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Mirrors > Home > MPE Home > Th. List > numclwlk1 | Structured version Visualization version GIF version |
Description: Statement 9 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v is kf(n-2)". Since πΊ is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0. (Contributed by AV, 23-May-2022.) |
Ref | Expression |
---|---|
numclwlk1.v | β’ π = (VtxβπΊ) |
numclwlk1.c | β’ πΆ = {π€ β (ClWalksβπΊ) β£ ((β―β(1st βπ€)) = π β§ ((2nd βπ€)β0) = π β§ ((2nd βπ€)β(π β 2)) = π)} |
numclwlk1.f | β’ πΉ = {π€ β (ClWalksβπΊ) β£ ((β―β(1st βπ€)) = (π β 2) β§ ((2nd βπ€)β0) = π)} |
Ref | Expression |
---|---|
numclwlk1 | β’ (((π β Fin β§ πΊ RegUSGraph πΎ) β§ (π β π β§ π β (β€β₯β2))) β (β―βπΆ) = (πΎ Β· (β―βπΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzp1 12811 | . . . 4 β’ (π β (β€β₯β2) β (π = 2 β¨ π β (β€β₯β(2 + 1)))) | |
2 | numclwlk1.v | . . . . . . . 8 β’ π = (VtxβπΊ) | |
3 | numclwlk1.c | . . . . . . . 8 β’ πΆ = {π€ β (ClWalksβπΊ) β£ ((β―β(1st βπ€)) = π β§ ((2nd βπ€)β0) = π β§ ((2nd βπ€)β(π β 2)) = π)} | |
4 | numclwlk1.f | . . . . . . . 8 β’ πΉ = {π€ β (ClWalksβπΊ) β£ ((β―β(1st βπ€)) = (π β 2) β§ ((2nd βπ€)β0) = π)} | |
5 | 2, 3, 4 | numclwlk1lem1 29355 | . . . . . . 7 β’ (((π β Fin β§ πΊ RegUSGraph πΎ) β§ (π β π β§ π = 2)) β (β―βπΆ) = (πΎ Β· (β―βπΉ))) |
6 | 5 | expcom 415 | . . . . . 6 β’ ((π β π β§ π = 2) β ((π β Fin β§ πΊ RegUSGraph πΎ) β (β―βπΆ) = (πΎ Β· (β―βπΉ)))) |
7 | 6 | expcom 415 | . . . . 5 β’ (π = 2 β (π β π β ((π β Fin β§ πΊ RegUSGraph πΎ) β (β―βπΆ) = (πΎ Β· (β―βπΉ))))) |
8 | 2, 3, 4 | numclwlk1lem2 29356 | . . . . . . . 8 β’ (((π β Fin β§ πΊ RegUSGraph πΎ) β§ (π β π β§ π β (β€β₯β3))) β (β―βπΆ) = (πΎ Β· (β―βπΉ))) |
9 | 8 | expcom 415 | . . . . . . 7 β’ ((π β π β§ π β (β€β₯β3)) β ((π β Fin β§ πΊ RegUSGraph πΎ) β (β―βπΆ) = (πΎ Β· (β―βπΉ)))) |
10 | 9 | expcom 415 | . . . . . 6 β’ (π β (β€β₯β3) β (π β π β ((π β Fin β§ πΊ RegUSGraph πΎ) β (β―βπΆ) = (πΎ Β· (β―βπΉ))))) |
11 | 2p1e3 12302 | . . . . . . 7 β’ (2 + 1) = 3 | |
12 | 11 | fveq2i 6850 | . . . . . 6 β’ (β€β₯β(2 + 1)) = (β€β₯β3) |
13 | 10, 12 | eleq2s 2856 | . . . . 5 β’ (π β (β€β₯β(2 + 1)) β (π β π β ((π β Fin β§ πΊ RegUSGraph πΎ) β (β―βπΆ) = (πΎ Β· (β―βπΉ))))) |
14 | 7, 13 | jaoi 856 | . . . 4 β’ ((π = 2 β¨ π β (β€β₯β(2 + 1))) β (π β π β ((π β Fin β§ πΊ RegUSGraph πΎ) β (β―βπΆ) = (πΎ Β· (β―βπΉ))))) |
15 | 1, 14 | syl 17 | . . 3 β’ (π β (β€β₯β2) β (π β π β ((π β Fin β§ πΊ RegUSGraph πΎ) β (β―βπΆ) = (πΎ Β· (β―βπΉ))))) |
16 | 15 | impcom 409 | . 2 β’ ((π β π β§ π β (β€β₯β2)) β ((π β Fin β§ πΊ RegUSGraph πΎ) β (β―βπΆ) = (πΎ Β· (β―βπΉ)))) |
17 | 16 | impcom 409 | 1 β’ (((π β Fin β§ πΊ RegUSGraph πΎ) β§ (π β π β§ π β (β€β₯β2))) β (β―βπΆ) = (πΎ Β· (β―βπΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β¨ wo 846 β§ w3a 1088 = wceq 1542 β wcel 2107 {crab 3410 class class class wbr 5110 βcfv 6501 (class class class)co 7362 1st c1st 7924 2nd c2nd 7925 Fincfn 8890 0cc0 11058 1c1 11059 + caddc 11061 Β· cmul 11063 β cmin 11392 2c2 12215 3c3 12216 β€β₯cuz 12770 β―chash 14237 Vtxcvtx 27989 RegUSGraph crusgr 28546 ClWalkscclwlks 28760 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-rp 12923 df-xadd 13041 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-word 14410 df-lsw 14458 df-concat 14466 df-s1 14491 df-substr 14536 df-pfx 14566 df-s2 14744 df-vtx 27991 df-iedg 27992 df-edg 28041 df-uhgr 28051 df-ushgr 28052 df-upgr 28075 df-umgr 28076 df-uspgr 28143 df-usgr 28144 df-fusgr 28307 df-nbgr 28323 df-vtxdg 28456 df-rgr 28547 df-rusgr 28548 df-wlks 28589 df-clwlks 28761 df-wwlks 28817 df-wwlksn 28818 df-clwwlk 28968 df-clwwlkn 29011 df-clwwlknon 29074 |
This theorem is referenced by: (None) |
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