Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mulextsr1lem | Unicode version |
Description: Lemma for mulextsr1 6865. (Contributed by Jim Kingdon, 17-Feb-2020.) |
Ref | Expression |
---|---|
mulextsr1lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcomprg 6676 | . . . . . . 7 | |
2 | 1 | adantl 262 | . . . . . 6 |
3 | addclpr 6635 | . . . . . . . 8 | |
4 | 3 | adantl 262 | . . . . . . 7 |
5 | simp2l 930 | . . . . . . . 8 | |
6 | simp3r 933 | . . . . . . . 8 | |
7 | mulclpr 6670 | . . . . . . . 8 | |
8 | 5, 6, 7 | syl2anc 391 | . . . . . . 7 |
9 | simp1r 929 | . . . . . . . 8 | |
10 | mulclpr 6670 | . . . . . . . 8 | |
11 | 9, 6, 10 | syl2anc 391 | . . . . . . 7 |
12 | 4, 8, 11 | caovcld 5654 | . . . . . 6 |
13 | simp1l 928 | . . . . . . . 8 | |
14 | simp3l 932 | . . . . . . . 8 | |
15 | mulclpr 6670 | . . . . . . . 8 | |
16 | 13, 14, 15 | syl2anc 391 | . . . . . . 7 |
17 | simp2r 931 | . . . . . . . 8 | |
18 | mulclpr 6670 | . . . . . . . 8 | |
19 | 17, 14, 18 | syl2anc 391 | . . . . . . 7 |
20 | 4, 16, 19 | caovcld 5654 | . . . . . 6 |
21 | 2, 12, 20 | caovcomd 5657 | . . . . 5 |
22 | addassprg 6677 | . . . . . . 7 | |
23 | 22 | adantl 262 | . . . . . 6 |
24 | 16, 11, 8, 2, 23, 19, 4 | caov411d 5686 | . . . . 5 |
25 | distrprg 6686 | . . . . . . . 8 | |
26 | 25 | adantl 262 | . . . . . . 7 |
27 | mulcomprg 6678 | . . . . . . . 8 | |
28 | 27 | adantl 262 | . . . . . . 7 |
29 | 26, 13, 17, 14, 4, 28 | caovdir2d 5677 | . . . . . 6 |
30 | 26, 5, 9, 6, 4, 28 | caovdir2d 5677 | . . . . . 6 |
31 | 29, 30 | oveq12d 5530 | . . . . 5 |
32 | 21, 24, 31 | 3eqtr4d 2082 | . . . 4 |
33 | mulclpr 6670 | . . . . . . 7 | |
34 | 13, 6, 33 | syl2anc 391 | . . . . . 6 |
35 | mulclpr 6670 | . . . . . . 7 | |
36 | 9, 14, 35 | syl2anc 391 | . . . . . 6 |
37 | mulclpr 6670 | . . . . . . 7 | |
38 | 5, 14, 37 | syl2anc 391 | . . . . . 6 |
39 | mulclpr 6670 | . . . . . . 7 | |
40 | 17, 6, 39 | syl2anc 391 | . . . . . 6 |
41 | 34, 36, 38, 2, 23, 40, 4 | caov411d 5686 | . . . . 5 |
42 | 26, 5, 9, 14, 4, 28 | caovdir2d 5677 | . . . . . 6 |
43 | 26, 13, 17, 6, 4, 28 | caovdir2d 5677 | . . . . . 6 |
44 | 42, 43 | oveq12d 5530 | . . . . 5 |
45 | 41, 44 | eqtr4d 2075 | . . . 4 |
46 | 32, 45 | breq12d 3777 | . . 3 |
47 | 29, 20 | eqeltrd 2114 | . . . . 5 |
48 | 30, 12 | eqeltrd 2114 | . . . . 5 |
49 | addclpr 6635 | . . . . . . 7 | |
50 | 5, 9, 49 | syl2anc 391 | . . . . . 6 |
51 | mulclpr 6670 | . . . . . 6 | |
52 | 50, 14, 51 | syl2anc 391 | . . . . 5 |
53 | addclpr 6635 | . . . . . . 7 | |
54 | 13, 17, 53 | syl2anc 391 | . . . . . 6 |
55 | mulclpr 6670 | . . . . . 6 | |
56 | 54, 6, 55 | syl2anc 391 | . . . . 5 |
57 | addextpr 6719 | . . . . 5 | |
58 | 47, 48, 52, 56, 57 | syl22anc 1136 | . . . 4 |
59 | mulcomprg 6678 | . . . . . . . . 9 | |
60 | 59 | 3adant2 923 | . . . . . . . 8 |
61 | mulcomprg 6678 | . . . . . . . . 9 | |
62 | 61 | 3adant1 922 | . . . . . . . 8 |
63 | 60, 62 | breq12d 3777 | . . . . . . 7 |
64 | ltmprr 6740 | . . . . . . 7 | |
65 | 63, 64 | sylbid 139 | . . . . . 6 |
66 | 54, 50, 14, 65 | syl3anc 1135 | . . . . 5 |
67 | mulcomprg 6678 | . . . . . . . 8 | |
68 | 50, 6, 67 | syl2anc 391 | . . . . . . 7 |
69 | mulcomprg 6678 | . . . . . . . 8 | |
70 | 54, 6, 69 | syl2anc 391 | . . . . . . 7 |
71 | 68, 70 | breq12d 3777 | . . . . . 6 |
72 | ltmprr 6740 | . . . . . . 7 | |
73 | 50, 54, 6, 72 | syl3anc 1135 | . . . . . 6 |
74 | 71, 73 | sylbid 139 | . . . . 5 |
75 | 66, 74 | orim12d 700 | . . . 4 |
76 | 58, 75 | syld 40 | . . 3 |
77 | 46, 76 | sylbid 139 | . 2 |
78 | addcomprg 6676 | . . . . 5 | |
79 | 5, 9, 78 | syl2anc 391 | . . . 4 |
80 | 79 | breq2d 3776 | . . 3 |
81 | addcomprg 6676 | . . . . 5 | |
82 | 13, 17, 81 | syl2anc 391 | . . . 4 |
83 | 82 | breq2d 3776 | . . 3 |
84 | 80, 83 | orbi12d 707 | . 2 |
85 | 77, 84 | sylibd 138 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 w3a 885 wceq 1243 wcel 1393 class class class wbr 3764 (class class class)co 5512 cnp 6389 cpp 6391 cmp 6392 cltp 6393 |
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-3or 886 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-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 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-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-imp 6567 df-iltp 6568 |
This theorem is referenced by: mulextsr1 6865 |
Copyright terms: Public domain | W3C validator |