Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > prmuloclemcalc | Unicode version |
Description: Calculations for prmuloc 7374. (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 5790 | . . . . . 6 |
3 | prmuloclemcalc.ru | . . . . . . . . 9 | |
4 | ltrelnq 7173 | . . . . . . . . . 10 | |
5 | 4 | brel 4591 | . . . . . . . . 9 |
6 | 3, 5 | syl 14 | . . . . . . . 8 |
7 | 6 | simprd 113 | . . . . . . 7 |
8 | prmuloclemcalc.a | . . . . . . 7 | |
9 | prmuloclemcalc.x | . . . . . . 7 | |
10 | distrnqg 7195 | . . . . . . 7 | |
11 | 7, 8, 9, 10 | syl3anc 1216 | . . . . . 6 |
12 | 2, 11 | eqtr3d 2174 | . . . . 5 |
13 | prmuloclemcalc.b | . . . . . . 7 | |
14 | mulcomnqg 7191 | . . . . . . 7 | |
15 | 13, 7, 14 | syl2anc 408 | . . . . . 6 |
16 | prmuloclemcalc.udp | . . . . . . . . . 10 | |
17 | ltmnqi 7211 | . . . . . . . . . 10 | |
18 | 16, 13, 17 | syl2anc 408 | . . . . . . . . 9 |
19 | prmuloclemcalc.d | . . . . . . . . . 10 | |
20 | prmuloclemcalc.p | . . . . . . . . . 10 | |
21 | distrnqg 7195 | . . . . . . . . . 10 | |
22 | 13, 19, 20, 21 | syl3anc 1216 | . . . . . . . . 9 |
23 | 18, 22 | breqtrd 3954 | . . . . . . . 8 |
24 | mulcomnqg 7191 | . . . . . . . . . . 11 | |
25 | 20, 13, 24 | syl2anc 408 | . . . . . . . . . 10 |
26 | prmuloclemcalc.pbrx | . . . . . . . . . 10 | |
27 | 25, 26 | eqbrtrrd 3952 | . . . . . . . . 9 |
28 | mulclnq 7184 | . . . . . . . . . 10 | |
29 | 13, 19, 28 | syl2anc 408 | . . . . . . . . 9 |
30 | ltanqi 7210 | . . . . . . . . 9 | |
31 | 27, 29, 30 | syl2anc 408 | . . . . . . . 8 |
32 | ltsonq 7206 | . . . . . . . . 9 | |
33 | 32, 4 | sotri 4934 | . . . . . . . 8 |
34 | 23, 31, 33 | syl2anc 408 | . . . . . . 7 |
35 | ltmnqi 7211 | . . . . . . . . . 10 | |
36 | 3, 9, 35 | syl2anc 408 | . . . . . . . . 9 |
37 | 6 | simpld 111 | . . . . . . . . . 10 |
38 | mulcomnqg 7191 | . . . . . . . . . 10 | |
39 | 9, 37, 38 | syl2anc 408 | . . . . . . . . 9 |
40 | mulcomnqg 7191 | . . . . . . . . . 10 | |
41 | 9, 7, 40 | syl2anc 408 | . . . . . . . . 9 |
42 | 36, 39, 41 | 3brtr3d 3959 | . . . . . . . 8 |
43 | ltanqi 7210 | . . . . . . . 8 | |
44 | 42, 29, 43 | syl2anc 408 | . . . . . . 7 |
45 | 32, 4 | sotri 4934 | . . . . . . 7 |
46 | 34, 44, 45 | syl2anc 408 | . . . . . 6 |
47 | 15, 46 | eqbrtrrd 3952 | . . . . 5 |
48 | 12, 47 | eqbrtrrd 3952 | . . . 4 |
49 | mulclnq 7184 | . . . . . 6 | |
50 | 7, 8, 49 | syl2anc 408 | . . . . 5 |
51 | mulclnq 7184 | . . . . . 6 | |
52 | 7, 9, 51 | syl2anc 408 | . . . . 5 |
53 | addcomnqg 7189 | . . . . 5 | |
54 | 50, 52, 53 | syl2anc 408 | . . . 4 |
55 | addcomnqg 7189 | . . . . 5 | |
56 | 29, 52, 55 | syl2anc 408 | . . . 4 |
57 | 48, 54, 56 | 3brtr3d 3959 | . . 3 |
58 | ltanqg 7208 | . . . 4 | |
59 | 50, 29, 52, 58 | syl3anc 1216 | . . 3 |
60 | 57, 59 | mpbird 166 | . 2 |
61 | mulcomnqg 7191 | . . 3 | |
62 | 13, 19, 61 | syl2anc 408 | . 2 |
63 | 60, 62 | breqtrd 3954 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cnq 7088 cplq 7090 cmq 7091 cltq 7093 |
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-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-ltnqqs 7161 |
This theorem is referenced by: prmuloc 7374 |
Copyright terms: Public domain | W3C validator |