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Mirrors > Home > MPE Home > Th. List > clwwlknlbonbgr1 | Structured version Visualization version GIF version |
Description: The last but one vertex in a closed walk is a neighbor of the first vertex of the closed walk. (Contributed by AV, 17-Feb-2022.) |
Ref | Expression |
---|---|
clwwlknlbonbgr1 | β’ ((πΊ β USGraph β§ π β (π ClWWalksN πΊ)) β (πβ(π β 1)) β (πΊ NeighbVtx (πβ0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . 5 β’ (VtxβπΊ) = (VtxβπΊ) | |
2 | eqid 2733 | . . . . 5 β’ (EdgβπΊ) = (EdgβπΊ) | |
3 | 1, 2 | clwwlknp 29030 | . . . 4 β’ (π β (π ClWWalksN πΊ) β ((π β Word (VtxβπΊ) β§ (β―βπ) = π) β§ βπ β (0..^(π β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ))) |
4 | lsw 14461 | . . . . . . . . . 10 β’ (π β Word (VtxβπΊ) β (lastSβπ) = (πβ((β―βπ) β 1))) | |
5 | fvoveq1 7384 | . . . . . . . . . 10 β’ ((β―βπ) = π β (πβ((β―βπ) β 1)) = (πβ(π β 1))) | |
6 | 4, 5 | sylan9eq 2793 | . . . . . . . . 9 β’ ((π β Word (VtxβπΊ) β§ (β―βπ) = π) β (lastSβπ) = (πβ(π β 1))) |
7 | 6 | preq1d 4704 | . . . . . . . 8 β’ ((π β Word (VtxβπΊ) β§ (β―βπ) = π) β {(lastSβπ), (πβ0)} = {(πβ(π β 1)), (πβ0)}) |
8 | 7 | eleq1d 2819 | . . . . . . 7 β’ ((π β Word (VtxβπΊ) β§ (β―βπ) = π) β ({(lastSβπ), (πβ0)} β (EdgβπΊ) β {(πβ(π β 1)), (πβ0)} β (EdgβπΊ))) |
9 | 8 | biimpd 228 | . . . . . 6 β’ ((π β Word (VtxβπΊ) β§ (β―βπ) = π) β ({(lastSβπ), (πβ0)} β (EdgβπΊ) β {(πβ(π β 1)), (πβ0)} β (EdgβπΊ))) |
10 | 9 | a1d 25 | . . . . 5 β’ ((π β Word (VtxβπΊ) β§ (β―βπ) = π) β (βπ β (0..^(π β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β ({(lastSβπ), (πβ0)} β (EdgβπΊ) β {(πβ(π β 1)), (πβ0)} β (EdgβπΊ)))) |
11 | 10 | 3imp 1112 | . . . 4 β’ (((π β Word (VtxβπΊ) β§ (β―βπ) = π) β§ βπ β (0..^(π β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ)) β {(πβ(π β 1)), (πβ0)} β (EdgβπΊ)) |
12 | 3, 11 | syl 17 | . . 3 β’ (π β (π ClWWalksN πΊ) β {(πβ(π β 1)), (πβ0)} β (EdgβπΊ)) |
13 | 12 | adantl 483 | . 2 β’ ((πΊ β USGraph β§ π β (π ClWWalksN πΊ)) β {(πβ(π β 1)), (πβ0)} β (EdgβπΊ)) |
14 | 2 | nbusgreledg 28350 | . . 3 β’ (πΊ β USGraph β ((πβ(π β 1)) β (πΊ NeighbVtx (πβ0)) β {(πβ(π β 1)), (πβ0)} β (EdgβπΊ))) |
15 | 14 | adantr 482 | . 2 β’ ((πΊ β USGraph β§ π β (π ClWWalksN πΊ)) β ((πβ(π β 1)) β (πΊ NeighbVtx (πβ0)) β {(πβ(π β 1)), (πβ0)} β (EdgβπΊ))) |
16 | 13, 15 | mpbird 257 | 1 β’ ((πΊ β USGraph β§ π β (π ClWWalksN πΊ)) β (πβ(π β 1)) β (πΊ NeighbVtx (πβ0))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 {cpr 4592 βcfv 6500 (class class class)co 7361 0cc0 11059 1c1 11060 + caddc 11062 β cmin 11393 ..^cfzo 13576 β―chash 14239 Word cword 14411 lastSclsw 14459 Vtxcvtx 27996 Edgcedg 28047 USGraphcusgr 28149 NeighbVtx cnbgr 28329 ClWWalksN cclwwlkn 29017 |
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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-oadd 8420 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-dju 9845 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-n0 12422 df-xnn0 12494 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-lsw 14460 df-edg 28048 df-upgr 28082 df-umgr 28083 df-usgr 28151 df-nbgr 28330 df-clwwlk 28975 df-clwwlkn 29018 |
This theorem is referenced by: extwwlkfab 29345 |
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