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Mirrors > Home > ILE Home > Th. List > mulcmpblnr | Unicode version |
Description: Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) |
Ref | Expression |
---|---|
mulcmpblnr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcmpblnrlemg 7661 | . . 3 | |
2 | simplrr 526 | . . . . 5 | |
3 | simprll 527 | . . . . 5 | |
4 | mulclpr 7493 | . . . . 5 | |
5 | 2, 3, 4 | syl2anc 409 | . . . 4 |
6 | simplll 523 | . . . . . . 7 | |
7 | mulclpr 7493 | . . . . . . 7 | |
8 | 6, 3, 7 | syl2anc 409 | . . . . . 6 |
9 | simpllr 524 | . . . . . . 7 | |
10 | simprlr 528 | . . . . . . 7 | |
11 | mulclpr 7493 | . . . . . . 7 | |
12 | 9, 10, 11 | syl2anc 409 | . . . . . 6 |
13 | addclpr 7458 | . . . . . 6 | |
14 | 8, 12, 13 | syl2anc 409 | . . . . 5 |
15 | simplrl 525 | . . . . . . 7 | |
16 | simprrr 530 | . . . . . . 7 | |
17 | mulclpr 7493 | . . . . . . 7 | |
18 | 15, 16, 17 | syl2anc 409 | . . . . . 6 |
19 | simprrl 529 | . . . . . . 7 | |
20 | mulclpr 7493 | . . . . . . 7 | |
21 | 2, 19, 20 | syl2anc 409 | . . . . . 6 |
22 | addclpr 7458 | . . . . . 6 | |
23 | 18, 21, 22 | syl2anc 409 | . . . . 5 |
24 | addclpr 7458 | . . . . 5 | |
25 | 14, 23, 24 | syl2anc 409 | . . . 4 |
26 | mulclpr 7493 | . . . . . . 7 | |
27 | 6, 10, 26 | syl2anc 409 | . . . . . 6 |
28 | mulclpr 7493 | . . . . . . 7 | |
29 | 9, 3, 28 | syl2anc 409 | . . . . . 6 |
30 | addclpr 7458 | . . . . . 6 | |
31 | 27, 29, 30 | syl2anc 409 | . . . . 5 |
32 | mulclpr 7493 | . . . . . . 7 | |
33 | 15, 19, 32 | syl2anc 409 | . . . . . 6 |
34 | mulclpr 7493 | . . . . . . 7 | |
35 | 2, 16, 34 | syl2anc 409 | . . . . . 6 |
36 | addclpr 7458 | . . . . . 6 | |
37 | 33, 35, 36 | syl2anc 409 | . . . . 5 |
38 | addclpr 7458 | . . . . 5 | |
39 | 31, 37, 38 | syl2anc 409 | . . . 4 |
40 | addcanprg 7537 | . . . 4 | |
41 | 5, 25, 39, 40 | syl3anc 1220 | . . 3 |
42 | 1, 41 | syld 45 | . 2 |
43 | enrbreq 7655 | . . 3 | |
44 | 14, 31, 37, 23, 43 | syl22anc 1221 | . 2 |
45 | 42, 44 | sylibrd 168 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 cop 3563 class class class wbr 3966 (class class class)co 5825 cnp 7212 cpp 7214 cmp 7215 cer 7217 |
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 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
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 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-eprel 4250 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-1o 6364 df-2o 6365 df-oadd 6368 df-omul 6369 df-er 6481 df-ec 6483 df-qs 6487 df-ni 7225 df-pli 7226 df-mi 7227 df-lti 7228 df-plpq 7265 df-mpq 7266 df-enq 7268 df-nqqs 7269 df-plqqs 7270 df-mqqs 7271 df-1nqqs 7272 df-rq 7273 df-ltnqqs 7274 df-enq0 7345 df-nq0 7346 df-0nq0 7347 df-plq0 7348 df-mq0 7349 df-inp 7387 df-iplp 7389 df-imp 7390 df-enr 7647 |
This theorem is referenced by: mulsrmo 7665 |
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