Theorem numclwwlk5 27246

Description: Statement 13 in [Huneke] p. 2: "Let p be a prime divisor of k-1; then f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 2-Jun-2021.)

Hypotheses

Ref	Expression
numclwwlk3.v	|- V = (Vtx ` G )
numclwwlk3.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )

Assertion

Ref	Expression
numclwwlk5	|- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 )

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

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

Proof of Theorem numclwwlk5

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> G RegUSGraph K )
2		simpr1 1067	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> X e. V )
3		numclwwlk3.v	. . . . . . . . . . . . 13 |- V = (Vtx ` G )
4	3	finrusgrfusgr 26461	. . . . . . . . . . . 12 |- ( ( G RegUSGraph K /\ V e. Fin ) -> G e. FinUSGraph )
5	4	3adant2 1080	. . . . . . . . . . 11 |- ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) -> G e. FinUSGraph )
6	5	adantl 482	. . . . . . . . . 10 |- ( ( X e. V /\ ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) ) -> G e. FinUSGraph )
7		simpr1 1067	. . . . . . . . . 10 |- ( ( X e. V /\ ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) ) -> G RegUSGraph K )
8		ne0i 3921	. . . . . . . . . . 11 |- ( X e. V -> V =/= (/) )
9	8	adantr 481	. . . . . . . . . 10 |- ( ( X e. V /\ ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) ) -> V =/= (/) )
10	3	frusgrnn0 26467	. . . . . . . . . 10 |- ( ( G e. FinUSGraph /\ G RegUSGraph K /\ V =/= (/) ) -> K e. NN0 )
11	6, 7, 9, 10	syl3anc 1326	. . . . . . . . 9 |- ( ( X e. V /\ ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) ) -> K e. NN0 )
12	11	ex 450	. . . . . . . 8 |- ( X e. V -> ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) -> K e. NN0 ) )
13	12	3ad2ant1 1082	. . . . . . 7 |- ( ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) -> ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) -> K e. NN0 ) )
14	13	impcom 446	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> K e. NN0 )
15	1, 2, 14	3jca 1242	. . . . 5 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> ( G RegUSGraph K /\ X e. V /\ K e. NN0 ) )
16		simpr3 1069	. . . . 5 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> 2 || ( K - 1 ) )
17		numclwwlk3.f	. . . . . 6 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
18	3, 17	numclwwlk5lem 27245	. . . . 5 |- ( ( G RegUSGraph K /\ X e. V /\ K e. NN0 ) -> ( 2 || ( K - 1 ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 ) )
19	15, 16, 18	sylc 65	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 )
20	19	a1i 11	. . 3 |- ( P = 2 -> ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 ) )
21		eleq1 2689	. . . . 5 |- ( P = 2 -> ( P e. Prime <-> 2 e. Prime ) )
22		breq1 4656	. . . . 5 |- ( P = 2 -> ( P || ( K - 1 ) <-> 2 || ( K - 1 ) ) )
23	21, 22	3anbi23d 1402	. . . 4 |- ( P = 2 -> ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) <-> ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) )
24	23	anbi2d 740	. . 3 |- ( P = 2 -> ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) <-> ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) ) )
25		oveq2 6658	. . . . . 6 |- ( P = 2 -> ( X F P ) = ( X F 2 ) )
26	25	fveq2d 6195	. . . . 5 |- ( P = 2 -> ( # ` ( X F P ) ) = ( # ` ( X F 2 ) ) )
27		id 22	. . . . 5 |- ( P = 2 -> P = 2 )
28	26, 27	oveq12d 6668	. . . 4 |- ( P = 2 -> ( ( # ` ( X F P ) ) mod P ) = ( ( # ` ( X F 2 ) ) mod 2 ) )
29	28	eqeq1d 2624	. . 3 |- ( P = 2 -> ( ( ( # ` ( X F P ) ) mod P ) = 1 <-> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 ) )
30	20, 24, 29	3imtr4d 283	. 2 |- ( P = 2 -> ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 ) )
31		3simpa 1058	. . . . . . . 8 |- ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) -> ( G RegUSGraph K /\ G e. FriendGraph ) )
32	31	adantr 481	. . . . . . 7 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( G RegUSGraph K /\ G e. FriendGraph ) )
33	32	adantl 482	. . . . . 6 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> ( G RegUSGraph K /\ G e. FriendGraph ) )
