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Mirrors > Home > ILE Home > Th. List > mulextsr1lem | Unicode version |
Description: Lemma for mulextsr1 7713. (Contributed by Jim Kingdon, 17-Feb-2020.) |
Ref | Expression |
---|---|
mulextsr1lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcomprg 7510 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | addclpr 7469 | . . . . . . . 8 | |
4 | 3 | adantl 275 | . . . . . . 7 |
5 | simp2l 1012 | . . . . . . . 8 | |
6 | simp3r 1015 | . . . . . . . 8 | |
7 | mulclpr 7504 | . . . . . . . 8 | |
8 | 5, 6, 7 | syl2anc 409 | . . . . . . 7 |
9 | simp1r 1011 | . . . . . . . 8 | |
10 | mulclpr 7504 | . . . . . . . 8 | |
11 | 9, 6, 10 | syl2anc 409 | . . . . . . 7 |
12 | 4, 8, 11 | caovcld 5986 | . . . . . 6 |
13 | simp1l 1010 | . . . . . . . 8 | |
14 | simp3l 1014 | . . . . . . . 8 | |
15 | mulclpr 7504 | . . . . . . . 8 | |
16 | 13, 14, 15 | syl2anc 409 | . . . . . . 7 |
17 | simp2r 1013 | . . . . . . . 8 | |
18 | mulclpr 7504 | . . . . . . . 8 | |
19 | 17, 14, 18 | syl2anc 409 | . . . . . . 7 |
20 | 4, 16, 19 | caovcld 5986 | . . . . . 6 |
21 | 2, 12, 20 | caovcomd 5989 | . . . . 5 |
22 | addassprg 7511 | . . . . . . 7 | |
23 | 22 | adantl 275 | . . . . . 6 |
24 | 16, 11, 8, 2, 23, 19, 4 | caov411d 6018 | . . . . 5 |
25 | distrprg 7520 | . . . . . . . 8 | |
26 | 25 | adantl 275 | . . . . . . 7 |
27 | mulcomprg 7512 | . . . . . . . 8 | |
28 | 27 | adantl 275 | . . . . . . 7 |
29 | 26, 13, 17, 14, 4, 28 | caovdir2d 6009 | . . . . . 6 |
30 | 26, 5, 9, 6, 4, 28 | caovdir2d 6009 | . . . . . 6 |
31 | 29, 30 | oveq12d 5854 | . . . . 5 |
32 | 21, 24, 31 | 3eqtr4d 2207 | . . . 4 |
33 | mulclpr 7504 | . . . . . . 7 | |
34 | 13, 6, 33 | syl2anc 409 | . . . . . 6 |
35 | mulclpr 7504 | . . . . . . 7 | |
36 | 9, 14, 35 | syl2anc 409 | . . . . . 6 |
37 | mulclpr 7504 | . . . . . . 7 | |
38 | 5, 14, 37 | syl2anc 409 | . . . . . 6 |
39 | mulclpr 7504 | . . . . . . 7 | |
40 | 17, 6, 39 | syl2anc 409 | . . . . . 6 |
41 | 34, 36, 38, 2, 23, 40, 4 | caov411d 6018 | . . . . 5 |
42 | 26, 5, 9, 14, 4, 28 | caovdir2d 6009 | . . . . . 6 |
43 | 26, 13, 17, 6, 4, 28 | caovdir2d 6009 | . . . . . 6 |
44 | 42, 43 | oveq12d 5854 | . . . . 5 |
45 | 41, 44 | eqtr4d 2200 | . . . 4 |
46 | 32, 45 | breq12d 3989 | . . 3 |
47 | 29, 20 | eqeltrd 2241 | . . . . 5 |
48 | 30, 12 | eqeltrd 2241 | . . . . 5 |
49 | addclpr 7469 | . . . . . . 7 | |
50 | 5, 9, 49 | syl2anc 409 | . . . . . 6 |
51 | mulclpr 7504 | . . . . . 6 | |
52 | 50, 14, 51 | syl2anc 409 | . . . . 5 |
53 | addclpr 7469 | . . . . . . 7 | |
54 | 13, 17, 53 | syl2anc 409 | . . . . . 6 |
55 | mulclpr 7504 | . . . . . 6 | |
56 | 54, 6, 55 | syl2anc 409 | . . . . 5 |
57 | addextpr 7553 | . . . . 5 | |
58 | 47, 48, 52, 56, 57 | syl22anc 1228 | . . . 4 |
59 | mulcomprg 7512 | . . . . . . . . 9 | |
60 | 59 | 3adant2 1005 | . . . . . . . 8 |
61 | mulcomprg 7512 | . . . . . . . . 9 | |
62 | 61 | 3adant1 1004 | . . . . . . . 8 |
63 | 60, 62 | breq12d 3989 | . . . . . . 7 |
64 | ltmprr 7574 | . . . . . . 7 | |
65 | 63, 64 | sylbid 149 | . . . . . 6 |
66 | 54, 50, 14, 65 | syl3anc 1227 | . . . . 5 |
67 | mulcomprg 7512 | . . . . . . . 8 | |
68 | 50, 6, 67 | syl2anc 409 | . . . . . . 7 |
69 | mulcomprg 7512 | . . . . . . . 8 | |
70 | 54, 6, 69 | syl2anc 409 | . . . . . . 7 |
71 | 68, 70 | breq12d 3989 | . . . . . 6 |
72 | ltmprr 7574 | . . . . . . 7 | |
73 | 50, 54, 6, 72 | syl3anc 1227 | . . . . . 6 |
74 | 71, 73 | sylbid 149 | . . . . 5 |
75 | 66, 74 | orim12d 776 | . . . 4 |
76 | 58, 75 | syld 45 | . . 3 |
77 | 46, 76 | sylbid 149 | . 2 |
78 | addcomprg 7510 | . . . . 5 | |
79 | 5, 9, 78 | syl2anc 409 | . . . 4 |
80 | 79 | breq2d 3988 | . . 3 |
81 | addcomprg 7510 | . . . . 5 | |
82 | 13, 17, 81 | syl2anc 409 | . . . 4 |
83 | 82 | breq2d 3988 | . . 3 |
84 | 80, 83 | orbi12d 783 | . 2 |
85 | 77, 84 | sylibd 148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 w3a 967 wceq 1342 wcel 2135 class class class wbr 3976 (class class class)co 5836 cnp 7223 cpp 7225 cmp 7226 cltp 7227 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-i1p 7399 df-iplp 7400 df-imp 7401 df-iltp 7402 |
This theorem is referenced by: mulextsr1 7713 |
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