![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 29091 | . . 3 β’ (π β β0 β (π WWalksN πΊ) = {π€ β (WWalksβπΊ) β£ (β―βπ€) = (π + 1)}) | |
2 | 1 | eleq2d 2820 | . 2 β’ (π β β0 β (π β (π WWalksN πΊ) β π β {π€ β (WWalksβπΊ) β£ (β―βπ€) = (π + 1)})) |
3 | fveqeq2 6901 | . . 3 β’ (π€ = π β ((β―βπ€) = (π + 1) β (β―βπ) = (π + 1))) | |
4 | 3 | elrab 3684 | . 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 287 | 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 397 = wceq 1542 β wcel 2107 {crab 3433 βcfv 6544 (class class class)co 7409 1c1 11111 + caddc 11113 β0cn0 12472 β―chash 14290 WWalkscwwlks 29079 WWalksN cwwlksn 29080 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-wwlksn 29085 |
This theorem is referenced by: wwlksnprcl 29093 iswwlksnx 29094 wwlknbp 29096 wwlknp 29097 wwlkswwlksn 29119 wlklnwwlkln1 29122 wlklnwwlkln2lem 29136 wlknewwlksn 29141 wwlksnred 29146 wwlksnext 29147 wwlksnextproplem3 29165 wspthsnonn0vne 29171 elwspths2spth 29221 rusgrnumwwlkl1 29222 clwwlkel 29299 clwwlkf 29300 clwwlknwwlksnb 29308 |
Copyright terms: Public domain | W3C validator |