Theorem numclwwlk1 27231

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 G 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, but only for finite graphs! (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.)

Hypotheses

Ref	Expression
extwwlkfab.v	|- V = (Vtx ` G )
extwwlkfab.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
extwwlkfab.c	|- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )

Assertion

Ref	Expression
numclwwlk1	|- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X C N ) ) = ( K x. ( # ` ( X F ( N - 2 ) ) ) ) )

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

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

Proof of Theorem numclwwlk1

Dummy variables f x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3 |- ( X C N ) e. _V
2		rusgrusgr 26460	. . . . 5 |- ( G RegUSGraph K -> G e. USGraph )
3	2	ad2antlr 763	. . . 4 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> G e. USGraph )
4		simprl 794	. . . 4 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> X e. V )
5		simprr 796	. . . 4 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> N e. ( ZZ>= ` 3 ) )
6		extwwlkfab.v	. . . . 5 |- V = (Vtx ` G )
7		extwwlkfab.f	. . . . 5 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
8		extwwlkfab.c	. . . . 5 |- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
9	6, 7, 8	numclwlk1lem2 27230	. . . 4 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> E. f f : ( X C N ) -1-1-onto-> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) )
10	3, 4, 5, 9	syl3anc 1326	. . 3 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> E. f f : ( X C N ) -1-1-onto-> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) )
11		hasheqf1oi 13140	. . 3 |- ( ( X C N ) e. _V -> ( E. f f : ( X C N ) -1-1-onto-> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) -> ( # ` ( X C N ) ) = ( # ` ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) ) ) )
12	1, 10, 11	mpsyl 68	. 2 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X C N ) ) = ( # ` ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) ) )
13		simpll 790	. . . 4 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> V e. Fin )
14		uz3m2nn 11731	. . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
15	14	adantl 482	. . . . 5 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( N - 2 ) e. NN )
16	15	adantl 482	. . . 4 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( N - 2 ) e. NN )
17	7, 6	numclwwlkffin 27214	. . . 4 |- ( ( V e. Fin /\ X e. V /\ ( N - 2 ) e. NN ) -> ( X F ( N - 2 ) ) e. Fin )
18	13, 4, 16, 17	syl3anc 1326	. . 3 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( X F ( N - 2 ) ) e. Fin )
19	6	finrusgrfusgr 26461	. . . . . . 7 |- ( ( G RegUSGraph K /\ V e. Fin ) -> G e. FinUSGraph )
20	19	ancoms 469	. . . . . 6 |- ( ( V e. Fin /\ G RegUSGraph K ) -> G e. FinUSGraph )
21		fusgrfis 26222	. . . . . 6 |- ( G e. FinUSGraph -> (Edg ` G ) e. Fin )
22	20, 21	syl 17	. . . . 5 |- ( ( V e. Fin /\ G RegUSGraph K ) -> (Edg ` G ) e. Fin )
23	22	adantr 481	. . . 4 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> (Edg ` G ) e. Fin )
24		eqid 2622	. . . . 5 |- (Edg ` G ) = (Edg ` G )
25	6, 24	nbusgrfi 26276	. . . 4 |- ( ( G e. USGraph /\ (Edg ` G ) e. Fin /\ X e. V ) -> ( G NeighbVtx X ) e. Fin )
26	3, 23, 4, 25	syl3anc 1326	. . 3 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( G NeighbVtx X ) e. Fin )
27		hashxp 13221	. . 3 |- ( ( ( X F ( N - 2 ) ) e. Fin /\ ( G NeighbVtx X ) e. Fin ) -> ( # ` ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) ) = ( ( # ` ( X F ( N - 2 ) ) ) x. ( # ` ( G NeighbVtx X ) ) ) )
28	18, 26, 27	syl2anc 693	. 2 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) ) = ( ( # ` ( X F ( N - 2 ) ) ) x. ( # ` ( G NeighbVtx X ) ) ) )
29	6	rusgrpropnb 26479	. . . . . . . . 9 |- ( G RegUSGraph K -> ( G e. USGraph /\ K e. NN0* /\ A. x e. V ( # ` ( G NeighbVtx x ) ) = K ) )
