![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iswwlks | Structured version Visualization version GIF version |
Description: A word over the set of vertices representing a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.) |
Ref | Expression |
---|---|
wwlks.v | β’ π = (VtxβπΊ) |
wwlks.e | β’ πΈ = (EdgβπΊ) |
Ref | Expression |
---|---|
iswwlks | β’ (π β (WWalksβπΊ) β (π β β β§ π β Word π β§ βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β πΈ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 3001 | . . . 4 β’ (π€ = π β (π€ β β β π β β )) | |
2 | fveq2 6890 | . . . . . . 7 β’ (π€ = π β (β―βπ€) = (β―βπ)) | |
3 | 2 | oveq1d 7426 | . . . . . 6 β’ (π€ = π β ((β―βπ€) β 1) = ((β―βπ) β 1)) |
4 | 3 | oveq2d 7427 | . . . . 5 β’ (π€ = π β (0..^((β―βπ€) β 1)) = (0..^((β―βπ) β 1))) |
5 | fveq1 6889 | . . . . . . 7 β’ (π€ = π β (π€βπ) = (πβπ)) | |
6 | fveq1 6889 | . . . . . . 7 β’ (π€ = π β (π€β(π + 1)) = (πβ(π + 1))) | |
7 | 5, 6 | preq12d 4744 | . . . . . 6 β’ (π€ = π β {(π€βπ), (π€β(π + 1))} = {(πβπ), (πβ(π + 1))}) |
8 | 7 | eleq1d 2816 | . . . . 5 β’ (π€ = π β ({(π€βπ), (π€β(π + 1))} β πΈ β {(πβπ), (πβ(π + 1))} β πΈ)) |
9 | 4, 8 | raleqbidv 3340 | . . . 4 β’ (π€ = π β (βπ β (0..^((β―βπ€) β 1)){(π€βπ), (π€β(π + 1))} β πΈ β βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β πΈ)) |
10 | 1, 9 | anbi12d 629 | . . 3 β’ (π€ = π β ((π€ β β β§ βπ β (0..^((β―βπ€) β 1)){(π€βπ), (π€β(π + 1))} β πΈ) β (π β β β§ βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β πΈ))) |
11 | 10 | elrab 3682 | . 2 β’ (π β {π€ β Word π β£ (π€ β β β§ βπ β (0..^((β―βπ€) β 1)){(π€βπ), (π€β(π + 1))} β πΈ)} β (π β Word π β§ (π β β β§ βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β πΈ))) |
12 | wwlks.v | . . . 4 β’ π = (VtxβπΊ) | |
13 | wwlks.e | . . . 4 β’ πΈ = (EdgβπΊ) | |
14 | 12, 13 | wwlks 29356 | . . 3 β’ (WWalksβπΊ) = {π€ β Word π β£ (π€ β β β§ βπ β (0..^((β―βπ€) β 1)){(π€βπ), (π€β(π + 1))} β πΈ)} |
15 | 14 | eleq2i 2823 | . 2 β’ (π β (WWalksβπΊ) β π β {π€ β Word π β£ (π€ β β β§ βπ β (0..^((β―βπ€) β 1)){(π€βπ), (π€β(π + 1))} β πΈ)}) |
16 | 3anan12 1094 | . 2 β’ ((π β β β§ π β Word π β§ βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β πΈ) β (π β Word π β§ (π β β β§ βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β πΈ))) | |
17 | 11, 15, 16 | 3bitr4i 302 | 1 β’ (π β (WWalksβπΊ) β (π β β β§ π β Word π β§ βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β πΈ)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 β wne 2938 βwral 3059 {crab 3430 β c0 4321 {cpr 4629 βcfv 6542 (class class class)co 7411 0cc0 11112 1c1 11113 + caddc 11115 β cmin 11448 ..^cfzo 13631 β―chash 14294 Word cword 14468 Vtxcvtx 28523 Edgcedg 28574 WWalkscwwlks 29346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-wwlks 29351 |
This theorem is referenced by: iswwlksnx 29361 wwlkbp 29362 wwlknp 29364 wwlksn0s 29382 0enwwlksnge1 29385 wlkiswwlks1 29388 wlkiswwlks2 29396 wlkiswwlksupgr2 29398 wwlksm1edg 29402 wlknewwlksn 29408 wwlksnred 29413 wwlksnext 29414 wwlksnfi 29427 rusgrnumwwlkl1 29489 clwwlkel 29566 clwwlkf 29567 clwwlkwwlksb 29574 |
Copyright terms: Public domain | W3C validator |