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Mirrors > Home > ILE Home > Th. List > mulcnsr | Unicode version |
Description: Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) |
Ref | Expression |
---|---|
mulcnsr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulclsr 7771 |
. . . . 5
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2 | 1 | ad2ant2r 509 |
. . . 4
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3 | m1r 7769 |
. . . . 5
![]() ![]() ![]() ![]() | |
4 | mulclsr 7771 |
. . . . . 6
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5 | 4 | ad2ant2l 508 |
. . . . 5
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6 | mulclsr 7771 |
. . . . 5
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7 | 3, 5, 6 | sylancr 414 |
. . . 4
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8 | addclsr 7770 |
. . . 4
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9 | 2, 7, 8 | syl2anc 411 |
. . 3
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10 | mulclsr 7771 |
. . . . 5
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11 | 10 | ad2ant2lr 510 |
. . . 4
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12 | mulclsr 7771 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 12 | ad2ant2rl 511 |
. . . 4
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14 | addclsr 7770 |
. . . 4
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15 | 11, 13, 14 | syl2anc 411 |
. . 3
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16 | opelxpi 4673 |
. . 3
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17 | 9, 15, 16 | syl2anc 411 |
. 2
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18 | simpll 527 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | simprl 529 |
. . . . 5
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20 | 18, 19 | oveq12d 5909 |
. . . 4
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21 | simplr 528 |
. . . . . 6
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22 | simprr 531 |
. . . . . 6
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23 | 21, 22 | oveq12d 5909 |
. . . . 5
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24 | 23 | oveq2d 5907 |
. . . 4
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25 | 20, 24 | oveq12d 5909 |
. . 3
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26 | 21, 19 | oveq12d 5909 |
. . . 4
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27 | 18, 22 | oveq12d 5909 |
. . . 4
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28 | 26, 27 | oveq12d 5909 |
. . 3
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29 | 25, 28 | opeq12d 3801 |
. 2
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30 | df-mul 7841 |
. . 3
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31 | df-c 7835 |
. . . . . . 7
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32 | 31 | eleq2i 2256 |
. . . . . 6
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33 | 31 | eleq2i 2256 |
. . . . . 6
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34 | 32, 33 | anbi12i 460 |
. . . . 5
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35 | 34 | anbi1i 458 |
. . . 4
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36 | 35 | oprabbii 5946 |
. . 3
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37 | 30, 36 | eqtri 2210 |
. 2
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38 | 17, 29, 37 | ovi3 6028 |
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-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 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-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4304 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-1o 6435 df-2o 6436 df-oadd 6439 df-omul 6440 df-er 6553 df-ec 6555 df-qs 6559 df-ni 7321 df-pli 7322 df-mi 7323 df-lti 7324 df-plpq 7361 df-mpq 7362 df-enq 7364 df-nqqs 7365 df-plqqs 7366 df-mqqs 7367 df-1nqqs 7368 df-rq 7369 df-ltnqqs 7370 df-enq0 7441 df-nq0 7442 df-0nq0 7443 df-plq0 7444 df-mq0 7445 df-inp 7483 df-i1p 7484 df-iplp 7485 df-imp 7486 df-enr 7743 df-nr 7744 df-plr 7745 df-mr 7746 df-m1r 7750 df-c 7835 df-mul 7841 |
This theorem is referenced by: mulresr 7855 mulcnsrec 7860 axmulcl 7883 axi2m1 7892 axcnre 7898 |
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