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Mirrors > Home > MPE Home > Th. List > clwwlksnscsh | Structured version Visualization version Unicode version |
Description: The set of cyclical shifts of a word representing a closed walk is the set of closed walks represented by cyclical shifts of a word. (Contributed by Alexander van der Vekens, 15-Jun-2018.) (Revised by AV, 30-Apr-2021.) |
Ref | Expression |
---|---|
clwwlksnscsh | ClWWalksN ClWWalksN cyclShift Word Vtx cyclShift |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2626 | . . . 4 cyclShift cyclShift | |
2 | 1 | rexbidv 3052 | . . 3 cyclShift cyclShift |
3 | 2 | cbvrabv 3199 | . 2 ClWWalksN cyclShift ClWWalksN cyclShift |
4 | eqid 2622 | . . . . . . . 8 Vtx Vtx | |
5 | 4 | clwwlksnwrd 26886 | . . . . . . 7 ClWWalksN Word Vtx |
6 | 5 | ad2antrl 764 | . . . . . 6 ClWWalksN ClWWalksN cyclShift Word Vtx |
7 | simprr 796 | . . . . . 6 ClWWalksN ClWWalksN cyclShift cyclShift | |
8 | 6, 7 | jca 554 | . . . . 5 ClWWalksN ClWWalksN cyclShift Word Vtx cyclShift |
9 | simprr 796 | . . . . . . . . . . . . 13 Word Vtx ClWWalksN ClWWalksN | |
10 | simpllr 799 | . . . . . . . . . . . . 13 Word Vtx ClWWalksN cyclShift | |
11 | clwwnisshclwwsn 26930 | . . . . . . . . . . . . 13 ClWWalksN cyclShift ClWWalksN | |
12 | 9, 10, 11 | syl2an2r 876 | . . . . . . . . . . . 12 Word Vtx ClWWalksN cyclShift cyclShift ClWWalksN |
13 | eleq1 2689 | . . . . . . . . . . . . 13 cyclShift ClWWalksN cyclShift ClWWalksN | |
14 | 13 | adantl 482 | . . . . . . . . . . . 12 Word Vtx ClWWalksN cyclShift ClWWalksN cyclShift ClWWalksN |
15 | 12, 14 | mpbird 247 | . . . . . . . . . . 11 Word Vtx ClWWalksN cyclShift ClWWalksN |
16 | 15 | exp31 630 | . . . . . . . . . 10 Word Vtx ClWWalksN cyclShift ClWWalksN |
17 | 16 | com23 86 | . . . . . . . . 9 Word Vtx cyclShift ClWWalksN ClWWalksN |
18 | 17 | rexlimdva 3031 | . . . . . . . 8 Word Vtx cyclShift ClWWalksN ClWWalksN |
19 | 18 | imp 445 | . . . . . . 7 Word Vtx cyclShift ClWWalksN ClWWalksN |
20 | 19 | impcom 446 | . . . . . 6 ClWWalksN Word Vtx cyclShift ClWWalksN |
21 | simprr 796 | . . . . . 6 ClWWalksN Word Vtx cyclShift cyclShift | |
22 | 20, 21 | jca 554 | . . . . 5 ClWWalksN Word Vtx cyclShift ClWWalksN cyclShift |
23 | 8, 22 | impbida 877 | . . . 4 ClWWalksN ClWWalksN cyclShift Word Vtx cyclShift |
24 | eqeq1 2626 | . . . . . 6 cyclShift cyclShift | |
25 | 24 | rexbidv 3052 | . . . . 5 cyclShift cyclShift |
26 | 25 | elrab 3363 | . . . 4 ClWWalksN cyclShift ClWWalksN cyclShift |
27 | eqeq1 2626 | . . . . . 6 cyclShift cyclShift | |
28 | 27 | rexbidv 3052 | . . . . 5 cyclShift cyclShift |
29 | 28 | elrab 3363 | . . . 4 Word Vtx cyclShift Word Vtx cyclShift |
30 | 23, 26, 29 | 3bitr4g 303 | . . 3 ClWWalksN ClWWalksN cyclShift Word Vtx cyclShift |
31 | 30 | eqrdv 2620 | . 2 ClWWalksN ClWWalksN cyclShift Word Vtx cyclShift |
32 | 3, 31 | syl5eq 2668 | 1 ClWWalksN ClWWalksN cyclShift Word Vtx cyclShift |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wrex 2913 crab 2916 cfv 5888 (class class class)co 6650 cc0 9936 cn0 11292 cfz 12326 Word cword 13291 cyclShift ccsh 13534 Vtxcvtx 25874 ClWWalksN cclwwlksn 26876 |
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-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-substr 13303 df-csh 13535 df-clwwlks 26877 df-clwwlksn 26878 |
This theorem is referenced by: hashecclwwlksn1 26954 umgrhashecclwwlk 26955 |
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