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Mirrors > Home > ILE Home > Th. List > mulextsr1lem | Unicode version |
Description: Lemma for mulextsr1 7589. (Contributed by Jim Kingdon, 17-Feb-2020.) |
Ref | Expression |
---|---|
mulextsr1lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcomprg 7386 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | addclpr 7345 | . . . . . . . 8 | |
4 | 3 | adantl 275 | . . . . . . 7 |
5 | simp2l 1007 | . . . . . . . 8 | |
6 | simp3r 1010 | . . . . . . . 8 | |
7 | mulclpr 7380 | . . . . . . . 8 | |
8 | 5, 6, 7 | syl2anc 408 | . . . . . . 7 |
9 | simp1r 1006 | . . . . . . . 8 | |
10 | mulclpr 7380 | . . . . . . . 8 | |
11 | 9, 6, 10 | syl2anc 408 | . . . . . . 7 |
12 | 4, 8, 11 | caovcld 5924 | . . . . . 6 |
13 | simp1l 1005 | . . . . . . . 8 | |
14 | simp3l 1009 | . . . . . . . 8 | |
15 | mulclpr 7380 | . . . . . . . 8 | |
16 | 13, 14, 15 | syl2anc 408 | . . . . . . 7 |
17 | simp2r 1008 | . . . . . . . 8 | |
18 | mulclpr 7380 | . . . . . . . 8 | |
19 | 17, 14, 18 | syl2anc 408 | . . . . . . 7 |
20 | 4, 16, 19 | caovcld 5924 | . . . . . 6 |
21 | 2, 12, 20 | caovcomd 5927 | . . . . 5 |
22 | addassprg 7387 | . . . . . . 7 | |
23 | 22 | adantl 275 | . . . . . 6 |
24 | 16, 11, 8, 2, 23, 19, 4 | caov411d 5956 | . . . . 5 |
25 | distrprg 7396 | . . . . . . . 8 | |
26 | 25 | adantl 275 | . . . . . . 7 |
27 | mulcomprg 7388 | . . . . . . . 8 | |
28 | 27 | adantl 275 | . . . . . . 7 |
29 | 26, 13, 17, 14, 4, 28 | caovdir2d 5947 | . . . . . 6 |
30 | 26, 5, 9, 6, 4, 28 | caovdir2d 5947 | . . . . . 6 |
31 | 29, 30 | oveq12d 5792 | . . . . 5 |
32 | 21, 24, 31 | 3eqtr4d 2182 | . . . 4 |
33 | mulclpr 7380 | . . . . . . 7 | |
34 | 13, 6, 33 | syl2anc 408 | . . . . . 6 |
35 | mulclpr 7380 | . . . . . . 7 | |
36 | 9, 14, 35 | syl2anc 408 | . . . . . 6 |
37 | mulclpr 7380 | . . . . . . 7 | |
38 | 5, 14, 37 | syl2anc 408 | . . . . . 6 |
39 | mulclpr 7380 | . . . . . . 7 | |
40 | 17, 6, 39 | syl2anc 408 | . . . . . 6 |
41 | 34, 36, 38, 2, 23, 40, 4 | caov411d 5956 | . . . . 5 |
42 | 26, 5, 9, 14, 4, 28 | caovdir2d 5947 | . . . . . 6 |
43 | 26, 13, 17, 6, 4, 28 | caovdir2d 5947 | . . . . . 6 |
44 | 42, 43 | oveq12d 5792 | . . . . 5 |
45 | 41, 44 | eqtr4d 2175 | . . . 4 |
46 | 32, 45 | breq12d 3942 | . . 3 |
47 | 29, 20 | eqeltrd 2216 | . . . . 5 |
48 | 30, 12 | eqeltrd 2216 | . . . . 5 |
49 | addclpr 7345 | . . . . . . 7 | |
50 | 5, 9, 49 | syl2anc 408 | . . . . . 6 |
51 | mulclpr 7380 | . . . . . 6 | |
52 | 50, 14, 51 | syl2anc 408 | . . . . 5 |
53 | addclpr 7345 | . . . . . . 7 | |
54 | 13, 17, 53 | syl2anc 408 | . . . . . 6 |
55 | mulclpr 7380 | . . . . . 6 | |
56 | 54, 6, 55 | syl2anc 408 | . . . . 5 |
57 | addextpr 7429 | . . . . 5 | |
58 | 47, 48, 52, 56, 57 | syl22anc 1217 | . . . 4 |
59 | mulcomprg 7388 | . . . . . . . . 9 | |
60 | 59 | 3adant2 1000 | . . . . . . . 8 |
61 | mulcomprg 7388 | . . . . . . . . 9 | |
62 | 61 | 3adant1 999 | . . . . . . . 8 |
63 | 60, 62 | breq12d 3942 | . . . . . . 7 |
64 | ltmprr 7450 | . . . . . . 7 | |
65 | 63, 64 | sylbid 149 | . . . . . 6 |
66 | 54, 50, 14, 65 | syl3anc 1216 | . . . . 5 |
67 | mulcomprg 7388 | . . . . . . . 8 | |
68 | 50, 6, 67 | syl2anc 408 | . . . . . . 7 |
69 | mulcomprg 7388 | . . . . . . . 8 | |
70 | 54, 6, 69 | syl2anc 408 | . . . . . . 7 |
71 | 68, 70 | breq12d 3942 | . . . . . 6 |
72 | ltmprr 7450 | . . . . . . 7 | |
73 | 50, 54, 6, 72 | syl3anc 1216 | . . . . . 6 |
74 | 71, 73 | sylbid 149 | . . . . 5 |
75 | 66, 74 | orim12d 775 | . . . 4 |
76 | 58, 75 | syld 45 | . . 3 |
77 | 46, 76 | sylbid 149 | . 2 |
78 | addcomprg 7386 | . . . . 5 | |
79 | 5, 9, 78 | syl2anc 408 | . . . 4 |
80 | 79 | breq2d 3941 | . . 3 |
81 | addcomprg 7386 | . . . . 5 | |
82 | 13, 17, 81 | syl2anc 408 | . . . 4 |
83 | 82 | breq2d 3941 | . . 3 |
84 | 80, 83 | orbi12d 782 | . 2 |
85 | 77, 84 | sylibd 148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cnp 7099 cpp 7101 cmp 7102 cltp 7103 |
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-3or 963 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-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 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-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-i1p 7275 df-iplp 7276 df-imp 7277 df-iltp 7278 |
This theorem is referenced by: mulextsr1 7589 |
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