Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlcone | Structured version Visualization version Unicode version |
Description: If two translations have different traces, the trace of their composition is also different. (Contributed by NM, 14-Jun-2013.) |
Ref | Expression |
---|---|
trlcone.b | |
trlcone.h | |
trlcone.t | |
trlcone.r |
Ref | Expression |
---|---|
trlcone |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3l 1116 | . . 3 | |
2 | simp11 1091 | . . . . . . . . 9 | |
3 | simp12l 1174 | . . . . . . . . . 10 | |
4 | trlcone.h | . . . . . . . . . . 11 | |
5 | trlcone.t | . . . . . . . . . . 11 | |
6 | 4, 5 | ltrncnv 35432 | . . . . . . . . . 10 |
7 | 2, 3, 6 | syl2anc 693 | . . . . . . . . 9 |
8 | simp12r 1175 | . . . . . . . . . 10 | |
9 | 4, 5 | ltrnco 36007 | . . . . . . . . . 10 |
10 | 2, 3, 8, 9 | syl3anc 1326 | . . . . . . . . 9 |
11 | eqid 2622 | . . . . . . . . . 10 | |
12 | eqid 2622 | . . . . . . . . . 10 | |
13 | trlcone.r | . . . . . . . . . 10 | |
14 | 11, 12, 4, 5, 13 | trlco 36015 | . . . . . . . . 9 |
15 | 2, 7, 10, 14 | syl3anc 1326 | . . . . . . . 8 |
16 | coass 5654 | . . . . . . . . . 10 | |
17 | trlcone.b | . . . . . . . . . . . . . . 15 | |
18 | 17, 4, 5 | ltrn1o 35410 | . . . . . . . . . . . . . 14 |
19 | 2, 3, 18 | syl2anc 693 | . . . . . . . . . . . . 13 |
20 | f1ococnv1 6165 | . . . . . . . . . . . . 13 | |
21 | 19, 20 | syl 17 | . . . . . . . . . . . 12 |
22 | 21 | coeq1d 5283 | . . . . . . . . . . 11 |
23 | 17, 4, 5 | ltrn1o 35410 | . . . . . . . . . . . . 13 |
24 | 2, 8, 23 | syl2anc 693 | . . . . . . . . . . . 12 |
25 | f1of 6137 | . . . . . . . . . . . 12 | |
26 | fcoi2 6079 | . . . . . . . . . . . 12 | |
27 | 24, 25, 26 | 3syl 18 | . . . . . . . . . . 11 |
28 | 22, 27 | eqtrd 2656 | . . . . . . . . . 10 |
29 | 16, 28 | syl5reqr 2671 | . . . . . . . . 9 |
30 | 29 | fveq2d 6195 | . . . . . . . 8 |
31 | simp11l 1172 | . . . . . . . . . 10 | |
32 | simp2 1062 | . . . . . . . . . 10 | |
33 | eqid 2622 | . . . . . . . . . . 11 | |
34 | 12, 33 | hlatjidm 34655 | . . . . . . . . . 10 |
35 | 31, 32, 34 | syl2anc 693 | . . . . . . . . 9 |
36 | 4, 5, 13 | trlcnv 35452 | . . . . . . . . . . . 12 |
37 | 2, 3, 36 | syl2anc 693 | . . . . . . . . . . 11 |
38 | 37 | eqcomd 2628 | . . . . . . . . . 10 |
39 | simp3 1063 | . . . . . . . . . 10 | |
40 | 38, 39 | oveq12d 6668 | . . . . . . . . 9 |
41 | 35, 40 | eqtr3d 2658 | . . . . . . . 8 |
42 | 15, 30, 41 | 3brtr4d 4685 | . . . . . . 7 |
43 | hlatl 34647 | . . . . . . . . 9 | |
44 | 31, 43 | syl 17 | . . . . . . . 8 |
45 | simp13r 1177 | . . . . . . . . 9 | |
46 | 17, 33, 4, 5, 13 | trlnidat 35460 | . . . . . . . . 9 |
47 | 2, 8, 45, 46 | syl3anc 1326 | . . . . . . . 8 |
48 | 11, 33 | atcmp 34598 | . . . . . . . 8 |
49 | 44, 47, 32, 48 | syl3anc 1326 | . . . . . . 7 |
50 | 42, 49 | mpbid 222 | . . . . . 6 |
51 | 50 | eqcomd 2628 | . . . . 5 |
52 | 51 | 3expia 1267 | . . . 4 |
53 | 52 | necon3d 2815 | . . 3 |
54 | 1, 53 | mpd 15 | . 2 |
55 | simpl3r 1117 | . . . . 5 | |
56 | simpl1 1064 | . . . . . . 7 | |
57 | simpl2r 1115 | . . . . . . 7 | |
58 | eqid 2622 | . . . . . . . 8 | |
59 | 17, 58, 4, 5, 13 | trlid0b 35465 | . . . . . . 7 |
60 | 56, 57, 59 | syl2anc 693 | . . . . . 6 |
61 | 60 | necon3bid 2838 | . . . . 5 |
62 | 55, 61 | mpbid 222 | . . . 4 |
63 | 62 | necomd 2849 | . . 3 |
64 | simpr 477 | . . 3 | |
65 | simpl2l 1114 | . . . . . . . 8 | |
66 | 17, 58, 4, 5, 13 | trlid0b 35465 | . . . . . . . 8 |
67 | 56, 65, 66 | syl2anc 693 | . . . . . . 7 |
68 | 64, 67 | mpbird 247 | . . . . . 6 |
69 | 68 | coeq1d 5283 | . . . . 5 |
70 | 56, 57, 23 | syl2anc 693 | . . . . . 6 |
71 | 70, 25, 26 | 3syl 18 | . . . . 5 |
72 | 69, 71 | eqtrd 2656 | . . . 4 |
73 | 72 | fveq2d 6195 | . . 3 |
74 | 63, 64, 73 | 3netr4d 2871 | . 2 |
75 | simp1 1061 | . . 3 | |
76 | simp2l 1087 | . . 3 | |
77 | 58, 33, 4, 5, 13 | trlator0 35458 | . . 3 |
78 | 75, 76, 77 | syl2anc 693 | . 2 |
79 | 54, 74, 78 | mpjaodan 827 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 class class class wbr 4653 cid 5023 ccnv 5113 cres 5116 ccom 5118 wf 5884 wf1o 5887 cfv 5888 (class class class)co 6650 cbs 15857 cple 15948 cjn 16944 cp0 17037 catm 34550 cal 34551 chlt 34637 clh 35270 cltrn 35387 ctrl 35445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-undef 7399 df-map 7859 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 |
This theorem is referenced by: trljco 36028 cdlemh2 36104 cdlemh 36105 cdlemk3 36121 cdlemk12 36138 cdlemk12u 36160 cdlemkfid1N 36209 cdlemk54 36246 |
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