34		simprl3 1108	. . . . . 6 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> V e. Fin )
35		simprr1 1109	. . . . . 6 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> X e. V )
36		eldifsn 4317	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
37		oddprmge3 15412	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) )
38	36, 37	sylbir 225	. . . . . . . . . 10 |- ( ( P e. Prime /\ P =/= 2 ) -> P e. ( ZZ>= ` 3 ) )
39	38	ex 450	. . . . . . . . 9 |- ( P e. Prime -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
40	39	3ad2ant2 1083	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
41	40	adantl 482	. . . . . . 7 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
42	41	impcom 446	. . . . . 6 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> P e. ( ZZ>= ` 3 ) )
43	3, 17	numclwwlk3 27243	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V e. Fin /\ X e. V /\ P e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X F P ) ) = ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) + ( K ^ ( P - 2 ) ) ) )
44	33, 34, 35, 42, 43	syl13anc 1328	. . . . 5 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> ( # ` ( X F P ) ) = ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) + ( K ^ ( P - 2 ) ) ) )
45	44	oveq1d 6665	. . . 4 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) + ( K ^ ( P - 2 ) ) ) mod P ) )
46	12	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) -> K e. NN0 ) )
47	46	impcom 446	. . . . . . . . . 10 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> K e. NN0 )
48	47	nn0zd 11480	. . . . . . . . 9 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> K e. ZZ )
49		peano2zm 11420	. . . . . . . . 9 |- ( K e. ZZ -> ( K - 1 ) e. ZZ )
50		zre 11381	. . . . . . . . 9 |- ( ( K - 1 ) e. ZZ -> ( K - 1 ) e. RR )
51	48, 49, 50	3syl 18	. . . . . . . 8 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( K - 1 ) e. RR )
52		simpl3 1066	. . . . . . . . . . 11 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> V e. Fin )
53		simpr1 1067	. . . . . . . . . . 11 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> X e. V )
54		prmm2nn0 15410	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P - 2 ) e. NN0 )
55	54	3ad2ant2 1083	. . . . . . . . . . . 12 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P - 2 ) e. NN0 )
56	55	adantl 482	. . . . . . . . . . 11 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( P - 2 ) e. NN0 )
57	52, 53, 56	3jca 1242	. . . . . . . . . 10 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( V e. Fin /\ X e. V /\ ( P - 2 ) e. NN0 ) )
58	17, 3	numclwwlkffin0 27215	. . . . . . . . . 10 |- ( ( V e. Fin /\ X e. V /\ ( P - 2 ) e. NN0 ) -> ( X F ( P - 2 ) ) e. Fin )
59		hashcl 13147	. . . . . . . . . 10 |- ( ( X F ( P - 2 ) ) e. Fin -> ( # ` ( X F ( P - 2 ) ) ) e. NN0 )
60	57, 58, 59	3syl 18	. . . . . . . . 9 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( # ` ( X F ( P - 2 ) ) ) e. NN0 )
61	60	nn0red 11352	. . . . . . . 8 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( # ` ( X F ( P - 2 ) ) ) e. RR )
62	51, 61	remulcld 10070	. . . . . . 7 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) e. RR )
63	47	nn0red 11352	. . . . . . . 8 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> K e. RR )
64	63, 56	reexpcld 13025	. . . . . . 7 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( K ^ ( P - 2 ) ) e. RR )
65		prmnn 15388	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
66	65	nnrpd 11870	. . . . . . . . 9 |- ( P e. Prime -> P e. RR+ )
67	66	3ad2ant2 1083	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. RR+ )
68	67	adantl 482	. . . . . . 7 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> P e. RR+ )
69	62, 64, 68	3jca 1242	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) e. RR /\ ( K ^ ( P - 2 ) ) e. RR /\ P e. RR+ ) )
70	69	adantl 482	. . . . 5 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) e. RR /\ ( K ^ ( P - 2 ) ) e. RR /\ P e. RR+ ) )
71		modaddabs 12708	. . . . . 6 |- ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) e. RR /\ ( K ^ ( P - 2 ) ) e. RR /\ P e. RR+ ) -> ( ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) + ( ( K ^ ( P - 2 ) ) mod P ) ) mod P ) = ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) + ( K ^ ( P - 2 ) ) ) mod P ) )
