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Mirrors > Home > ILE Home > Th. List > addcmpblnq | Unicode version |
Description: Lemma showing compatibility of addition. (Contributed by NM, 27-Aug-1995.) |
Ref | Expression |
---|---|
addcmpblnq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distrpig 6431 | . . . . . . . 8 | |
2 | 1 | adantl 262 | . . . . . . 7 |
3 | simplll 485 | . . . . . . . 8 | |
4 | simprlr 490 | . . . . . . . 8 | |
5 | mulclpi 6426 | . . . . . . . 8 | |
6 | 3, 4, 5 | syl2anc 391 | . . . . . . 7 |
7 | simpllr 486 | . . . . . . . 8 | |
8 | simprll 489 | . . . . . . . 8 | |
9 | mulclpi 6426 | . . . . . . . 8 | |
10 | 7, 8, 9 | syl2anc 391 | . . . . . . 7 |
11 | mulclpi 6426 | . . . . . . . . 9 | |
12 | 11 | ad2ant2l 477 | . . . . . . . 8 |
13 | 12 | ad2ant2l 477 | . . . . . . 7 |
14 | addclpi 6425 | . . . . . . . 8 | |
15 | 14 | adantl 262 | . . . . . . 7 |
16 | mulcompig 6429 | . . . . . . . 8 | |
17 | 16 | adantl 262 | . . . . . . 7 |
18 | 2, 6, 10, 13, 15, 17 | caovdir2d 5677 | . . . . . 6 |
19 | simplrr 488 | . . . . . . . 8 | |
20 | mulasspig 6430 | . . . . . . . . 9 | |
21 | 20 | adantl 262 | . . . . . . . 8 |
22 | simprrr 492 | . . . . . . . 8 | |
23 | mulclpi 6426 | . . . . . . . . 9 | |
24 | 23 | adantl 262 | . . . . . . . 8 |
25 | 3, 4, 19, 17, 21, 22, 24 | caov4d 5685 | . . . . . . 7 |
26 | 7, 8, 19, 17, 21, 22, 24 | caov4d 5685 | . . . . . . 7 |
27 | 25, 26 | oveq12d 5530 | . . . . . 6 |
28 | 18, 27 | eqtrd 2072 | . . . . 5 |
29 | oveq1 5519 | . . . . . 6 | |
30 | oveq2 5520 | . . . . . 6 | |
31 | 29, 30 | oveqan12d 5531 | . . . . 5 |
32 | 28, 31 | sylan9eq 2092 | . . . 4 |
33 | mulclpi 6426 | . . . . . . . 8 | |
34 | 7, 4, 33 | syl2anc 391 | . . . . . . 7 |
35 | simplrl 487 | . . . . . . . 8 | |
36 | mulclpi 6426 | . . . . . . . 8 | |
37 | 35, 22, 36 | syl2anc 391 | . . . . . . 7 |
38 | simprrl 491 | . . . . . . . 8 | |
39 | mulclpi 6426 | . . . . . . . 8 | |
40 | 19, 38, 39 | syl2anc 391 | . . . . . . 7 |
41 | distrpig 6431 | . . . . . . 7 | |
42 | 34, 37, 40, 41 | syl3anc 1135 | . . . . . 6 |
43 | 7, 4, 35, 17, 21, 22, 24 | caov4d 5685 | . . . . . . 7 |
44 | 7, 4, 19, 17, 21, 38, 24 | caov4d 5685 | . . . . . . 7 |
45 | 43, 44 | oveq12d 5530 | . . . . . 6 |
46 | 42, 45 | eqtrd 2072 | . . . . 5 |
47 | 46 | adantr 261 | . . . 4 |
48 | 32, 47 | eqtr4d 2075 | . . 3 |
49 | addclpi 6425 | . . . . . . . . . 10 | |
50 | 5, 9, 49 | syl2an 273 | . . . . . . . . 9 |
51 | 50 | an42s 523 | . . . . . . . 8 |
52 | 33 | ad2ant2l 477 | . . . . . . . 8 |
53 | 51, 52 | jca 290 | . . . . . . 7 |
54 | addclpi 6425 | . . . . . . . . . 10 | |
55 | 36, 39, 54 | syl2an 273 | . . . . . . . . 9 |
56 | 55 | an42s 523 | . . . . . . . 8 |
57 | 56, 12 | jca 290 | . . . . . . 7 |
58 | 53, 57 | anim12i 321 | . . . . . 6 |
59 | 58 | an4s 522 | . . . . 5 |
60 | enqbreq 6454 | . . . . 5 | |
61 | 59, 60 | syl 14 | . . . 4 |
62 | 61 | adantr 261 | . . 3 |
63 | 48, 62 | mpbird 156 | . 2 |
64 | 63 | ex 108 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 cop 3378 class class class wbr 3764 (class class class)co 5512 cnpi 6370 cpli 6371 cmi 6372 ceq 6377 |
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-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-id 4030 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-oadd 6005 df-omul 6006 df-ni 6402 df-pli 6403 df-mi 6404 df-enq 6445 |
This theorem is referenced by: addpipqqs 6468 |
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