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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 29740, 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 12838 | . . . . . . . 8 β’ (π β (β€β₯β3) β π β β) | |
2 | 2cnd 12294 | . . . . . . . 8 β’ (π β (β€β₯β3) β 2 β β) | |
3 | 1, 2 | npcand 11579 | . . . . . . 7 β’ (π β (β€β₯β3) β ((π β 2) + 2) = π) |
4 | 3 | eqcomd 2732 | . . . . . 6 β’ (π β (β€β₯β3) β π = ((π β 2) + 2)) |
5 | 4 | 3ad2ant3 1132 | . . . . 5 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β π = ((π β 2) + 2)) |
6 | 5 | adantl 481 | . . . 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 7421 | . . 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 6889 | . 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 766 | . . 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 1192 | . . 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 12879 | . . . . 5 β’ (π β (β€β₯β3) β (π β 2) β β) | |
12 | 11 | 3ad2ant3 1132 | . . . 4 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β (π β 2) β β) |
13 | 12 | adantl 481 | . . 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 30142 | . . 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 1368 | . 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 482 | . . . 4 β’ ((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β πΊ RegUSGraph πΎ) | |
20 | simp1 1133 | . . . 4 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β π β Fin) | |
21 | 19, 20 | anim12i 612 | . . 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 616 | . . . . 5 β’ ((π β π β§ π β (β€β₯β3)) β (π β π β§ (π β 2) β β)) |
23 | 22 | 3adant1 1127 | . . . 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 481 | . . 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 30137 | . . 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 583 | . 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 2772 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 {crab 3426 class class class wbr 5141 βcfv 6537 (class class class)co 7405 β cmpo 7407 Fincfn 8941 0cc0 11112 + caddc 11115 β cmin 11448 βcn 12216 2c2 12271 3c3 12272 β€β₯cuz 12826 βcexp 14032 β―chash 14295 lastSclsw 14518 Vtxcvtx 28764 RegUSGraph crusgr 29322 WWalksN cwwlksn 29589 ClWWalksNOncclwwlknon 29849 FriendGraph cfrgr 30020 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 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 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-dju 9898 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-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-rp 12981 df-xadd 13099 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-word 14471 df-lsw 14519 df-concat 14527 df-s1 14552 df-substr 14597 df-pfx 14627 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-vtx 28766 df-iedg 28767 df-edg 28816 df-uhgr 28826 df-ushgr 28827 df-upgr 28850 df-umgr 28851 df-uspgr 28918 df-usgr 28919 df-fusgr 29082 df-nbgr 29098 df-vtxdg 29232 df-rgr 29323 df-rusgr 29324 df-wwlks 29593 df-wwlksn 29594 df-wwlksnon 29595 df-clwwlk 29744 df-clwwlkn 29787 df-clwwlknon 29850 df-frgr 30021 |
This theorem is referenced by: numclwwlk3 30147 |
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