Theorem numclwwlk5 24984

Description: Huneke: "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.)

Hypotheses

Ref	Expression
numclwwlk.c	|- C = ( n e. NN0 |-> ( ( V ClWWalksN E ) ` n ) )
numclwwlk.f	|- F = ( v e. V , n e. NN0 |-> { w e. ( C ` n ) | ( w ` 0 ) = v } )

Assertion

Ref	Expression
numclwwlk5	|- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 )

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

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

Proof of Theorem numclwwlk5

Dummy variables m u z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1000	. . . . 5 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> <. V , E >. RegUSGrph K )
2		simpr3 1005	. . . . 5 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> 2 || ( K - 1 ) )
3		simpr1 1003	. . . . 5 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> X e. V )
4		numclwwlk.c	. . . . . 6 |- C = ( n e. NN0 |-> ( ( V ClWWalksN E ) ` n ) )
5		numclwwlk.f	. . . . . 6 |- F = ( v e. V , n e. NN0 |-> { w e. ( C ` n ) | ( w ` 0 ) = v } )
6	4, 5	numclwwlk5lem 24983	. . . . 5 |- ( ( <. V , E >. RegUSGrph K /\ 2 || ( K - 1 ) /\ X e. V ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 )
7	1, 2, 3, 6	syl3anc 1229	. . . 4 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 )
8	7	a1i 11	. . 3 |- ( P = 2 -> ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 ) )
9		eleq1 2515	. . . . 5 |- ( P = 2 -> ( P e. Prime <-> 2 e. Prime ) )
10		breq1 4440	. . . . 5 |- ( P = 2 -> ( P || ( K - 1 ) <-> 2 || ( K - 1 ) ) )
11	9, 10	3anbi23d 1303	. . . 4 |- ( P = 2 -> ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) <-> ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) )
12	11	anbi2d 703	. . 3 |- ( P = 2 -> ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) <-> ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) ) )
13		oveq2 6289	. . . . . 6 |- ( P = 2 -> ( X F P ) = ( X F 2 ) )
14	13	fveq2d 5860	. . . . 5 |- ( P = 2 -> ( # ` ( X F P ) ) = ( # ` ( X F 2 ) ) )
15		id 22	. . . . 5 |- ( P = 2 -> P = 2 )
16	14, 15	oveq12d 6299	. . . 4 |- ( P = 2 -> ( ( # ` ( X F P ) ) mod P ) = ( ( # ` ( X F 2 ) ) mod 2 ) )
17	16	eqeq1d 2445	. . 3 |- ( P = 2 -> ( ( ( # ` ( X F P ) ) mod P ) = 1 <-> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 ) )
18	8, 12, 17	3imtr4d 268	. 2 |- ( P = 2 -> ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 ) )
19		3simpa 994	. . . . . . . 8 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( <. V , E >. RegUSGrph K /\ V FriendGrph E ) )
20	19	adantr 465	. . . . . . 7 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( <. V , E >. RegUSGrph K /\ V FriendGrph E ) )
21	20	adantl 466	. . . . . 6 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> ( <. V , E >. RegUSGrph K /\ V FriendGrph E ) )
22		simprl3 1044	. . . . . 6 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> V e. Fin )
23		simprr1 1045	. . . . . 6 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> X e. V )
24		prmn2uzge3 14114	. . . . . . . . . 10 |- ( ( P e. Prime /\ P =/= 2 ) -> P e. ( ZZ>= ` 3 ) )
25	24	ex 434	. . . . . . . . 9 |- ( P e. Prime -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
26	25	3ad2ant2 1019	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
27	26	adantl 466	. . . . . . 7 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
28	27	impcom 430	. . . . . 6 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> P e. ( ZZ>= ` 3 ) )
29		fveq1 5855	. . . . . . . . . . . 12 |- ( u = w -> ( u ` 0 ) = ( w ` 0 ) )
30	29	eqeq1d 2445	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` 0 ) = v <-> ( w ` 0 ) = v ) )
31		fveq1 5855	. . . . . . . . . . . 12 |- ( u = w -> ( u ` ( n - 2 ) ) = ( w ` ( n - 2 ) ) )
32	31, 29	eqeq12d 2465	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` ( n - 2 ) ) = ( u ` 0 ) <-> ( w ` ( n - 2 ) ) = ( w ` 0 ) ) )
33	30, 32	anbi12d 710	. . . . . . . . . 10 |- ( u = w -> ( ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) = ( u ` 0 ) ) <-> ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) ) )
34	33	cbvrabv 3094	. . . . . . . . 9 |- { u e. ( C ` n ) | ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) = ( u ` 0 ) ) } = { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) }
35	34	a1i 11	. . . . . . . 8 |- ( ( v e. V /\ n e. ( ZZ>= ` 2 ) ) -> { u e. ( C ` n ) | ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) = ( u ` 0 ) ) } = { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
