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Mirrors > Home > MPE Home > Th. List > numclwwlk2 | Structured version Visualization version GIF version |
Description: Statement 10 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 k^(n-2) - f(n-2)." According to rusgrnumwlkg 28964, we have k^(n-2) different walks of length (n-2): v(0) ... v(n-2). From this number, the number of closed walks of length (n-2), which is f(n-2) per definition, must be subtracted, because for these walks v(n-2) =/= v(0) = v would hold. Because of the friendship condition, there is exactly one vertex v(n-1) which is a neighbor of v(n-2) as well as of v(n)=v=v(0), because v(n-2) and v(n)=v are different, so the number of walks v(0) ... v(n-2) is identical with the number of walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n-2) can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-May-2022.) |
Ref | Expression |
---|---|
numclwwlk.v | β’ π = (VtxβπΊ) |
numclwwlk.q | β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) |
numclwwlk.h | β’ π» = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) |
Ref | Expression |
---|---|
numclwwlk2 | β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ»π)) = ((πΎβ(π β 2)) β (β―β(π(ClWWalksNOnβπΊ)(π β 2))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelcn 12782 | . . . . . . . 8 β’ (π β (β€β₯β3) β π β β) | |
2 | 2cnd 12238 | . . . . . . . 8 β’ (π β (β€β₯β3) β 2 β β) | |
3 | 1, 2 | npcand 11523 | . . . . . . 7 β’ (π β (β€β₯β3) β ((π β 2) + 2) = π) |
4 | 3 | eqcomd 2743 | . . . . . 6 β’ (π β (β€β₯β3) β π = ((π β 2) + 2)) |
5 | 4 | 3ad2ant3 1136 | . . . . 5 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β π = ((π β 2) + 2)) |
6 | 5 | adantl 483 | . . . 4 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β π = ((π β 2) + 2)) |
7 | 6 | oveq2d 7378 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (ππ»π) = (ππ»((π β 2) + 2))) |
8 | 7 | fveq2d 6851 | . 2 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ»π)) = (β―β(ππ»((π β 2) + 2)))) |
9 | simplr 768 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β πΊ β FriendGraph ) | |
10 | simpr2 1196 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β π β π) | |
11 | uz3m2nn 12823 | . . . . 5 β’ (π β (β€β₯β3) β (π β 2) β β) | |
12 | 11 | 3ad2ant3 1136 | . . . 4 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β (π β 2) β β) |
13 | 12 | adantl 483 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (π β 2) β β) |
14 | numclwwlk.v | . . . 4 β’ π = (VtxβπΊ) | |
15 | numclwwlk.q | . . . 4 β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) | |
16 | numclwwlk.h | . . . 4 β’ π» = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) | |
17 | 14, 15, 16 | numclwwlk2lem3 29366 | . . 3 β’ ((πΊ β FriendGraph β§ π β π β§ (π β 2) β β) β (β―β(ππ(π β 2))) = (β―β(ππ»((π β 2) + 2)))) |
18 | 9, 10, 13, 17 | syl3anc 1372 | . 2 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ(π β 2))) = (β―β(ππ»((π β 2) + 2)))) |
19 | simpl 484 | . . . 4 β’ ((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β πΊ RegUSGraph πΎ) | |
20 | simp1 1137 | . . . 4 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β π β Fin) | |
21 | 19, 20 | anim12i 614 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (πΊ RegUSGraph πΎ β§ π β Fin)) |
22 | 11 | anim2i 618 | . . . . 5 β’ ((π β π β§ π β (β€β₯β3)) β (π β π β§ (π β 2) β β)) |
23 | 22 | 3adant1 1131 | . . . 4 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β (π β π β§ (π β 2) β β)) |
24 | 23 | adantl 483 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (π β π β§ (π β 2) β β)) |
25 | 14, 15 | numclwwlkqhash 29361 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ π β Fin) β§ (π β π β§ (π β 2) β β)) β (β―β(ππ(π β 2))) = ((πΎβ(π β 2)) β (β―β(π(ClWWalksNOnβπΊ)(π β 2))))) |
26 | 21, 24, 25 | syl2anc 585 | . 2 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ(π β 2))) = ((πΎβ(π β 2)) β (β―β(π(ClWWalksNOnβπΊ)(π β 2))))) |
27 | 8, 18, 26 | 3eqtr2d 2783 | 1 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ»π)) = ((πΎβ(π β 2)) β (β―β(π(ClWWalksNOnβπΊ)(π β 2))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 {crab 3410 class class class wbr 5110 βcfv 6501 (class class class)co 7362 β cmpo 7364 Fincfn 8890 0cc0 11058 + caddc 11061 β cmin 11392 βcn 12160 2c2 12215 3c3 12216 β€β₯cuz 12770 βcexp 13974 β―chash 14237 lastSclsw 14457 Vtxcvtx 27989 RegUSGraph crusgr 28546 WWalksN cwwlksn 28813 ClWWalksNOncclwwlknon 29073 FriendGraph cfrgr 29244 |
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-inf2 9584 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 ax-pre-sup 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 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-disj 5076 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-se 5594 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-isom 6510 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-sup 9385 df-oi 9453 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-div 11820 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-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 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-wwlks 28817 df-wwlksn 28818 df-wwlksnon 28819 df-clwwlk 28968 df-clwwlkn 29011 df-clwwlknon 29074 df-frgr 29245 |
This theorem is referenced by: numclwwlk3 29371 |
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