Theorem numclwwlk5 31831

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 994	. . . . 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 999	. . . . 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 997	. . . . 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 31830	. . . . 5 |- ( ( <. V , E >. RegUSGrph K /\ 2 || ( K - 1 ) /\ X e. V ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 )
7	1, 2, 3, 6	syl3anc 1223	. . . 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 2532	. . . . 5 |- ( P = 2 -> ( P e. Prime <-> 2 e. Prime ) )
10		breq1 4443	. . . . 5 |- ( P = 2 -> ( P || ( K - 1 ) <-> 2 || ( K - 1 ) ) )
11	9, 10	3anbi23d 1297	. . . 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 6283	. . . . . 6 |- ( P = 2 -> ( X F P ) = ( X F 2 ) )
14	13	fveq2d 5861	. . . . 5 |- ( P = 2 -> ( # ` ( X F P ) ) = ( # ` ( X F 2 ) ) )
15		id 22	. . . . 5 |- ( P = 2 -> P = 2 )
16	14, 15	oveq12d 6293	. . . 4 |- ( P = 2 -> ( ( # ` ( X F P ) ) mod P ) = ( ( # ` ( X F 2 ) ) mod 2 ) )
17	16	eqeq1d 2462	. . 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 988	. . . . . . . 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 1038	. . . . . 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 1039	. . . . . 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 14085	. . . . . . . . . 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 1013	. . . . . . . 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 5856	. . . . . . . . . . . 12 |- ( u = w -> ( u ` 0 ) = ( w ` 0 ) )
30	29	eqeq1d 2462	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` 0 ) = v <-> ( w ` 0 ) = v ) )
31		fveq1 5856	. . . . . . . . . . . 12 |- ( u = w -> ( u ` ( n - 2 ) ) = ( w ` ( n - 2 ) ) )
32	31, 29	eqeq12d 2482	. . . . . . . . . . 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 3105	. . . . . . . . 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 6337	. . . . . . 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 5857	. . . . . . . . . . . 12 |- ( u = w -> ( lastS ` u ) = ( lastS ` w ) )
38	37	neeq1d 2737	. . . . . . . . . . 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 3105	. . . . . . . . 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 6337	. . . . . . 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 2475	. . . . . . . . . 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 3098	. . . . . . . 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 5857	. . . . . . . . . 10 |- ( m = n -> ( C ` m ) = ( C ` n ) )
47		oveq1 6282	. . . . . . . . . . . . 13 |- ( m = n -> ( m - 2 ) = ( n - 2 ) )
48	47	fveq2d 5861	. . . . . . . . . . . 12 |- ( m = n -> ( u ` ( m - 2 ) ) = ( u ` ( n - 2 ) ) )
49	48	neeq1d 2737	. . . . . . . . . . 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 3101	. . . . . . . . 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 2739	. . . . . . . . . . 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 3105	. . . . . . . . 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 2517	. . . . . . . 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 6352	. . . . . . 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 31828	. . . . . 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 1225	. . . . 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 6290	. . . 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 24591	. . . . . . . . . . . . 13 |- ( <. V , E >. RegUSGrph K -> ( V USGrph E /\ K e. NN0 /\ A. v e. V ( ( V VDeg E ) ` v ) = K ) )
61		nn0z 10876	. . . . . . . . . . . . . 14 |- ( K e. NN0 -> K e. ZZ )
62	61	3ad2ant2 1013	. . . . . . . . . . . . 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 10955	. . . . . . . . . . 11 |- ( <. V , E >. RegUSGrph K -> K e. RR )
65		peano2rem 9875	. . . . . . . . . . 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 1012	. . . . . . . . 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 24593	. . . . . . . . . . . . . . 15 |- ( <. V , E >. RegUSGrph K -> V USGrph E )
70		usgrav 24001	. . . . . . . . . . . . . . . 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 1010	. . . . . . . . . . 11 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( V e. Fin /\ E e. _V ) )
76		prmm2nn0 14086	. . . . . . . . . . . . 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 1011	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( X e. V /\ ( P - 2 ) e. NN0 ) )
79	4, 5	numclwwlkffin 31801	. . . . . . . . . . 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 12383	. . . . . . . . . 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 10842	. . . . . . . 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 9613	. . . . . . 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 1012	. . . . . . . 8 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. RR )
86	76	3ad2ant2 1013	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P - 2 ) e. NN0 )
87		reexpcl 12139	. . . . . . . 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 14068	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
90	89	nnrpd 11244	. . . . . . . . 9 |- ( P e. Prime -> P e. RR+ )
91	90	3ad2ant2 1013	. . . . . . . 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 1171	. . . . . 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 11990	. . . . . 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 2468	. . . . 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 1013	. . . . . . . . . . 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 10895	. . . . . . . . . . . . 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 1012	. . . . . . . . . . 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 10953	. . . . . . . . . 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 1171	. . . . . . . . 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 999	. . . . . . . . 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 13892	. . . . . . . . 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 1012	. . . . . . . . . 10 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. ZZ )
110		simp2 992	. . . . . . . . . 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 14176	. . . . . . . . 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 6293	. . . . . . 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 6290	. . . . . 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 10636	. . . . . . . . . 10 |- ( 0 + 1 ) = 1
117	116	oveq1i 6285	. . . . . . . . 9 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
118	89	nnred 10540	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR )
119		prmgt1 14084	. . . . . . . . . 10 |- ( P e. Prime -> 1 < P )
120		1mod 11984	. . . . . . . . . 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 2513	. . . . . . . 8 |- ( P e. Prime -> ( ( 0 + 1 ) mod P ) = 1 )
123	122	3ad2ant2 1013	. . . . . . 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 2501	. . . . 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 2505	. . 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 2773	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 968   = wceq 1374   e. wcel 1762   =/= wne 2655   A.wral 2807   {crab 2811   _Vcvv 3106   <.cop 4026   class class class wbr 4440   |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   |-> cmpt2 6277   Fincfn 7506   RRcr 9480   0cc0 9481   1c1 9482   + caddc 9484   x. cmul 9486   < clt 9617   - cmin 9794   NNcn 10525   2c2 10574   3c3 10575   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   RR+crp 11209   mod cmo 11952   ^cexp 12122   #chash 12360   lastS clsw 12488   || cdivides 13836   Primecprime 14065   USGrph cusg 23993   WWalksN cwwlkn 24340   ClWWalksN cclwwlkn 24411   VDeg cvdg 24555   RegUSGrph crusgra 24585   FriendGrph cfrgra 31706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-xadd 11308  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-hash 12361  df-word 12495  df-lsw 12496  df-concat 12497  df-s1 12498  df-substr 12499  df-s2 12763  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-dvds 13837  df-gcd 13993  df-prm 14066  df-phi 14144  df-usgra 23996  df-nbgra 24082  df-wlk 24170  df-wwlk 24341  df-wwlkn 24342  df-clwwlk 24413  df-clwwlkn 24414  df-vdgr 24556  df-rgra 24586  df-rusgra 24587  df-frgra 31707

This theorem is referenced by:  numclwwlk6  31832