36	35	mpt2eq3ia 6347	. . . . . . 7 |- ( v e. V , n e. ( ZZ>= ` 2 ) |-> { u e. ( C ` n ) | ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) = ( u ` 0 ) ) } ) = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
37		fveq2 5856	. . . . . . . . . . . 12 |- ( u = w -> ( lastS ` u ) = ( lastS ` w ) )
38	37	neeq1d 2720	. . . . . . . . . . 11 |- ( u = w -> ( ( lastS ` u ) =/= v <-> ( lastS ` w ) =/= v ) )
39	30, 38	anbi12d 710	. . . . . . . . . 10 |- ( u = w -> ( ( ( u ` 0 ) = v /\ ( lastS ` u ) =/= v ) <-> ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) ) )
40	39	cbvrabv 3094	. . . . . . . . 9 |- { u e. ( ( V WWalksN E ) ` n ) | ( ( u ` 0 ) = v /\ ( lastS ` u ) =/= v ) } = { w e. ( ( V WWalksN E ) ` n ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) }
41	40	a1i 11	. . . . . . . 8 |- ( ( v e. V /\ n e. NN0 ) -> { u e. ( ( V WWalksN E ) ` n ) | ( ( u ` 0 ) = v /\ ( lastS ` u ) =/= v ) } = { w e. ( ( V WWalksN E ) ` n ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
42	41	mpt2eq3ia 6347	. . . . . . 7 |- ( v e. V , n e. NN0 |-> { u e. ( ( V WWalksN E ) ` n ) | ( ( u ` 0 ) = v /\ ( lastS ` u ) =/= v ) } ) = ( v e. V , n e. NN0 |-> { w e. ( ( V WWalksN E ) ` n ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
43		eqeq2 2458	. . . . . . . . . 10 |- ( z = v -> ( ( u ` 0 ) = z <-> ( u ` 0 ) = v ) )
44	43	anbi1d 704	. . . . . . . . 9 |- ( z = v -> ( ( ( u ` 0 ) = z /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) <-> ( ( u ` 0 ) = v /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) ) )
45	44	rabbidv 3087	. . . . . . . 8 |- ( z = v -> { u e. ( C ` m ) | ( ( u ` 0 ) = z /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) } = { u e. ( C ` m ) | ( ( u ` 0 ) = v /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) } )
46		fveq2 5856	. . . . . . . . . 10 |- ( m = n -> ( C ` m ) = ( C ` n ) )
47		oveq1 6288	. . . . . . . . . . . . 13 |- ( m = n -> ( m - 2 ) = ( n - 2 ) )
48	47	fveq2d 5860	. . . . . . . . . . . 12 |- ( m = n -> ( u ` ( m - 2 ) ) = ( u ` ( n - 2 ) ) )
49	48	neeq1d 2720	. . . . . . . . . . 11 |- ( m = n -> ( ( u ` ( m - 2 ) ) =/= ( u ` 0 ) <-> ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) )
50	49	anbi2d 703	. . . . . . . . . 10 |- ( m = n -> ( ( ( u ` 0 ) = v /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) <-> ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) ) )
51	46, 50	rabeqbidv 3090	. . . . . . . . 9 |- ( m = n -> { u e. ( C ` m ) | ( ( u ` 0 ) = v /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) } = { u e. ( C ` n ) | ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) } )
52	31, 29	neeq12d 2722	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` ( n - 2 ) ) =/= ( u ` 0 ) <-> ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) )
53	30, 52	anbi12d 710	. . . . . . . . . 10 |- ( u = w -> ( ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) <-> ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) ) )
54	53	cbvrabv 3094	. . . . . . . . 9 |- { u e. ( C ` n ) | ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) } = { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) }
55	51, 54	syl6eq 2500	. . . . . . . 8 |- ( m = n -> { u e. ( C ` m ) | ( ( u ` 0 ) = v /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) } = { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )
56	45, 55	cbvmpt2v 6362	. . . . . . 7 |- ( z e. V , m e. NN0 |-> { u e. ( C ` m ) | ( ( u ` 0 ) = z /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) } ) = ( v e. V , n e. NN0 |-> { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )
57	4, 5, 36, 42, 56	numclwwlk3 24981	. . . . . 6 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E ) /\ ( V e. Fin /\ X e. V /\ P e. ( ZZ>= ` 3 ) ) ) -> ( # ` ( X F P ) ) = ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) + ( K ^ ( P - 2 ) ) ) )
58	21, 22, 23, 28, 57	syl13anc 1231	. . . . 5 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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 ) ) ) )
59	58	oveq1d 6296	. . . 4 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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 ) )
60		rusgraprop 24801	. . . . . . . . . . . . 13 |- ( <. V , E >. RegUSGrph K -> ( V USGrph E /\ K e. NN0 /\ A. v e. V ( ( V VDeg E ) ` v ) = K ) )
