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Mirrors > Home > MPE Home > Th. List > clwwnisshclwwsn | Structured version Visualization version GIF version |
Description: Cyclically shifting a closed walk as word of fixed length results in a closed walk as word of the same length (in an undirected graph). (Contributed by Alexander van der Vekens, 10-Jun-2018.) (Revised by AV, 29-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) |
Ref | Expression |
---|---|
clwwnisshclwwsn | β’ ((π β (π ClWWalksN πΊ) β§ π β (0...π)) β (π cyclShift π) β (π ClWWalksN πΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlkclwwlkn 29814 | . . 3 β’ (π β (π ClWWalksN πΊ) β π β (ClWWalksβπΊ)) | |
2 | clwwlknlen 29816 | . . . . . . 7 β’ (π β (π ClWWalksN πΊ) β (β―βπ) = π) | |
3 | 2 | eqcomd 2733 | . . . . . 6 β’ (π β (π ClWWalksN πΊ) β π = (β―βπ)) |
4 | 3 | oveq2d 7430 | . . . . 5 β’ (π β (π ClWWalksN πΊ) β (0...π) = (0...(β―βπ))) |
5 | 4 | eleq2d 2814 | . . . 4 β’ (π β (π ClWWalksN πΊ) β (π β (0...π) β π β (0...(β―βπ)))) |
6 | 5 | biimpa 476 | . . 3 β’ ((π β (π ClWWalksN πΊ) β§ π β (0...π)) β π β (0...(β―βπ))) |
7 | clwwisshclwwsn 29800 | . . 3 β’ ((π β (ClWWalksβπΊ) β§ π β (0...(β―βπ))) β (π cyclShift π) β (ClWWalksβπΊ)) | |
8 | 1, 6, 7 | syl2an2r 684 | . 2 β’ ((π β (π ClWWalksN πΊ) β§ π β (0...π)) β (π cyclShift π) β (ClWWalksβπΊ)) |
9 | eqid 2727 | . . . . 5 β’ (VtxβπΊ) = (VtxβπΊ) | |
10 | 9 | clwwlknwrd 29818 | . . . 4 β’ (π β (π ClWWalksN πΊ) β π β Word (VtxβπΊ)) |
11 | elfzelz 13519 | . . . 4 β’ (π β (0...π) β π β β€) | |
12 | cshwlen 14767 | . . . 4 β’ ((π β Word (VtxβπΊ) β§ π β β€) β (β―β(π cyclShift π)) = (β―βπ)) | |
13 | 10, 11, 12 | syl2an 595 | . . 3 β’ ((π β (π ClWWalksN πΊ) β§ π β (0...π)) β (β―β(π cyclShift π)) = (β―βπ)) |
14 | 2 | adantr 480 | . . 3 β’ ((π β (π ClWWalksN πΊ) β§ π β (0...π)) β (β―βπ) = π) |
15 | 13, 14 | eqtrd 2767 | . 2 β’ ((π β (π ClWWalksN πΊ) β§ π β (0...π)) β (β―β(π cyclShift π)) = π) |
16 | isclwwlkn 29811 | . 2 β’ ((π cyclShift π) β (π ClWWalksN πΊ) β ((π cyclShift π) β (ClWWalksβπΊ) β§ (β―β(π cyclShift π)) = π)) | |
17 | 8, 15, 16 | sylanbrc 582 | 1 β’ ((π β (π ClWWalksN πΊ) β§ π β (0...π)) β (π cyclShift π) β (π ClWWalksN πΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 0cc0 11124 β€cz 12574 ...cfz 13502 β―chash 14307 Word cword 14482 cyclShift ccsh 14756 Vtxcvtx 28783 ClWWalkscclwwlk 29765 ClWWalksN cclwwlkn 29808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-ico 13348 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-hash 14308 df-word 14483 df-lsw 14531 df-concat 14539 df-substr 14609 df-pfx 14639 df-csh 14757 df-clwwlk 29766 df-clwwlkn 29809 |
This theorem is referenced by: clwwlknscsh 29846 |
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