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Mirrors > Home > MPE Home > Th. List > qerclwwlknfi | Structured version Visualization version GIF version |
Description: The quotient set of the set of closed walks (defined as words) with a fixed length according to the equivalence relation ⌠is finite. (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
Ref | Expression |
---|---|
erclwwlkn.w | ⢠ð = (ð ClWWalksN ðº) |
erclwwlkn.r | ⢠⌠= {âšð¡, ð¢â© ⣠(ð¡ â ð â§ ð¢ â ð â§ âð â (0...ð)ð¡ = (ð¢ cyclShift ð))} |
Ref | Expression |
---|---|
qerclwwlknfi | ⢠((Vtxâðº) â Fin â (ð / ⌠) â Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erclwwlkn.w | . . . 4 ⢠ð = (ð ClWWalksN ðº) | |
2 | clwwlknfi 29894 | . . . 4 ⢠((Vtxâðº) â Fin â (ð ClWWalksN ðº) â Fin) | |
3 | 1, 2 | eqeltrid 2829 | . . 3 ⢠((Vtxâðº) â Fin â ð â Fin) |
4 | pwfi 9196 | . . 3 ⢠(ð â Fin â ð« ð â Fin) | |
5 | 3, 4 | sylib 217 | . 2 ⢠((Vtxâðº) â Fin â ð« ð â Fin) |
6 | erclwwlkn.r | . . . . 5 ⢠⌠= {âšð¡, ð¢â© ⣠(ð¡ â ð â§ ð¢ â ð â§ âð â (0...ð)ð¡ = (ð¢ cyclShift ð))} | |
7 | 1, 6 | erclwwlkn 29921 | . . . 4 ⢠⌠Er ð |
8 | 7 | a1i 11 | . . 3 ⢠((Vtxâðº) â Fin â ⌠Er ð) |
9 | 8 | qsss 8790 | . 2 ⢠((Vtxâðº) â Fin â (ð / ⌠) â ð« ð) |
10 | 5, 9 | ssfid 9285 | 1 ⢠((Vtxâðº) â Fin â (ð / ⌠) â Fin) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ w3a 1084 = wceq 1533 â wcel 2098 âwrex 3060 ð« cpw 4599 {copab 5206 âcfv 6543 (class class class)co 7413 Er wer 8715 / cqs 8717 Fincfn 8957 0cc0 11133 ...cfz 13511 cyclShift ccsh 14765 Vtxcvtx 28848 ClWWalksN cclwwlkn 29873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-ec 8720 df-qs 8724 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-hash 14317 df-word 14492 df-concat 14548 df-substr 14618 df-pfx 14648 df-csh 14766 df-clwwlk 29831 df-clwwlkn 29874 |
This theorem is referenced by: fusgrhashclwwlkn 29928 clwwlkndivn 29929 |
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