61		nn0z 10893	. . . . . . . . . . . . . 14 |- ( K e. NN0 -> K e. ZZ )
62	61	3ad2ant2 1019	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ K e. NN0 /\ A. v e. V ( ( V VDeg E ) ` v ) = K ) -> K e. ZZ )
63	60, 62	syl 16	. . . . . . . . . . . 12 |- ( <. V , E >. RegUSGrph K -> K e. ZZ )
64	63	zred 10974	. . . . . . . . . . 11 |- ( <. V , E >. RegUSGrph K -> K e. RR )
65		peano2rem 9891	. . . . . . . . . . 11 |- ( K e. RR -> ( K - 1 ) e. RR )
66	64, 65	syl 16	. . . . . . . . . 10 |- ( <. V , E >. RegUSGrph K -> ( K - 1 ) e. RR )
67	66	3ad2ant1 1018	. . . . . . . . 9 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( K - 1 ) e. RR )
68	67	adantr 465	. . . . . . . 8 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( K - 1 ) e. RR )
69		rusisusgra 24803	. . . . . . . . . . . . . . 15 |- ( <. V , E >. RegUSGrph K -> V USGrph E )
70		usgrav 24210	. . . . . . . . . . . . . . . 16 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
71	70	simprd 463	. . . . . . . . . . . . . . 15 |- ( V USGrph E -> E e. _V )
72	69, 71	syl 16	. . . . . . . . . . . . . 14 |- ( <. V , E >. RegUSGrph K -> E e. _V )
73	72	anim1i 568	. . . . . . . . . . . . 13 |- ( ( <. V , E >. RegUSGrph K /\ V e. Fin ) -> ( E e. _V /\ V e. Fin ) )
74	73	ancomd 451	. . . . . . . . . . . 12 |- ( ( <. V , E >. RegUSGrph K /\ V e. Fin ) -> ( V e. Fin /\ E e. _V ) )
75	74	3adant2 1016	. . . . . . . . . . 11 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( V e. Fin /\ E e. _V ) )
76		prmm2nn0 14115	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P - 2 ) e. NN0 )
77	76	anim2i 569	. . . . . . . . . . . 12 |- ( ( X e. V /\ P e. Prime ) -> ( X e. V /\ ( P - 2 ) e. NN0 ) )
78	77	3adant3 1017	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( X e. V /\ ( P - 2 ) e. NN0 ) )
79	4, 5	numclwwlkffin 24954	. . . . . . . . . . 11 |- ( ( ( V e. Fin /\ E e. _V ) /\ ( X e. V /\ ( P - 2 ) e. NN0 ) ) -> ( X F ( P - 2 ) ) e. Fin )
80	75, 78, 79	syl2an 477	. . . . . . . . . 10 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( X F ( P - 2 ) ) e. Fin )
81		hashcl 12407	. . . . . . . . . 10 |- ( ( X F ( P - 2 ) ) e. Fin -> ( # ` ( X F ( P - 2 ) ) ) e. NN0 )
82	80, 81	syl 16	. . . . . . . . 9 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( # ` ( X F ( P - 2 ) ) ) e. NN0 )
83	82	nn0red 10859	. . . . . . . 8 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( # ` ( X F ( P - 2 ) ) ) e. RR )
84	68, 83	remulcld 9627	. . . . . . 7 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) e. RR )
85	64	3ad2ant1 1018	. . . . . . . 8 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. RR )
86	76	3ad2ant2 1019	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P - 2 ) e. NN0 )
87		reexpcl 12162	. . . . . . . 8 |- ( ( K e. RR /\ ( P - 2 ) e. NN0 ) -> ( K ^ ( P - 2 ) ) e. RR )
88	85, 86, 87	syl2an 477	. . . . . . 7 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( K ^ ( P - 2 ) ) e. RR )
89		prmnn 14097	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
90	89	nnrpd 11264	. . . . . . . . 9 |- ( P e. Prime -> P e. RR+ )
91	90	3ad2ant2 1019	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. RR+ )
92	91	adantl 466	. . . . . . 7 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> P e. RR+ )
93	84, 88, 92	3jca 1177	. . . . . 6 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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+ ) )
94	93	adantl 466	. . . . 5 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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+ ) )
95		modaddabs 12013	. . . . . 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 ) )
96	95	eqcomd 2451	. . . . 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 ) )
97	94, 96	syl 16	. . . 4 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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 ) )
98	89	3ad2ant2 1019	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. NN )
99	98	adantl 466	. . . . . . . . . 10 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> P e. NN )
100		peano2zm 10913	. . . . . . . . . . . . 13 |- ( K e. ZZ -> ( K - 1 ) e. ZZ )
101	63, 100	syl 16	. . . . . . . . . . . 12 |- ( <. V , E >. RegUSGrph K -> ( K - 1 ) e. ZZ )
102	101	3ad2ant1 1018	. . . . . . . . . . 11 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( K - 1 ) e. ZZ )
