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Mirrors > Home > MPE Home > Th. List > extwwlkfabel | Structured version Visualization version GIF version |
Description: Characterization of an element of the set (ππΆπ), i.e., a double loop of length π on vertex π with a construction from the set πΉ of closed walks on π with length smaller by 2 than the fixed length by appending a neighbor of the last vertex and afterwards the last vertex (which is the first vertex) itself ("walking forth and back" from the last vertex). (Contributed by AV, 22-Feb-2022.) (Revised by AV, 31-Oct-2022.) |
Ref | Expression |
---|---|
extwwlkfab.v | β’ π = (VtxβπΊ) |
extwwlkfab.c | β’ πΆ = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£}) |
extwwlkfab.f | β’ πΉ = (π(ClWWalksNOnβπΊ)(π β 2)) |
Ref | Expression |
---|---|
extwwlkfabel | β’ ((πΊ β USGraph β§ π β π β§ π β (β€β₯β3)) β (π β (ππΆπ) β (π β (π ClWWalksN πΊ) β§ ((π prefix (π β 2)) β πΉ β§ (πβ(π β 1)) β (πΊ NeighbVtx π) β§ (πβ(π β 2)) = π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | extwwlkfab.v | . . . 4 β’ π = (VtxβπΊ) | |
2 | extwwlkfab.c | . . . 4 β’ πΆ = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£}) | |
3 | extwwlkfab.f | . . . 4 β’ πΉ = (π(ClWWalksNOnβπΊ)(π β 2)) | |
4 | 1, 2, 3 | extwwlkfab 30201 | . . 3 β’ ((πΊ β USGraph β§ π β π β§ π β (β€β₯β3)) β (ππΆπ) = {π€ β (π ClWWalksN πΊ) β£ ((π€ prefix (π β 2)) β πΉ β§ (π€β(π β 1)) β (πΊ NeighbVtx π) β§ (π€β(π β 2)) = π)}) |
5 | 4 | eleq2d 2811 | . 2 β’ ((πΊ β USGraph β§ π β π β§ π β (β€β₯β3)) β (π β (ππΆπ) β π β {π€ β (π ClWWalksN πΊ) β£ ((π€ prefix (π β 2)) β πΉ β§ (π€β(π β 1)) β (πΊ NeighbVtx π) β§ (π€β(π β 2)) = π)})) |
6 | oveq1 7420 | . . . . 5 β’ (π€ = π β (π€ prefix (π β 2)) = (π prefix (π β 2))) | |
7 | 6 | eleq1d 2810 | . . . 4 β’ (π€ = π β ((π€ prefix (π β 2)) β πΉ β (π prefix (π β 2)) β πΉ)) |
8 | fveq1 6889 | . . . . 5 β’ (π€ = π β (π€β(π β 1)) = (πβ(π β 1))) | |
9 | 8 | eleq1d 2810 | . . . 4 β’ (π€ = π β ((π€β(π β 1)) β (πΊ NeighbVtx π) β (πβ(π β 1)) β (πΊ NeighbVtx π))) |
10 | fveq1 6889 | . . . . 5 β’ (π€ = π β (π€β(π β 2)) = (πβ(π β 2))) | |
11 | 10 | eqeq1d 2727 | . . . 4 β’ (π€ = π β ((π€β(π β 2)) = π β (πβ(π β 2)) = π)) |
12 | 7, 9, 11 | 3anbi123d 1432 | . . 3 β’ (π€ = π β (((π€ prefix (π β 2)) β πΉ β§ (π€β(π β 1)) β (πΊ NeighbVtx π) β§ (π€β(π β 2)) = π) β ((π prefix (π β 2)) β πΉ β§ (πβ(π β 1)) β (πΊ NeighbVtx π) β§ (πβ(π β 2)) = π))) |
13 | 12 | elrab 3676 | . 2 β’ (π β {π€ β (π ClWWalksN πΊ) β£ ((π€ prefix (π β 2)) β πΉ β§ (π€β(π β 1)) β (πΊ NeighbVtx π) β§ (π€β(π β 2)) = π)} β (π β (π ClWWalksN πΊ) β§ ((π prefix (π β 2)) β πΉ β§ (πβ(π β 1)) β (πΊ NeighbVtx π) β§ (πβ(π β 2)) = π))) |
14 | 5, 13 | bitrdi 286 | 1 β’ ((πΊ β USGraph β§ π β π β§ π β (β€β₯β3)) β (π β (ππΆπ) β (π β (π ClWWalksN πΊ) β§ ((π prefix (π β 2)) β πΉ β§ (πβ(π β 1)) β (πΊ NeighbVtx π) β§ (πβ(π β 2)) = π)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 {crab 3419 βcfv 6543 (class class class)co 7413 β cmpo 7415 1c1 11134 β cmin 11469 2c2 12292 3c3 12293 β€β₯cuz 12847 prefix cpfx 14647 Vtxcvtx 28848 USGraphcusgr 29001 NeighbVtx cnbgr 29184 ClWWalksN cclwwlkn 29873 ClWWalksNOncclwwlknon 29936 |
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 |
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-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-2o 8481 df-oadd 8484 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 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-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-lsw 14540 df-substr 14618 df-pfx 14648 df-edg 28900 df-upgr 28934 df-umgr 28935 df-usgr 29003 df-nbgr 29185 df-wwlks 29680 df-wwlksn 29681 df-clwwlk 29831 df-clwwlkn 29874 df-clwwlknon 29937 |
This theorem is referenced by: numclwwlk1lem2foa 30203 numclwwlk1lem2f 30204 |
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