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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 7141 | . . . . . . . 8 | |
2 | 1 | adantl 275 | . . . . . . 7 |
3 | simplll 522 | . . . . . . . 8 | |
4 | simprlr 527 | . . . . . . . 8 | |
5 | mulclpi 7136 | . . . . . . . 8 | |
6 | 3, 4, 5 | syl2anc 408 | . . . . . . 7 |
7 | simpllr 523 | . . . . . . . 8 | |
8 | simprll 526 | . . . . . . . 8 | |
9 | mulclpi 7136 | . . . . . . . 8 | |
10 | 7, 8, 9 | syl2anc 408 | . . . . . . 7 |
11 | mulclpi 7136 | . . . . . . . . 9 | |
12 | 11 | ad2ant2l 499 | . . . . . . . 8 |
13 | 12 | ad2ant2l 499 | . . . . . . 7 |
14 | addclpi 7135 | . . . . . . . 8 | |
15 | 14 | adantl 275 | . . . . . . 7 |
16 | mulcompig 7139 | . . . . . . . 8 | |
17 | 16 | adantl 275 | . . . . . . 7 |
18 | 2, 6, 10, 13, 15, 17 | caovdir2d 5947 | . . . . . 6 |
19 | simplrr 525 | . . . . . . . 8 | |
20 | mulasspig 7140 | . . . . . . . . 9 | |
21 | 20 | adantl 275 | . . . . . . . 8 |
22 | simprrr 529 | . . . . . . . 8 | |
23 | mulclpi 7136 | . . . . . . . . 9 | |
24 | 23 | adantl 275 | . . . . . . . 8 |
25 | 3, 4, 19, 17, 21, 22, 24 | caov4d 5955 | . . . . . . 7 |
26 | 7, 8, 19, 17, 21, 22, 24 | caov4d 5955 | . . . . . . 7 |
27 | 25, 26 | oveq12d 5792 | . . . . . 6 |
28 | 18, 27 | eqtrd 2172 | . . . . 5 |
29 | oveq1 5781 | . . . . . 6 | |
30 | oveq2 5782 | . . . . . 6 | |
31 | 29, 30 | oveqan12d 5793 | . . . . 5 |
32 | 28, 31 | sylan9eq 2192 | . . . 4 |
33 | mulclpi 7136 | . . . . . . . 8 | |
34 | 7, 4, 33 | syl2anc 408 | . . . . . . 7 |
35 | simplrl 524 | . . . . . . . 8 | |
36 | mulclpi 7136 | . . . . . . . 8 | |
37 | 35, 22, 36 | syl2anc 408 | . . . . . . 7 |
38 | simprrl 528 | . . . . . . . 8 | |
39 | mulclpi 7136 | . . . . . . . 8 | |
40 | 19, 38, 39 | syl2anc 408 | . . . . . . 7 |
41 | distrpig 7141 | . . . . . . 7 | |
42 | 34, 37, 40, 41 | syl3anc 1216 | . . . . . 6 |
43 | 7, 4, 35, 17, 21, 22, 24 | caov4d 5955 | . . . . . . 7 |
44 | 7, 4, 19, 17, 21, 38, 24 | caov4d 5955 | . . . . . . 7 |
45 | 43, 44 | oveq12d 5792 | . . . . . 6 |
46 | 42, 45 | eqtrd 2172 | . . . . 5 |
47 | 46 | adantr 274 | . . . 4 |
48 | 32, 47 | eqtr4d 2175 | . . 3 |
49 | addclpi 7135 | . . . . . . . . . 10 | |
50 | 5, 9, 49 | syl2an 287 | . . . . . . . . 9 |
51 | 50 | an42s 578 | . . . . . . . 8 |
52 | 33 | ad2ant2l 499 | . . . . . . . 8 |
53 | 51, 52 | jca 304 | . . . . . . 7 |
54 | addclpi 7135 | . . . . . . . . . 10 | |
55 | 36, 39, 54 | syl2an 287 | . . . . . . . . 9 |
56 | 55 | an42s 578 | . . . . . . . 8 |
57 | 56, 12 | jca 304 | . . . . . . 7 |
58 | 53, 57 | anim12i 336 | . . . . . 6 |
59 | 58 | an4s 577 | . . . . 5 |
60 | enqbreq 7164 | . . . . 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 962 wceq 1331 wcel 1480 cop 3530 class class class wbr 3929 (class class class)co 5774 cnpi 7080 cpli 7081 cmi 7082 ceq 7087 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-ni 7112 df-pli 7113 df-mi 7114 df-enq 7155 |
This theorem is referenced by: addpipqqs 7178 |
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