 | . . . 4 โข (๐ด โ โ โ (๐ด / i) = ((1 / i) ยท ๐ด)) |
6 | irec 10619 | . . . . 5 โข (1 / i) = -i | |
7 | 6 | oveq1i 5884 | . . . 4 โข ((1 / i) ยท ๐ด) = (-i ยท ๐ด) |
8 | 5, 7 | eqtrdi 2226 | . . 3 โข (๐ด โ โ โ (๐ด / i) = (-i ยท ๐ด)) |
9 | 8 | fveq2d 5519 | . 2 โข (๐ด โ โ โ (โโ(๐ด / i)) = (โโ(-i ยท ๐ด))) |
10 | 1, 9 | eqtrd 2210 | 1 โข (๐ด โ โ โ (โโ๐ด) = (โโ(-i ยท ๐ด))) |
Colors of variables: wff set class |
Syntax hints: โ wi 4 = wceq 1353 โ wcel 2148 class class class wbr 4003 โcfv 5216 (class class class)co 5874 โcc 7808 0cc0 7810 1c1 7811 ici 7812 ยท cmul 7815 -cneg 8128 # cap 8537 / cdiv 8628 โcre 10848 โcim 10849 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-2 8977 df-cj 10850 df-re 10851 df-im 10852 |
This theorem is referenced by: imcl 10862 absimle 11092 recan 11117 |
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