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Mirrors > Home > MPE Home > Th. List > imcl | Structured version Visualization version GIF version |
Description: The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
imcl | โข (๐ด โ โ โ (โโ๐ด) โ โ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imre 15051 | . 2 โข (๐ด โ โ โ (โโ๐ด) = (โโ(-i ยท ๐ด))) | |
2 | negicn 11457 | . . . 4 โข -i โ โ | |
3 | mulcl 11190 | . . . 4 โข ((-i โ โ โง ๐ด โ โ) โ (-i ยท ๐ด) โ โ) | |
4 | 2, 3 | mpan 688 | . . 3 โข (๐ด โ โ โ (-i ยท ๐ด) โ โ) |
5 | recl 15053 | . . 3 โข ((-i ยท ๐ด) โ โ โ (โโ(-i ยท ๐ด)) โ โ) | |
6 | 4, 5 | syl 17 | . 2 โข (๐ด โ โ โ (โโ(-i ยท ๐ด)) โ โ) |
7 | 1, 6 | eqeltrd 2833 | 1 โข (๐ด โ โ โ (โโ๐ด) โ โ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2106 โcfv 6540 (class class class)co 7405 โcc 11104 โcr 11105 ici 11108 ยท cmul 11111 -cneg 11441 โcre 15040 โcim 15041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-2 12271 df-cj 15042 df-re 15043 df-im 15044 |
This theorem is referenced by: imf 15056 remim 15060 mulre 15064 cjreb 15066 recj 15067 reneg 15068 readd 15069 remullem 15071 remul2 15073 imcj 15075 imneg 15076 imadd 15077 imsub 15078 immul2 15080 imdiv 15081 cjcj 15083 cjadd 15084 ipcnval 15086 cjmulval 15088 cjmulge0 15089 cjneg 15090 imval2 15094 cnrecnv 15108 imcli 15111 imcld 15138 absrele 15251 efeul 16101 absef 16136 absefib 16137 efieq1re 16138 cnsubrg 20997 mbfconst 25141 itgconst 25327 tanregt0 26039 ellogrn 26059 argimgt0 26111 argimlt0 26112 logneg2 26114 tanarg 26118 logf1o2 26149 logreclem 26256 asinlem3a 26364 asinlem3 26365 zetacvg 26508 ccfldextdgrr 32734 sqrtcval 42377 sigarls 45559 |
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