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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 15081 | . 2 โข (๐ด โ โ โ (โโ๐ด) = (โโ(-i ยท ๐ด))) | |
2 | negicn 11485 | . . . 4 โข -i โ โ | |
3 | mulcl 11216 | . . . 4 โข ((-i โ โ โง ๐ด โ โ) โ (-i ยท ๐ด) โ โ) | |
4 | 2, 3 | mpan 689 | . . 3 โข (๐ด โ โ โ (-i ยท ๐ด) โ โ) |
5 | recl 15083 | . . 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 2829 | 1 โข (๐ด โ โ โ (โโ๐ด) โ โ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2099 โcfv 6542 (class class class)co 7414 โcc 11130 โcr 11131 ici 11134 ยท cmul 11137 -cneg 11469 โcre 15070 โcim 15071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-2 12299 df-cj 15072 df-re 15073 df-im 15074 |
This theorem is referenced by: imf 15086 remim 15090 mulre 15094 cjreb 15096 recj 15097 reneg 15098 readd 15099 remullem 15101 remul2 15103 imcj 15105 imneg 15106 imadd 15107 imsub 15108 immul2 15110 imdiv 15111 cjcj 15113 cjadd 15114 ipcnval 15116 cjmulval 15118 cjmulge0 15119 cjneg 15120 imval2 15124 cnrecnv 15138 imcli 15141 imcld 15168 absrele 15281 efeul 16132 absef 16167 absefib 16168 efieq1re 16169 cnsubrg 21353 mbfconst 25555 itgconst 25741 tanregt0 26466 ellogrn 26486 argimgt0 26539 argimlt0 26540 logneg2 26542 tanarg 26546 logf1o2 26577 logreclem 26687 asinlem3a 26795 asinlem3 26796 zetacvg 26940 ccfldextdgrr 33350 sqrtcval 43065 sigarls 46239 |
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