Theorem rusgrnumwlkg 26872

Description: In a k-regular graph, the number of walks of a fixed length n from a fixed vertex is k to the power of n. This theorem corresponds to statement 11 in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular.". This theorem even holds for n=0: then the walk consists of only one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.)

Hypothesis

Ref	Expression
rusgrnumwwlkg.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
rusgrnumwlkg	|- ( ( G RegUSGraph K /\ ( V e. Fin /\ P e. V /\ N e. NN0 ) ) -> ( # ` { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } ) = ( K ^ N ) )

Distinct variable groups:   w, G   w, K   w, N   w, P   w, V

Proof of Theorem rusgrnumwlkg

Dummy variables f g p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . . 4 |- ( N WWalksN G ) e. _V
2	1	rabex 4813	. . 3 |- { p e. ( N WWalksN G ) | ( p ` 0 ) = P } e. _V
3		rusgrusgr 26460	. . . . . 6 |- ( G RegUSGraph K -> G e. USGraph )
4	3	adantr 481	. . . . 5 |- ( ( G RegUSGraph K /\ ( V e. Fin /\ P e. V /\ N e. NN0 ) ) -> G e. USGraph )
5		simpr3 1069	. . . . 5 |- ( ( G RegUSGraph K /\ ( V e. Fin /\ P e. V /\ N e. NN0 ) ) -> N e. NN0 )
6		rusgrnumwwlkg.v	. . . . . . . . 9 |- V = (Vtx ` G )
7	6	eleq2i 2693	. . . . . . . 8 |- ( P e. V <-> P e. (Vtx ` G ) )
8	7	biimpi 206	. . . . . . 7 |- ( P e. V -> P e. (Vtx ` G ) )
9	8	3ad2ant2 1083	. . . . . 6 |- ( ( V e. Fin /\ P e. V /\ N e. NN0 ) -> P e. (Vtx ` G ) )
10	9	adantl 482	. . . . 5 |- ( ( G RegUSGraph K /\ ( V e. Fin /\ P e. V /\ N e. NN0 ) ) -> P e. (Vtx ` G ) )
11		wlksnwwlknvbij 26803	. . . . 5 |- ( ( G e. USGraph /\ N e. NN0 /\ P e. (Vtx ` G ) ) -> E. f f : { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } -1-1-onto-> { p e. ( N WWalksN G ) | ( p ` 0 ) = P } )
12	4, 5, 10, 11	syl3anc 1326	. . . 4 |- ( ( G RegUSGraph K /\ ( V e. Fin /\ P e. V /\ N e. NN0 ) ) -> E. f f : { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } -1-1-onto-> { p e. ( N WWalksN G ) | ( p ` 0 ) = P } )
13		f1oexbi 7116	. . . 4 |- ( E. g g : { p e. ( N WWalksN G ) | ( p ` 0 ) = P } -1-1-onto-> { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } <-> E. f f : { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } -1-1-onto-> { p e. ( N WWalksN G ) | ( p ` 0 ) = P } )
14	12, 13	sylibr 224	. . 3 |- ( ( G RegUSGraph K /\ ( V e. Fin /\ P e. V /\ N e. NN0 ) ) -> E. g g : { p e. ( N WWalksN G ) | ( p ` 0 ) = P } -1-1-onto-> { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } )
15		hasheqf1oi 13140	. . 3 |- ( { p e. ( N WWalksN G ) | ( p ` 0 ) = P } e. _V -> ( E. g g : { p e. ( N WWalksN G ) | ( p ` 0 ) = P } -1-1-onto-> { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } -> ( # ` { p e. ( N WWalksN G ) | ( p ` 0 ) = P } ) = ( # ` { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } ) ) )
16	2, 14, 15	mpsyl 68	. 2 |- ( ( G RegUSGraph K /\ ( V e. Fin /\ P e. V /\ N e. NN0 ) ) -> ( # ` { p e. ( N WWalksN G ) | ( p ` 0 ) = P } ) = ( # ` { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } ) )
17	6	rusgrnumwwlkg 26871	. 2 |- ( ( G RegUSGraph K /\ ( V e. Fin /\ P e. V /\ N e. NN0 ) ) -> ( # ` { p e. ( N WWalksN G ) | ( p ` 0 ) = P } ) = ( K ^ N ) )
18	16, 17	eqtr3d 2658	1 |- ( ( G RegUSGraph K /\ ( V e. Fin /\ P e. V /\ N e. NN0 ) ) -> ( # ` { w e. (Walks ` G ) | ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) } ) = ( K ^ N ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Fincfn 7955   0cc0 9936   NN0cn0 11292   ^cexp 12860   #chash 13117  Vtxcvtx 25874   USGraph cusgr 26044   RegUSGraph crusgr 26452  Walkscwlks 26492   WWalksN cwwlksn 26718

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-ifp 1013  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-wlks 26495  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by: (None)