Theorem numclwwlk7 27249

Description: Statement 14 in [Huneke] p. 2: "The total number of closed walks of length p [in a friendship graph] is (k(k-1)+1)f(p)=1 (mod p)", since the number of vertices in a friendship graph is (k(k-1)+1), see frrusgrord0 27204 or frrusgrord 27205, and p divides (k-1), i.e. (k-1) mod p = 0 => k(k-1) mod p = 0 => k(k-1)+1 mod p = 1. Since the null graph is a friendship graph, see frgr0 27128, as well as k-regular (for any k), see 0vtxrgr 26472, but has no closed walk, see 0clwlk0 26992, this theorem would be false for a null graph: ( ( # ` ( P ClWWalksN G ) ) mod P ) = 0 =/= 1, so this case must be excluded (by assuming V =/= (/)). (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 3-Jun-2021.)

Hypothesis

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

Assertion

Ref	Expression
numclwwlk7	|- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( P ClWWalksN G ) ) mod P ) = 1 )

Proof of Theorem numclwwlk7

Step	Hyp	Ref	Expression
1		simpll 790	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) ) -> G RegUSGraph K )
2		simplr 792	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) ) -> G e. FriendGraph )
3		simprr 796	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) ) -> V e. Fin )
4	1, 2, 3	3jca 1242	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) ) -> ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) )
5		numclwwlk6.v	. . . 4 |- V = (Vtx ` G )
6	5	numclwwlk6 27248	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( P ClWWalksN G ) ) mod P ) = ( ( # ` V ) mod P ) )
7	4, 6	stoic3 1701	. 2 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( P ClWWalksN G ) ) mod P ) = ( ( # ` V ) mod P ) )
8		simp2 1062	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( V =/= (/) /\ V e. Fin ) )
9	8	ancomd 467	. . . . 5 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( V e. Fin /\ V =/= (/) ) )
10		simp1 1061	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( G RegUSGraph K /\ G e. FriendGraph ) )
11	10	ancomd 467	. . . . 5 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( G e. FriendGraph /\ G RegUSGraph K ) )
12	5	frrusgrord 27205	. . . . 5 |- ( ( V e. Fin /\ V =/= (/) ) -> ( ( G e. FriendGraph /\ G RegUSGraph K ) -> ( # ` V ) = ( ( K x. ( K - 1 ) ) + 1 ) ) )
13	9, 11, 12	sylc 65	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( # ` V ) = ( ( K x. ( K - 1 ) ) + 1 ) )
14	13	oveq1d 6665	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` V ) mod P ) = ( ( ( K x. ( K - 1 ) ) + 1 ) mod P ) )
15	5	numclwwlk7lem 27247	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) ) -> K e. NN0 )
16		nn0cn 11302	. . . . . . . . . . 11 |- ( K e. NN0 -> K e. CC )
17		peano2cnm 10347	. . . . . . . . . . . 12 |- ( K e. CC -> ( K - 1 ) e. CC )
18	16, 17	syl 17	. . . . . . . . . . 11 |- ( K e. NN0 -> ( K - 1 ) e. CC )
19	16, 18	mulcomd 10061	. . . . . . . . . 10 |- ( K e. NN0 -> ( K x. ( K - 1 ) ) = ( ( K - 1 ) x. K ) )
20	19	oveq1d 6665	. . . . . . . . 9 |- ( K e. NN0 -> ( ( K x. ( K - 1 ) ) mod P ) = ( ( ( K - 1 ) x. K ) mod P ) )
21	20	adantr 481	. . . . . . . 8 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( K x. ( K - 1 ) ) mod P ) = ( ( ( K - 1 ) x. K ) mod P ) )
22		prmnn 15388	. . . . . . . . . . 11 |- ( P e. Prime -> P e. NN )
23	22	ad2antrl 764	. . . . . . . . . 10 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> P e. NN )
24		nn0z 11400	. . . . . . . . . . . 12 |- ( K e. NN0 -> K e. ZZ )
25		peano2zm 11420	. . . . . . . . . . . 12 |- ( K e. ZZ -> ( K - 1 ) e. ZZ )
26	24, 25	syl 17	. . . . . . . . . . 11 |- ( K e. NN0 -> ( K - 1 ) e. ZZ )
27	26	adantr 481	. . . . . . . . . 10 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( K - 1 ) e. ZZ )
28	24	adantr 481	. . . . . . . . . 10 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> K e. ZZ )
29	23, 27, 28	3jca 1242	. . . . . . . . 9 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( P e. NN /\ ( K - 1 ) e. ZZ /\ K e. ZZ ) )
