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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 7255 | . . . . . . . 8 | |
2 | 1 | adantl 275 | . . . . . . 7 |
3 | simplll 523 | . . . . . . . 8 | |
4 | simprlr 528 | . . . . . . . 8 | |
5 | mulclpi 7250 | . . . . . . . 8 | |
6 | 3, 4, 5 | syl2anc 409 | . . . . . . 7 |
7 | simpllr 524 | . . . . . . . 8 | |
8 | simprll 527 | . . . . . . . 8 | |
9 | mulclpi 7250 | . . . . . . . 8 | |
10 | 7, 8, 9 | syl2anc 409 | . . . . . . 7 |
11 | mulclpi 7250 | . . . . . . . . 9 | |
12 | 11 | ad2ant2l 500 | . . . . . . . 8 |
13 | 12 | ad2ant2l 500 | . . . . . . 7 |
14 | addclpi 7249 | . . . . . . . 8 | |
15 | 14 | adantl 275 | . . . . . . 7 |
16 | mulcompig 7253 | . . . . . . . 8 | |
17 | 16 | adantl 275 | . . . . . . 7 |
18 | 2, 6, 10, 13, 15, 17 | caovdir2d 5999 | . . . . . 6 |
19 | simplrr 526 | . . . . . . . 8 | |
20 | mulasspig 7254 | . . . . . . . . 9 | |
21 | 20 | adantl 275 | . . . . . . . 8 |
22 | simprrr 530 | . . . . . . . 8 | |
23 | mulclpi 7250 | . . . . . . . . 9 | |
24 | 23 | adantl 275 | . . . . . . . 8 |
25 | 3, 4, 19, 17, 21, 22, 24 | caov4d 6007 | . . . . . . 7 |
26 | 7, 8, 19, 17, 21, 22, 24 | caov4d 6007 | . . . . . . 7 |
27 | 25, 26 | oveq12d 5844 | . . . . . 6 |
28 | 18, 27 | eqtrd 2190 | . . . . 5 |
29 | oveq1 5833 | . . . . . 6 | |
30 | oveq2 5834 | . . . . . 6 | |
31 | 29, 30 | oveqan12d 5845 | . . . . 5 |
32 | 28, 31 | sylan9eq 2210 | . . . 4 |
33 | mulclpi 7250 | . . . . . . . 8 | |
34 | 7, 4, 33 | syl2anc 409 | . . . . . . 7 |
35 | simplrl 525 | . . . . . . . 8 | |
36 | mulclpi 7250 | . . . . . . . 8 | |
37 | 35, 22, 36 | syl2anc 409 | . . . . . . 7 |
38 | simprrl 529 | . . . . . . . 8 | |
39 | mulclpi 7250 | . . . . . . . 8 | |
40 | 19, 38, 39 | syl2anc 409 | . . . . . . 7 |
41 | distrpig 7255 | . . . . . . 7 | |
42 | 34, 37, 40, 41 | syl3anc 1220 | . . . . . 6 |
43 | 7, 4, 35, 17, 21, 22, 24 | caov4d 6007 | . . . . . . 7 |
44 | 7, 4, 19, 17, 21, 38, 24 | caov4d 6007 | . . . . . . 7 |
45 | 43, 44 | oveq12d 5844 | . . . . . 6 |
46 | 42, 45 | eqtrd 2190 | . . . . 5 |
47 | 46 | adantr 274 | . . . 4 |
48 | 32, 47 | eqtr4d 2193 | . . 3 |
49 | addclpi 7249 | . . . . . . . . . 10 | |
50 | 5, 9, 49 | syl2an 287 | . . . . . . . . 9 |
51 | 50 | an42s 579 | . . . . . . . 8 |
52 | 33 | ad2ant2l 500 | . . . . . . . 8 |
53 | 51, 52 | jca 304 | . . . . . . 7 |
54 | addclpi 7249 | . . . . . . . . . 10 | |
55 | 36, 39, 54 | syl2an 287 | . . . . . . . . 9 |
56 | 55 | an42s 579 | . . . . . . . 8 |
57 | 56, 12 | jca 304 | . . . . . . 7 |
58 | 53, 57 | anim12i 336 | . . . . . 6 |
59 | 58 | an4s 578 | . . . . 5 |
60 | enqbreq 7278 | . . . . 5 | |
61 | 59, 60 | syl 14 | . . . 4 |
62 | 61 | adantr 274 | . . 3 |
63 | 48, 62 | mpbird 166 | . 2 |
64 | 63 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 cop 3564 class class class wbr 3967 (class class class)co 5826 cnpi 7194 cpli 7195 cmi 7196 ceq 7201 |
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 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 |
This theorem depends on definitions: df-bi 116 df-dc 821 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 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-irdg 6319 df-oadd 6369 df-omul 6370 df-ni 7226 df-pli 7227 df-mi 7228 df-enq 7269 |
This theorem is referenced by: addpipqqs 7292 |
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