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Mirrors > Home > MPE Home > Th. List > imre | Structured version Visualization version 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 15080 | . 2 โข (๐ด โ โ โ (โโ๐ด) = (โโ(๐ด / i))) | |
2 | ax-icn 11191 | . . . . 5 โข i โ โ | |
3 | ine0 11673 | . . . . 5 โข i โ 0 | |
4 | divrec2 11913 | . . . . 5 โข ((๐ด โ โ โง i โ โ โง i โ 0) โ (๐ด / i) = ((1 / i) ยท ๐ด)) | |
5 | 2, 3, 4 | mp3an23 1450 | . . . 4 โข (๐ด โ โ โ (๐ด / i) = ((1 / i) ยท ๐ด)) |
6 | irec 14190 | . . . . 5 โข (1 / i) = -i | |
7 | 6 | oveq1i 7424 | . . . 4 โข ((1 / i) ยท ๐ด) = (-i ยท ๐ด) |
8 | 5, 7 | eqtrdi 2784 | . . 3 โข (๐ด โ โ โ (๐ด / i) = (-i ยท ๐ด)) |
9 | 8 | fveq2d 6895 | . 2 โข (๐ด โ โ โ (โโ(๐ด / i)) = (โโ(-i ยท ๐ด))) |
10 | 1, 9 | eqtrd 2768 | 1 โข (๐ด โ โ โ (โโ๐ด) = (โโ(-i ยท ๐ด))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1534 โ wcel 2099 โ wne 2936 โcfv 6542 (class class class)co 7414 โcc 11130 0cc0 11132 1c1 11133 ici 11134 ยท cmul 11137 -cneg 11469 / cdiv 11895 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-im 15074 |
This theorem is referenced by: imcl 15084 cnpart 15213 sqrtneglem 15239 absimle 15282 recan 15309 tanregt0 26466 asinlem3a 26795 asinsinlem 26816 asinsin 26817 asinbnd 26824 atanbndlem 26850 ftc1anclem6 37165 |
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