Theorem numclwwlk2 27240

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 26872, 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.)

Hypotheses

Ref	Expression
numclwwlk.v	|- V = (Vtx ` G )
numclwwlk.q	|- Q = ( v e. V , n e. NN |-> { w e. ( n WWalksN G ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
numclwwlk.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
numclwwlk.h	|- H = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )

Assertion

Ref	Expression
numclwwlk2	|- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X H N ) ) = ( ( K ^ ( N - 2 ) ) - ( # ` ( X F ( N - 2 ) ) ) ) )

Distinct variable groups:   n, G, v, w   n, N, v, w   n, V, v   n, X, v, w   w, K   w, V

Allowed substitution hints:   Q( w, v, n)   F( w, v, n)   H( w, v, n)   K( v, n)

Proof of Theorem numclwwlk2

Step	Hyp	Ref	Expression
1		eluzelcn 11699	. . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
2		2cnd 11093	. . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
3	1, 2	npcand 10396	. . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) + 2 ) = N )
4	3	eqcomd 2628	. . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> N = ( ( N - 2 ) + 2 ) )
5	4	3ad2ant3 1084	. . . . 5 |- ( ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> N = ( ( N - 2 ) + 2 ) )
6	5	adantl 482	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> N = ( ( N - 2 ) + 2 ) )
7	6	oveq2d 6666	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( X H N ) = ( X H ( ( N - 2 ) + 2 ) ) )
8	7	fveq2d 6195	. 2 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X H N ) ) = ( # ` ( X H ( ( N - 2 ) + 2 ) ) ) )
9		simplr 792	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> G e. FriendGraph )
10		simpr2 1068	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> X e. V )
11		uz3m2nn 11731	. . . . 5 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
12	11	3ad2ant3 1084	. . . 4 |- ( ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( N - 2 ) e. NN )
13	12	adantl 482	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( N - 2 ) e. NN )
14		numclwwlk.v	. . . 4 |- V = (Vtx ` G )
15		numclwwlk.q	. . . 4 |- Q = ( v e. V , n e. NN |-> { w e. ( n WWalksN G ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
16		numclwwlk.f	. . . 4 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
17		numclwwlk.h	. . . 4 |- H = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )
18	14, 15, 16, 17	numclwwlk2lem3 27239	. . 3 |- ( ( G e. FriendGraph /\ X e. V /\ ( N - 2 ) e. NN ) -> ( # ` ( X Q ( N - 2 ) ) ) = ( # ` ( X H ( ( N - 2 ) + 2 ) ) ) )
19	9, 10, 13, 18	syl3anc 1326	. 2 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X Q ( N - 2 ) ) ) = ( # ` ( X H ( ( N - 2 ) + 2 ) ) ) )
20		simpl 473	. . . 4 |- ( ( G RegUSGraph K /\ G e. FriendGraph ) -> G RegUSGraph K )
21		simp1 1061	. . . 4 |- ( ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> V e. Fin )
22	20, 21	anim12i 590	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( G RegUSGraph K /\ V e. Fin ) )
23	11	anim2i 593	. . . . 5 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X e. V /\ ( N - 2 ) e. NN ) )
24	23	3adant1 1079	. . . 4 |- ( ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X e. V /\ ( N - 2 ) e. NN ) )
25	24	adantl 482	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( X e. V /\ ( N - 2 ) e. NN ) )
26	14, 15, 16	numclwwlkqhash 27233	. . 3 |- ( ( ( G RegUSGraph K /\ V e. Fin ) /\ ( X e. V /\ ( N - 2 ) e. NN ) ) -> ( # ` ( X Q ( N - 2 ) ) ) = ( ( K ^ ( N - 2 ) ) - ( # ` ( X F ( N - 2 ) ) ) ) )
27	22, 25, 26	syl2anc 693	. 2 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X Q ( N - 2 ) ) ) = ( ( K ^ ( N - 2 ) ) - ( # ` ( X F ( N - 2 ) ) ) ) )
28	8, 19, 27	3eqtr2d 2662	1 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X H N ) ) = ( ( K ^ ( N - 2 ) ) - ( # ` ( X F ( N - 2 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   0cc0 9936   + caddc 9939   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   ZZ>=cuz 11687   ^cexp 12860   #chash 13117   lastS clsw 13292  Vtxcvtx 25874   RegUSGraph crusgr 26452   WWalksN cwwlksn 26718   ClWWalksN cclwwlksn 26876   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-nbgr 26228  df-vtxdg 26362  df-rgr 26453  df-rusgr 26454  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878  df-frgr 27121

This theorem is referenced by:  numclwwlk3OLD  27242  numclwwlk3  27243