103	102	adantr 465	. . . . . . . . . 10 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( K - 1 ) e. ZZ )
104	82	nn0zd 10972	. . . . . . . . . 10 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( # ` ( X F ( P - 2 ) ) ) e. ZZ )
105	99, 103, 104	3jca 1177	. . . . . . . . 9 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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 ) )
106		simpr3 1005	. . . . . . . . 9 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> P || ( K - 1 ) )
107		mulmoddvds 13921	. . . . . . . . 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 ) )
108	105, 106, 107	sylc 60	. . . . . . . 8 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( ( K - 1 ) x. ( # ` ( X F ( P - 2 ) ) ) ) mod P ) = 0 )
109	63	3ad2ant1 1018	. . . . . . . . . 10 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. ZZ )
110		simp2 998	. . . . . . . . . 10 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. Prime )
111	109, 110	anim12ci 567	. . . . . . . . 9 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( P e. Prime /\ K e. ZZ ) )
112		powm2modprm 14205	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. ZZ ) -> ( P || ( K - 1 ) -> ( ( K ^ ( P - 2 ) ) mod P ) = 1 ) )
113	111, 106, 112	sylc 60	. . . . . . . 8 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( K ^ ( P - 2 ) ) mod P ) = 1 )
114	108, 113	oveq12d 6299	. . . . . . 7 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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 ) )
115	114	oveq1d 6296	. . . . . 6 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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 ) )
116		0p1e1 10653	. . . . . . . . . 10 |- ( 0 + 1 ) = 1
117	116	oveq1i 6291	. . . . . . . . 9 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
118	89	nnred 10557	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR )
119		prmgt1 14113	. . . . . . . . . 10 |- ( P e. Prime -> 1 < P )
120		1mod 12007	. . . . . . . . . 10 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
121	118, 119, 120	syl2anc 661	. . . . . . . . 9 |- ( P e. Prime -> ( 1 mod P ) = 1 )
122	117, 121	syl5eq 2496	. . . . . . . 8 |- ( P e. Prime -> ( ( 0 + 1 ) mod P ) = 1 )
123	122	3ad2ant2 1019	. . . . . . 7 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( ( 0 + 1 ) mod P ) = 1 )
124	123	adantl 466	. . . . . 6 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( 0 + 1 ) mod P ) = 1 )
125	115, 124	eqtrd 2484	. . . . 5 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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 )
126	125	adantl 466	. . . 4 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ 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 )
127	59, 97, 126	3eqtrd 2488	. . 3 |- ( ( P =/= 2 /\ ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 )
128	127	ex 434	. 2 |- ( P =/= 2 -> ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 ) )
129	18, 128	pm2.61ine 2756	1 |- ( ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) /\ ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) ) -> ( ( # ` ( X F P ) ) mod P ) = 1 )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   =/= wne 2638   A.wral 2793   {crab 2797   _Vcvv 3095   <.cop 4020   class class class wbr 4437   |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   |-> cmpt2 6283   Fincfn 7518   RRcr 9494   0cc0 9495   1c1 9496   + caddc 9498   x. cmul 9500   < clt 9631   - cmin 9810   NNcn 10542   2c2 10591   3c3 10592   NN0cn0 10801   ZZcz 10870   ZZ>=cuz 11090   RR+crp 11229   mod cmo 11975   ^cexp 12145   #chash 12384   lastS clsw 12514   || cdvds 13863   Primecprime 14094   USGrph cusg 24202   WWalksN cwwlkn 24550   ClWWalksN cclwwlkn 24621   VDeg cvdg 24765   RegUSGrph crusgra 24795   FriendGrph cfrgra 24860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-xadd 11328  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-hash 12385  df-word 12521  df-lsw 12522  df-concat 12523  df-s1 12524  df-substr 12525  df-s2 12792  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-sum 13488  df-dvds 13864  df-gcd 14022  df-prm 14095  df-phi 14173  df-usgra 24205  df-nbgra 24292  df-wlk 24380  df-wwlk 24551  df-wwlkn 24552  df-clwwlk 24623  df-clwwlkn 24624  df-vdgr 24766  df-rgra 24796  df-rusgra 24797  df-frgra 24861

This theorem is referenced by:  numclwwlk6  24985