Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > prmuloclemcalc | Unicode version |
Description: Calculations for prmuloc 6756. (Contributed by Jim Kingdon, 9-Dec-2019.) |
Ref | Expression |
---|---|
prmuloclemcalc.ru | |
prmuloclemcalc.udp | |
prmuloclemcalc.axb | |
prmuloclemcalc.pbrx | |
prmuloclemcalc.a | |
prmuloclemcalc.b | |
prmuloclemcalc.d | |
prmuloclemcalc.p | |
prmuloclemcalc.x |
Ref | Expression |
---|---|
prmuloclemcalc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmuloclemcalc.axb | . . . . . . 7 | |
2 | 1 | oveq2d 5548 | . . . . . 6 |
3 | prmuloclemcalc.ru | . . . . . . . . 9 | |
4 | ltrelnq 6555 | . . . . . . . . . 10 | |
5 | 4 | brel 4410 | . . . . . . . . 9 |
6 | 3, 5 | syl 14 | . . . . . . . 8 |
7 | 6 | simprd 112 | . . . . . . 7 |
8 | prmuloclemcalc.a | . . . . . . 7 | |
9 | prmuloclemcalc.x | . . . . . . 7 | |
10 | distrnqg 6577 | . . . . . . 7 | |
11 | 7, 8, 9, 10 | syl3anc 1169 | . . . . . 6 |
12 | 2, 11 | eqtr3d 2115 | . . . . 5 |
13 | prmuloclemcalc.b | . . . . . . 7 | |
14 | mulcomnqg 6573 | . . . . . . 7 | |
15 | 13, 7, 14 | syl2anc 403 | . . . . . 6 |
16 | prmuloclemcalc.udp | . . . . . . . . . 10 | |
17 | ltmnqi 6593 | . . . . . . . . . 10 | |
18 | 16, 13, 17 | syl2anc 403 | . . . . . . . . 9 |
19 | prmuloclemcalc.d | . . . . . . . . . 10 | |
20 | prmuloclemcalc.p | . . . . . . . . . 10 | |
21 | distrnqg 6577 | . . . . . . . . . 10 | |
22 | 13, 19, 20, 21 | syl3anc 1169 | . . . . . . . . 9 |
23 | 18, 22 | breqtrd 3809 | . . . . . . . 8 |
24 | mulcomnqg 6573 | . . . . . . . . . . 11 | |
25 | 20, 13, 24 | syl2anc 403 | . . . . . . . . . 10 |
26 | prmuloclemcalc.pbrx | . . . . . . . . . 10 | |
27 | 25, 26 | eqbrtrrd 3807 | . . . . . . . . 9 |
28 | mulclnq 6566 | . . . . . . . . . 10 | |
29 | 13, 19, 28 | syl2anc 403 | . . . . . . . . 9 |
30 | ltanqi 6592 | . . . . . . . . 9 | |
31 | 27, 29, 30 | syl2anc 403 | . . . . . . . 8 |
32 | ltsonq 6588 | . . . . . . . . 9 | |
33 | 32, 4 | sotri 4740 | . . . . . . . 8 |
34 | 23, 31, 33 | syl2anc 403 | . . . . . . 7 |
35 | ltmnqi 6593 | . . . . . . . . . 10 | |
36 | 3, 9, 35 | syl2anc 403 | . . . . . . . . 9 |
37 | 6 | simpld 110 | . . . . . . . . . 10 |
38 | mulcomnqg 6573 | . . . . . . . . . 10 | |
39 | 9, 37, 38 | syl2anc 403 | . . . . . . . . 9 |
40 | mulcomnqg 6573 | . . . . . . . . . 10 | |
41 | 9, 7, 40 | syl2anc 403 | . . . . . . . . 9 |
42 | 36, 39, 41 | 3brtr3d 3814 | . . . . . . . 8 |
43 | ltanqi 6592 | . . . . . . . 8 | |
44 | 42, 29, 43 | syl2anc 403 | . . . . . . 7 |
45 | 32, 4 | sotri 4740 | . . . . . . 7 |
46 | 34, 44, 45 | syl2anc 403 | . . . . . 6 |
47 | 15, 46 | eqbrtrrd 3807 | . . . . 5 |
48 | 12, 47 | eqbrtrrd 3807 | . . . 4 |
49 | mulclnq 6566 | . . . . . 6 | |
50 | 7, 8, 49 | syl2anc 403 | . . . . 5 |
51 | mulclnq 6566 | . . . . . 6 | |
52 | 7, 9, 51 | syl2anc 403 | . . . . 5 |
53 | addcomnqg 6571 | . . . . 5 | |
54 | 50, 52, 53 | syl2anc 403 | . . . 4 |
55 | addcomnqg 6571 | . . . . 5 | |
56 | 29, 52, 55 | syl2anc 403 | . . . 4 |
57 | 48, 54, 56 | 3brtr3d 3814 | . . 3 |
58 | ltanqg 6590 | . . . 4 | |
59 | 50, 29, 52, 58 | syl3anc 1169 | . . 3 |
60 | 57, 59 | mpbird 165 | . 2 |
61 | mulcomnqg 6573 | . . 3 | |
62 | 13, 19, 61 | syl2anc 403 | . 2 |
63 | 60, 62 | breqtrd 3809 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 class class class wbr 3785 (class class class)co 5532 cnq 6470 cplq 6472 cmq 6473 cltq 6475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-ltnqqs 6543 |
This theorem is referenced by: prmuloc 6756 |
Copyright terms: Public domain | W3C validator |