72	71	eqcomd 2628	. . . . 5 |- ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) e. RR /\ ( K ^ ( P - 2 ) ) e. RR /\ P e. RR+ ) -> ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) + ( K ^ ( P - 2 ) ) ) mod P ) = ( ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) + ( ( K ^ ( P - 2 ) ) mod P ) ) mod P ) )
73	70, 72	syl 17	. . . 4 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) + ( K ^ ( P - 2 ) ) ) mod P ) = ( ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) + ( ( K ^ ( P - 2 ) ) mod P ) ) mod P ) )
74	65	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. NN )
75	74	adantl 482	. . . . . . . . . 10 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> P e. NN )
76		nn0z 11400	. . . . . . . . . . 11 |- ( K e. NN0 -> K e. ZZ )
77	47, 76, 49	3syl 18	. . . . . . . . . 10 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( K - 1 ) e. ZZ )
78	60	nn0zd 11480	. . . . . . . . . 10 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( # ` ( X F ( P - 2 ) ) ) e. ZZ )
79	75, 77, 78	3jca 1242	. . . . . . . . 9 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( P e. NN /\ ( K - 1 ) e. ZZ /\ ( # ` ( X F ( P - 2 ) ) ) e. ZZ ) )
80		simpr3 1069	. . . . . . . . 9 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> P || ( K - 1 ) )
81		mulmoddvds 15051	. . . . . . . . 9 |- ( ( P e. NN /\ ( K - 1 ) e. ZZ /\ ( # ` ( X F ( P - 2 ) ) ) e. ZZ ) -> ( P || ( K - 1 ) -> ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) = 0 ) )
82	79, 80, 81	sylc 65	. . . . . . . 8 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) = 0 )
83		simpr2 1068	. . . . . . . . . 10 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> P e. Prime )
84	83, 48	jca 554	. . . . . . . . 9 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( P e. Prime /\ K e. ZZ ) )
85		powm2modprm 15508	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. ZZ ) -> ( P || ( K - 1 ) -> ( ( K ^ ( P - 2 ) ) mod P ) = 1 ) )
86	84, 80, 85	sylc 65	. . . . . . . 8 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( K ^ ( P - 2 ) ) mod P ) = 1 )
87	82, 86	oveq12d 6668	. . . . . . 7 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) + ( ( K ^ ( P - 2 ) ) mod P ) ) = ( 0 + 1 ) )
88	87	oveq1d 6665	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) + ( ( K ^ ( P - 2 ) ) mod P ) ) mod P ) = ( ( 0 + 1 ) mod P ) )
89		0p1e1 11132	. . . . . . . . . 10 |- ( 0 + 1 ) = 1
90	89	oveq1i 6660	. . . . . . . . 9 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
91	65	nnred 11035	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR )
92		prmgt1 15409	. . . . . . . . . 10 |- ( P e. Prime -> 1 < P )
93		1mod 12702	. . . . . . . . . 10 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
94	91, 92, 93	syl2anc 693	. . . . . . . . 9 |- ( P e. Prime -> ( 1 mod P ) = 1 )
95	90, 94	syl5eq 2668	. . . . . . . 8 |- ( P e. Prime -> ( ( 0 + 1 ) mod P ) = 1 )
96	95	3ad2ant2 1083	. . . . . . 7 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( ( 0 + 1 ) mod P ) = 1 )
97	96	adantl 482	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( 0 + 1 ) mod P ) = 1 )
98	88, 97	eqtrd 2656	. . . . 5 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) + ( ( K ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 )
99	98	adantl 482	. . . 4 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> ( ( ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) + ( ( K ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 )
100	45, 73, 99	3eqtrd 2660	. . 3 |- ( ( P =/= 2 /\ ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 )
101	100	ex 450	. 2 |- ( P =/= 2 -> ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 ) )
102	30, 101	pm2.61ine 2877	1 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   \ cdif 3571   (/)c0 3915   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   mod cmo 12668   ^cexp 12860   #chash 13117   || 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:  numclwwlk6  27248