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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 7716 | . . . . 5 | |
2 | 1 | ad2ant2r 506 | . . . 4 |
3 | m1r 7714 | . . . . 5 | |
4 | mulclsr 7716 | . . . . . 6 | |
5 | 4 | ad2ant2l 505 | . . . . 5 |
6 | mulclsr 7716 | . . . . 5 | |
7 | 3, 5, 6 | sylancr 412 | . . . 4 |
8 | addclsr 7715 | . . . 4 | |
9 | 2, 7, 8 | syl2anc 409 | . . 3 |
10 | mulclsr 7716 | . . . . 5 | |
11 | 10 | ad2ant2lr 507 | . . . 4 |
12 | mulclsr 7716 | . . . . 5 | |
13 | 12 | ad2ant2rl 508 | . . . 4 |
14 | addclsr 7715 | . . . 4 | |
15 | 11, 13, 14 | syl2anc 409 | . . 3 |
16 | opelxpi 4643 | . . 3 | |
17 | 9, 15, 16 | syl2anc 409 | . 2 |
18 | simpll 524 | . . . . 5 | |
19 | simprl 526 | . . . . 5 | |
20 | 18, 19 | oveq12d 5871 | . . . 4 |
21 | simplr 525 | . . . . . 6 | |
22 | simprr 527 | . . . . . 6 | |
23 | 21, 22 | oveq12d 5871 | . . . . 5 |
24 | 23 | oveq2d 5869 | . . . 4 |
25 | 20, 24 | oveq12d 5871 | . . 3 |
26 | 21, 19 | oveq12d 5871 | . . . 4 |
27 | 18, 22 | oveq12d 5871 | . . . 4 |
28 | 26, 27 | oveq12d 5871 | . . 3 |
29 | 25, 28 | opeq12d 3773 | . 2 |
30 | df-mul 7786 | . . 3 | |
31 | df-c 7780 | . . . . . . 7 | |
32 | 31 | eleq2i 2237 | . . . . . 6 |
33 | 31 | eleq2i 2237 | . . . . . 6 |
34 | 32, 33 | anbi12i 457 | . . . . 5 |
35 | 34 | anbi1i 455 | . . . 4 |
36 | 35 | oprabbii 5908 | . . 3 |
37 | 30, 36 | eqtri 2191 | . 2 |
38 | 17, 29, 37 | ovi3 5989 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wex 1485 wcel 2141 cop 3586 cxp 4609 (class class class)co 5853 coprab 5854 cnr 7259 cm1r 7262 cplr 7263 cmr 7264 cc 7772 cmul 7779 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-2o 6396 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-i1p 7429 df-iplp 7430 df-imp 7431 df-enr 7688 df-nr 7689 df-plr 7690 df-mr 7691 df-m1r 7695 df-c 7780 df-mul 7786 |
This theorem is referenced by: mulresr 7800 mulcnsrec 7805 axmulcl 7828 axi2m1 7837 axcnre 7843 |
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