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Mirrors > Home > MPE Home > Th. List > iswwlksn | Structured version Visualization version GIF version |
Description: A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.) |
Ref | Expression |
---|---|
iswwlksn | β’ (π β β0 β (π β (π WWalksN πΊ) β (π β (WWalksβπΊ) β§ (β―βπ) = (π + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlksn 29129 | . . 3 β’ (π β β0 β (π WWalksN πΊ) = {π€ β (WWalksβπΊ) β£ (β―βπ€) = (π + 1)}) | |
2 | 1 | eleq2d 2819 | . 2 β’ (π β β0 β (π β (π WWalksN πΊ) β π β {π€ β (WWalksβπΊ) β£ (β―βπ€) = (π + 1)})) |
3 | fveqeq2 6900 | . . 3 β’ (π€ = π β ((β―βπ€) = (π + 1) β (β―βπ) = (π + 1))) | |
4 | 3 | elrab 3683 | . 2 β’ (π β {π€ β (WWalksβπΊ) β£ (β―βπ€) = (π + 1)} β (π β (WWalksβπΊ) β§ (β―βπ) = (π + 1))) |
5 | 2, 4 | bitrdi 286 | 1 β’ (π β β0 β (π β (π WWalksN πΊ) β (π β (WWalksβπΊ) β§ (β―βπ) = (π + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 {crab 3432 βcfv 6543 (class class class)co 7411 1c1 11113 + caddc 11115 β0cn0 12474 β―chash 14292 WWalkscwwlks 29117 WWalksN cwwlksn 29118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-wwlksn 29123 |
This theorem is referenced by: wwlksnprcl 29131 iswwlksnx 29132 wwlknbp 29134 wwlknp 29135 wwlkswwlksn 29157 wlklnwwlkln1 29160 wlklnwwlkln2lem 29174 wlknewwlksn 29179 wwlksnred 29184 wwlksnext 29185 wwlksnextproplem3 29203 wspthsnonn0vne 29209 elwspths2spth 29259 rusgrnumwwlkl1 29260 clwwlkel 29337 clwwlkf 29338 clwwlknwwlksnb 29346 |
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