30		simprr 796	. . . . . . . . 9 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> P || ( K - 1 ) )
31		mulmoddvds 15051	. . . . . . . . 9 |- ( ( P e. NN /\ ( K - 1 ) e. ZZ /\ K e. ZZ ) -> ( P || ( K - 1 ) -> ( ( ( K - 1 ) x. K ) mod P ) = 0 ) )
32	29, 30, 31	sylc 65	. . . . . . . 8 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( K - 1 ) x. K ) mod P ) = 0 )
33	21, 32	eqtrd 2656	. . . . . . 7 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( K x. ( K - 1 ) ) mod P ) = 0 )
34	22	nnred 11035	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR )
35		prmgt1 15409	. . . . . . . . . 10 |- ( P e. Prime -> 1 < P )
36	34, 35	jca 554	. . . . . . . . 9 |- ( P e. Prime -> ( P e. RR /\ 1 < P ) )
37	36	ad2antrl 764	. . . . . . . 8 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( P e. RR /\ 1 < P ) )
38		1mod 12702	. . . . . . . 8 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
39	37, 38	syl 17	. . . . . . 7 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( 1 mod P ) = 1 )
40	33, 39	oveq12d 6668	. . . . . 6 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( K x. ( K - 1 ) ) mod P ) + ( 1 mod P ) ) = ( 0 + 1 ) )
41	40	oveq1d 6665	. . . . 5 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( ( K x. ( K - 1 ) ) mod P ) + ( 1 mod P ) ) mod P ) = ( ( 0 + 1 ) mod P ) )
42		nn0re 11301	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
43		peano2rem 10348	. . . . . . . . 9 |- ( K e. RR -> ( K - 1 ) e. RR )
44	42, 43	syl 17	. . . . . . . 8 |- ( K e. NN0 -> ( K - 1 ) e. RR )
45	42, 44	remulcld 10070	. . . . . . 7 |- ( K e. NN0 -> ( K x. ( K - 1 ) ) e. RR )
46	45	adantr 481	. . . . . 6 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( K x. ( K - 1 ) ) e. RR )
47		1red 10055	. . . . . 6 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> 1 e. RR )
48	22	nnrpd 11870	. . . . . . 7 |- ( P e. Prime -> P e. RR+ )
49	48	ad2antrl 764	. . . . . 6 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> P e. RR+ )
50		modaddabs 12708	. . . . . 6 |- ( ( ( K x. ( K - 1 ) ) e. RR /\ 1 e. RR /\ P e. RR+ ) -> ( ( ( ( K x. ( K - 1 ) ) mod P ) + ( 1 mod P ) ) mod P ) = ( ( ( K x. ( K - 1 ) ) + 1 ) mod P ) )
51	46, 47, 49, 50	syl3anc 1326	. . . . 5 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( ( K x. ( K - 1 ) ) mod P ) + ( 1 mod P ) ) mod P ) = ( ( ( K x. ( K - 1 ) ) + 1 ) mod P ) )
52		0p1e1 11132	. . . . . . 7 |- ( 0 + 1 ) = 1
53	52	oveq1i 6660	. . . . . 6 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
54	34, 35, 38	syl2anc 693	. . . . . . 7 |- ( P e. Prime -> ( 1 mod P ) = 1 )
55	54	ad2antrl 764	. . . . . 6 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( 1 mod P ) = 1 )
56	53, 55	syl5eq 2668	. . . . 5 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( 0 + 1 ) mod P ) = 1 )
57	41, 51, 56	3eqtr3d 2664	. . . 4 |- ( ( K e. NN0 /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( K x. ( K - 1 ) ) + 1 ) mod P ) = 1 )
58	15, 57	stoic3 1701	. . 3 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( K x. ( K - 1 ) ) + 1 ) mod P ) = 1 )
59	14, 58	eqtrd 2656	. 2 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` V ) mod P ) = 1 )
60	7, 59	eqtrd 2656	1 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( P ClWWalksN G ) ) mod P ) = 1 )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   RR+crp 11832   mod cmo 12668   #chash 13117   || cdvds 14983   Primecprime 15385  Vtxcvtx 25874   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-ac2 9285  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-inf 8349  df-oi 8415  df-card 8765  df-ac 8939  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-s3 13594  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-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsn 26725  df-wspthsnon 26726  df-clwwlks 26877  df-clwwlksn 26878  df-frgr 27121

This theorem is referenced by:  frgrreggt1  27251