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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulclsr 7674 | . . . . 5 | |
2 | 1 | ad2ant2r 501 | . . . 4 |
3 | m1r 7672 | . . . . 5 | |
4 | mulclsr 7674 | . . . . . 6 | |
5 | 4 | ad2ant2l 500 | . . . . 5 |
6 | mulclsr 7674 | . . . . 5 | |
7 | 3, 5, 6 | sylancr 411 | . . . 4 |
8 | addclsr 7673 | . . . 4 | |
9 | 2, 7, 8 | syl2anc 409 | . . 3 |
10 | mulclsr 7674 | . . . . 5 | |
11 | 10 | ad2ant2lr 502 | . . . 4 |
12 | mulclsr 7674 | . . . . 5 | |
13 | 12 | ad2ant2rl 503 | . . . 4 |
14 | addclsr 7673 | . . . 4 | |
15 | 11, 13, 14 | syl2anc 409 | . . 3 |
16 | opelxpi 4618 | . . 3 | |
17 | 9, 15, 16 | syl2anc 409 | . 2 |
18 | simpll 519 | . . . . 5 | |
19 | simprl 521 | . . . . 5 | |
20 | 18, 19 | oveq12d 5842 | . . . 4 |
21 | simplr 520 | . . . . . 6 | |
22 | simprr 522 | . . . . . 6 | |
23 | 21, 22 | oveq12d 5842 | . . . . 5 |
24 | 23 | oveq2d 5840 | . . . 4 |
25 | 20, 24 | oveq12d 5842 | . . 3 |
26 | 21, 19 | oveq12d 5842 | . . . 4 |
27 | 18, 22 | oveq12d 5842 | . . . 4 |
28 | 26, 27 | oveq12d 5842 | . . 3 |
29 | 25, 28 | opeq12d 3749 | . 2 |
30 | df-mul 7744 | . . 3 | |
31 | df-c 7738 | . . . . . . 7 | |
32 | 31 | eleq2i 2224 | . . . . . 6 |
33 | 31 | eleq2i 2224 | . . . . . 6 |
34 | 32, 33 | anbi12i 456 | . . . . 5 |
35 | 34 | anbi1i 454 | . . . 4 |
36 | 35 | oprabbii 5876 | . . 3 |
37 | 30, 36 | eqtri 2178 | . 2 |
38 | 17, 29, 37 | ovi3 5957 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wex 1472 wcel 2128 cop 3563 cxp 4584 (class class class)co 5824 coprab 5825 cnr 7217 cm1r 7220 cplr 7221 cmr 7222 cc 7730 cmul 7737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-1o 6363 df-2o 6364 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-lti 7227 df-plpq 7264 df-mpq 7265 df-enq 7267 df-nqqs 7268 df-plqqs 7269 df-mqqs 7270 df-1nqqs 7271 df-rq 7272 df-ltnqqs 7273 df-enq0 7344 df-nq0 7345 df-0nq0 7346 df-plq0 7347 df-mq0 7348 df-inp 7386 df-i1p 7387 df-iplp 7388 df-imp 7389 df-enr 7646 df-nr 7647 df-plr 7648 df-mr 7649 df-m1r 7653 df-c 7738 df-mul 7744 |
This theorem is referenced by: mulresr 7758 mulcnsrec 7763 axmulcl 7786 axi2m1 7795 axcnre 7801 |
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