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Mirrors > Home > ILE Home > Th. List > imval2 | Unicode version |
Description: The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
imval2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imcl 10898 |
. . . 4
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2 | 1 | recnd 8017 |
. . 3
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3 | 2mulicn 9172 |
. . . 4
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4 | 2muliap0 9174 |
. . . 4
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5 | divcanap4 8687 |
. . . 4
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6 | 3, 4, 5 | mp3an23 1340 |
. . 3
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7 | 2, 6 | syl 14 |
. 2
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8 | recl 10897 |
. . . . . . 7
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9 | 8 | recnd 8017 |
. . . . . 6
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10 | ax-icn 7937 |
. . . . . . 7
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11 | mulcl 7969 |
. . . . . . 7
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12 | 10, 2, 11 | sylancr 414 |
. . . . . 6
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13 | 9, 12 | addcld 8008 |
. . . . 5
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14 | 13, 9, 12 | subsubd 8327 |
. . . 4
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15 | replim 10903 |
. . . . 5
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16 | remim 10904 |
. . . . 5
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17 | 15, 16 | oveq12d 5915 |
. . . 4
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18 | 12 | 2timesd 9192 |
. . . . 5
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19 | mulcom 7971 |
. . . . . . . 8
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20 | 3, 19 | mpan2 425 |
. . . . . . 7
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21 | 2cn 9021 |
. . . . . . . 8
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22 | mulass 7973 |
. . . . . . . 8
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23 | 21, 10, 22 | mp3an12 1338 |
. . . . . . 7
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24 | 20, 23 | eqtrd 2222 |
. . . . . 6
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25 | 2, 24 | syl 14 |
. . . . 5
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26 | 9, 12 | pncan2d 8301 |
. . . . . 6
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27 | 26 | oveq1d 5912 |
. . . . 5
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28 | 18, 25, 27 | 3eqtr4d 2232 |
. . . 4
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29 | 14, 17, 28 | 3eqtr4rd 2233 |
. . 3
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30 | 29 | oveq1d 5912 |
. 2
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31 | 7, 30 | eqtr3d 2224 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-2 9009 df-cj 10886 df-re 10887 df-im 10888 |
This theorem is referenced by: resinval 11758 |
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