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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 6839 | . . . . 5 | |
2 | 1 | ad2ant2r 478 | . . . 4 |
3 | m1r 6837 | . . . . 5 | |
4 | mulclsr 6839 | . . . . . 6 | |
5 | 4 | ad2ant2l 477 | . . . . 5 |
6 | mulclsr 6839 | . . . . 5 | |
7 | 3, 5, 6 | sylancr 393 | . . . 4 |
8 | addclsr 6838 | . . . 4 | |
9 | 2, 7, 8 | syl2anc 391 | . . 3 |
10 | mulclsr 6839 | . . . . 5 | |
11 | 10 | ad2ant2lr 479 | . . . 4 |
12 | mulclsr 6839 | . . . . 5 | |
13 | 12 | ad2ant2rl 480 | . . . 4 |
14 | addclsr 6838 | . . . 4 | |
15 | 11, 13, 14 | syl2anc 391 | . . 3 |
16 | opelxpi 4376 | . . 3 | |
17 | 9, 15, 16 | syl2anc 391 | . 2 |
18 | simpll 481 | . . . . 5 | |
19 | simprl 483 | . . . . 5 | |
20 | 18, 19 | oveq12d 5530 | . . . 4 |
21 | simplr 482 | . . . . . 6 | |
22 | simprr 484 | . . . . . 6 | |
23 | 21, 22 | oveq12d 5530 | . . . . 5 |
24 | 23 | oveq2d 5528 | . . . 4 |
25 | 20, 24 | oveq12d 5530 | . . 3 |
26 | 21, 19 | oveq12d 5530 | . . . 4 |
27 | 18, 22 | oveq12d 5530 | . . . 4 |
28 | 26, 27 | oveq12d 5530 | . . 3 |
29 | 25, 28 | opeq12d 3557 | . 2 |
30 | df-mul 6901 | . . 3 | |
31 | df-c 6895 | . . . . . . 7 | |
32 | 31 | eleq2i 2104 | . . . . . 6 |
33 | 31 | eleq2i 2104 | . . . . . 6 |
34 | 32, 33 | anbi12i 433 | . . . . 5 |
35 | 34 | anbi1i 431 | . . . 4 |
36 | 35 | oprabbii 5560 | . . 3 |
37 | 30, 36 | eqtri 2060 | . 2 |
38 | 17, 29, 37 | ovi3 5637 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cop 3378 cxp 4343 (class class class)co 5512 coprab 5513 cnr 6395 cm1r 6398 cplr 6399 cmr 6400 cc 6887 cmul 6894 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-imp 6567 df-enr 6811 df-nr 6812 df-plr 6813 df-mr 6814 df-m1r 6818 df-c 6895 df-mul 6901 |
This theorem is referenced by: mulresr 6914 mulcnsrec 6919 axmulcl 6942 axi2m1 6949 axcnre 6955 |
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