Theorem numclwwlk6 27248

Description: For a prime divisor P of K - 1, the total number of closed walks of length P in a K-regular friendship graph is equal modulo P to the number of vertices. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.)

Hypothesis

Ref	Expression
numclwwlk6.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
numclwwlk6	|- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( P ClWWalksN G ) ) mod P ) = ( ( # ` V ) mod P ) )

Proof of Theorem numclwwlk6

Dummy variables n u v w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		numclwwlk6.v	. . . . . 6 |- V = (Vtx ` G )
2	1	finrusgrfusgr 26461	. . . . 5 |- ( ( G RegUSGraph K /\ V e. Fin ) -> G e. FinUSGraph )
3	2	3adant2 1080	. . . 4 |- ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) -> G e. FinUSGraph )
4		prmnn 15388	. . . . 5 |- ( P e. Prime -> P e. NN )
5	4	adantr 481	. . . 4 |- ( ( P e. Prime /\ P || ( K - 1 ) ) -> P e. NN )
6		eqid 2622	. . . . 5 |- ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
7	1, 6	numclwwlk4 27244	. . . 4 |- ( ( G e. FinUSGraph /\ P e. NN ) -> ( # ` ( P ClWWalksN G ) ) = sum_ x e. V ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) )
8	3, 5, 7	syl2an 494	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( # ` ( P ClWWalksN G ) ) = sum_ x e. V ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) )
9	8	oveq1d 6665	. 2 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( P ClWWalksN G ) ) mod P ) = ( sum_ x e. V ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) mod P ) )
10	5	adantl 482	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> P e. NN )
11		simp3 1063	. . . . 5 |- ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) -> V e. Fin )
12	11	adantr 481	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> V e. Fin )
13	12	adantr 481	. . . . . . . 8 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> V e. Fin )
14		simpr 477	. . . . . . . 8 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> x e. V )
15	10	adantr 481	. . . . . . . 8 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> P e. NN )
16	6, 1	numclwwlkffin 27214	. . . . . . . 8 |- ( ( V e. Fin /\ x e. V /\ P e. NN ) -> ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) e. Fin )
17	13, 14, 15, 16	syl3anc 1326	. . . . . . 7 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) e. Fin )
18		hashcl 13147	. . . . . . 7 |- ( ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) e. Fin -> ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) e. NN0 )
19	17, 18	syl 17	. . . . . 6 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) e. NN0 )
20	19	nn0zd 11480	. . . . 5 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) e. ZZ )
21	20	ralrimiva 2966	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> A. x e. V ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) e. ZZ )
22	10, 12, 21	modfsummod 14526	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( sum_ x e. V ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) mod P ) = ( sum_ x e. V ( ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) mod P ) mod P ) )
23		simpl 473	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) )
24		simpr 477	. . . . . . . . 9 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( P e. Prime /\ P || ( K - 1 ) ) )
25	24	anim1i 592	. . . . . . . 8 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> ( ( P e. Prime /\ P || ( K - 1 ) ) /\ x e. V ) )
26	25	ancomd 467	. . . . . . 7 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> ( x e. V /\ ( P e. Prime /\ P || ( K - 1 ) ) ) )
27		3anass 1042	. . . . . . 7 |- ( ( x e. V /\ P e. Prime /\ P || ( K - 1 ) ) <-> ( x e. V /\ ( P e. Prime /\ P || ( K - 1 ) ) ) )
28	26, 27	sylibr 224	. . . . . 6 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> ( x e. V /\ P e. Prime /\ P || ( K - 1 ) ) )
29		fveq1 6190	. . . . . . . . . . 11 |- ( w = u -> ( w ` 0 ) = ( u ` 0 ) )
30	29	eqeq1d 2624	. . . . . . . . . 10 |- ( w = u -> ( ( w ` 0 ) = v <-> ( u ` 0 ) = v ) )
31	30	cbvrabv 3199	. . . . . . . . 9 |- { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } = { u e. ( n ClWWalksN G ) | ( u ` 0 ) = v }
32	31	a1i 11	. . . . . . . 8 |- ( ( v e. V /\ n e. NN ) -> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } = { u e. ( n ClWWalksN G ) | ( u ` 0 ) = v } )
33	32	mpt2eq3ia 6720	. . . . . . 7 |- ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) = ( v e. V , n e. NN |-> { u e. ( n ClWWalksN G ) | ( u ` 0 ) = v } )
34	1, 33	numclwwlk5 27246	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( x e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) mod P ) = 1 )
35	23, 28, 34	syl2an2r 876	. . . . 5 |- ( ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) /\ x e. V ) -> ( ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) mod P ) = 1 )
36	35	sumeq2dv 14433	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> sum_ x e. V ( ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) mod P ) = sum_ x e. V 1 )
37	36	oveq1d 6665	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( sum_ x e. V ( ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) mod P ) mod P ) = ( sum_ x e. V 1 mod P ) )
38	22, 37	eqtrd 2656	. 2 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( sum_ x e. V ( # ` ( x ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) P ) ) mod P ) = ( sum_ x e. V 1 mod P ) )
39		1cnd 10056	. . . . 5 |- ( ( P e. Prime /\ P || ( K - 1 ) ) -> 1 e. CC )
40		fsumconst 14522	. . . . 5 |- ( ( V e. Fin /\ 1 e. CC ) -> sum_ x e. V 1 = ( ( # ` V ) x. 1 ) )
41	11, 39, 40	syl2an 494	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> sum_ x e. V 1 = ( ( # ` V ) x. 1 ) )
42		hashcl 13147	. . . . . . . 8 |- ( V e. Fin -> ( # ` V ) e. NN0 )
43	42	nn0red 11352	. . . . . . 7 |- ( V e. Fin -> ( # ` V ) e. RR )
44		ax-1rid 10006	. . . . . . 7 |- ( ( # ` V ) e. RR -> ( ( # ` V ) x. 1 ) = ( # ` V ) )
45	43, 44	syl 17	. . . . . 6 |- ( V e. Fin -> ( ( # ` V ) x. 1 ) = ( # ` V ) )
46	45	3ad2ant3 1084	. . . . 5 |- ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) -> ( ( # ` V ) x. 1 ) = ( # ` V ) )
47	46	adantr 481	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` V ) x. 1 ) = ( # ` V ) )
48	41, 47	eqtrd 2656	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> sum_ x e. V 1 = ( # ` V ) )
49	48	oveq1d 6665	. 2 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( sum_ x e. V 1 mod P ) = ( ( # ` V ) mod P ) )
50	9, 38, 49	3eqtrd 2660	1 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( P ClWWalksN G ) ) mod P ) = ( ( # ` V ) mod P ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   mod cmo 12668   #chash 13117   sum_csu 14416   || cdvds 14983   Primecprime 15385  Vtxcvtx 25874   FinUSGraph cfusgr 26208   RegUSGraph crusgr 26452   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-inf 8349  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-fl 12593  df-mod 12669  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-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  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:  numclwwlk7  27249