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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 6523 | . . . . . . . 8 | |
2 | 1 | adantl 271 | . . . . . . 7 |
3 | simplll 499 | . . . . . . . 8 | |
4 | simprlr 504 | . . . . . . . 8 | |
5 | mulclpi 6518 | . . . . . . . 8 | |
6 | 3, 4, 5 | syl2anc 403 | . . . . . . 7 |
7 | simpllr 500 | . . . . . . . 8 | |
8 | simprll 503 | . . . . . . . 8 | |
9 | mulclpi 6518 | . . . . . . . 8 | |
10 | 7, 8, 9 | syl2anc 403 | . . . . . . 7 |
11 | mulclpi 6518 | . . . . . . . . 9 | |
12 | 11 | ad2ant2l 491 | . . . . . . . 8 |
13 | 12 | ad2ant2l 491 | . . . . . . 7 |
14 | addclpi 6517 | . . . . . . . 8 | |
15 | 14 | adantl 271 | . . . . . . 7 |
16 | mulcompig 6521 | . . . . . . . 8 | |
17 | 16 | adantl 271 | . . . . . . 7 |
18 | 2, 6, 10, 13, 15, 17 | caovdir2d 5697 | . . . . . 6 |
19 | simplrr 502 | . . . . . . . 8 | |
20 | mulasspig 6522 | . . . . . . . . 9 | |
21 | 20 | adantl 271 | . . . . . . . 8 |
22 | simprrr 506 | . . . . . . . 8 | |
23 | mulclpi 6518 | . . . . . . . . 9 | |
24 | 23 | adantl 271 | . . . . . . . 8 |
25 | 3, 4, 19, 17, 21, 22, 24 | caov4d 5705 | . . . . . . 7 |
26 | 7, 8, 19, 17, 21, 22, 24 | caov4d 5705 | . . . . . . 7 |
27 | 25, 26 | oveq12d 5550 | . . . . . 6 |
28 | 18, 27 | eqtrd 2113 | . . . . 5 |
29 | oveq1 5539 | . . . . . 6 | |
30 | oveq2 5540 | . . . . . 6 | |
31 | 29, 30 | oveqan12d 5551 | . . . . 5 |
32 | 28, 31 | sylan9eq 2133 | . . . 4 |
33 | mulclpi 6518 | . . . . . . . 8 | |
34 | 7, 4, 33 | syl2anc 403 | . . . . . . 7 |
35 | simplrl 501 | . . . . . . . 8 | |
36 | mulclpi 6518 | . . . . . . . 8 | |
37 | 35, 22, 36 | syl2anc 403 | . . . . . . 7 |
38 | simprrl 505 | . . . . . . . 8 | |
39 | mulclpi 6518 | . . . . . . . 8 | |
40 | 19, 38, 39 | syl2anc 403 | . . . . . . 7 |
41 | distrpig 6523 | . . . . . . 7 | |
42 | 34, 37, 40, 41 | syl3anc 1169 | . . . . . 6 |
43 | 7, 4, 35, 17, 21, 22, 24 | caov4d 5705 | . . . . . . 7 |
44 | 7, 4, 19, 17, 21, 38, 24 | caov4d 5705 | . . . . . . 7 |
45 | 43, 44 | oveq12d 5550 | . . . . . 6 |
46 | 42, 45 | eqtrd 2113 | . . . . 5 |
47 | 46 | adantr 270 | . . . 4 |
48 | 32, 47 | eqtr4d 2116 | . . 3 |
49 | addclpi 6517 | . . . . . . . . . 10 | |
50 | 5, 9, 49 | syl2an 283 | . . . . . . . . 9 |
51 | 50 | an42s 553 | . . . . . . . 8 |
52 | 33 | ad2ant2l 491 | . . . . . . . 8 |
53 | 51, 52 | jca 300 | . . . . . . 7 |
54 | addclpi 6517 | . . . . . . . . . 10 | |
55 | 36, 39, 54 | syl2an 283 | . . . . . . . . 9 |
56 | 55 | an42s 553 | . . . . . . . 8 |
57 | 56, 12 | jca 300 | . . . . . . 7 |
58 | 53, 57 | anim12i 331 | . . . . . 6 |
59 | 58 | an4s 552 | . . . . 5 |
60 | enqbreq 6546 | . . . . 5 | |
61 | 59, 60 | syl 14 | . . . 4 |
62 | 61 | adantr 270 | . . 3 |
63 | 48, 62 | mpbird 165 | . 2 |
64 | 63 | ex 113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 cop 3401 class class class wbr 3785 (class class class)co 5532 cnpi 6462 cpli 6463 cmi 6464 ceq 6469 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 df-ni 6494 df-pli 6495 df-mi 6496 df-enq 6537 |
This theorem is referenced by: addpipqqs 6560 |
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