30		oveq2 6658	. . . . . . . . . . . . 13 |- ( x = X -> ( G NeighbVtx x ) = ( G NeighbVtx X ) )
31	30	fveq2d 6195	. . . . . . . . . . . 12 |- ( x = X -> ( # ` ( G NeighbVtx x ) ) = ( # ` ( G NeighbVtx X ) ) )
32	31	eqeq1d 2624	. . . . . . . . . . 11 |- ( x = X -> ( ( # ` ( G NeighbVtx x ) ) = K <-> ( # ` ( G NeighbVtx X ) ) = K ) )
33	32	rspccv 3306	. . . . . . . . . 10 |- ( A. x e. V ( # ` ( G NeighbVtx x ) ) = K -> ( X e. V -> ( # ` ( G NeighbVtx X ) ) = K ) )
34	33	3ad2ant3 1084	. . . . . . . . 9 |- ( ( G e. USGraph /\ K e. NN0* /\ A. x e. V ( # ` ( G NeighbVtx x ) ) = K ) -> ( X e. V -> ( # ` ( G NeighbVtx X ) ) = K ) )
35	29, 34	syl 17	. . . . . . . 8 |- ( G RegUSGraph K -> ( X e. V -> ( # ` ( G NeighbVtx X ) ) = K ) )
36	35	adantl 482	. . . . . . 7 |- ( ( V e. Fin /\ G RegUSGraph K ) -> ( X e. V -> ( # ` ( G NeighbVtx X ) ) = K ) )
37	36	com12 32	. . . . . 6 |- ( X e. V -> ( ( V e. Fin /\ G RegUSGraph K ) -> ( # ` ( G NeighbVtx X ) ) = K ) )
38	37	adantr 481	. . . . 5 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( V e. Fin /\ G RegUSGraph K ) -> ( # ` ( G NeighbVtx X ) ) = K ) )
39	38	impcom 446	. . . 4 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( G NeighbVtx X ) ) = K )
40	39	oveq2d 6666	. . 3 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( ( # ` ( X F ( N - 2 ) ) ) x. ( # ` ( G NeighbVtx X ) ) ) = ( ( # ` ( X F ( N - 2 ) ) ) x. K ) )
41		hashcl 13147	. . . . 5 |- ( ( X F ( N - 2 ) ) e. Fin -> ( # ` ( X F ( N - 2 ) ) ) e. NN0 )
42		nn0cn 11302	. . . . 5 |- ( ( # ` ( X F ( N - 2 ) ) ) e. NN0 -> ( # ` ( X F ( N - 2 ) ) ) e. CC )
43	18, 41, 42	3syl 18	. . . 4 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X F ( N - 2 ) ) ) e. CC )
44	20	adantr 481	. . . . . 6 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> G e. FinUSGraph )
45		simplr 792	. . . . . 6 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> G RegUSGraph K )
46		ne0i 3921	. . . . . . . 8 |- ( X e. V -> V =/= (/) )
47	46	adantr 481	. . . . . . 7 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> V =/= (/) )
48	47	adantl 482	. . . . . 6 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> V =/= (/) )
49	6	frusgrnn0 26467	. . . . . 6 |- ( ( G e. FinUSGraph /\ G RegUSGraph K /\ V =/= (/) ) -> K e. NN0 )
50	44, 45, 48, 49	syl3anc 1326	. . . . 5 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> K e. NN0 )
51	50	nn0cnd 11353	. . . 4 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> K e. CC )
52	43, 51	mulcomd 10061	. . 3 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( ( # ` ( X F ( N - 2 ) ) ) x. K ) = ( K x. ( # ` ( X F ( N - 2 ) ) ) ) )
53	40, 52	eqtrd 2656	. 2 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( ( # ` ( X F ( N - 2 ) ) ) x. ( # ` ( G NeighbVtx X ) ) ) = ( K x. ( # ` ( X F ( N - 2 ) ) ) ) )
54	12, 28, 53	3eqtrd 2660	1 |- ( ( ( V e. Fin /\ G RegUSGraph K ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X C N ) ) = ( K x. ( # ` ( X F ( N - 2 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   X. cxp 5112   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   CCcc 9934   0cc0 9936   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   NN0cn0 11292  NN0*cxnn0 11363   ZZ>=cuz 11687   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FinUSGraph cfusgr 26208   NeighbVtx cnbgr 26224   RegUSGraph crusgr 26452   ClWWalksN cclwwlksn 26876

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-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

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-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-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-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-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-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-s2 13593  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

This theorem is referenced by:  numclwwlk3OLD  27242  numclwwlk3  27243