Theorem numclwwlk5 25919

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.)

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 1033	. . . . 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 1038	. . . . 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 1036	. . . . 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 25918	. . . . 5 |- ( ( <. V , E >. RegUSGrph K /\ 2 || ( K - 1 ) /\ X e. V ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 )
7	1, 2, 3, 6	syl3anc 1292	. . . 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 2537	. . . . 5 |- ( P = 2 -> ( P e. Prime <-> 2 e. Prime ) )
10		breq1 4398	. . . . 5 |- ( P = 2 -> ( P || ( K - 1 ) <-> 2 || ( K - 1 ) ) )
11	9, 10	3anbi23d 1368	. . . 4 |- ( P = 2 -> ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) <-> ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) )
12	11	anbi2d 718	. . 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 6316	. . . . . 6 |- ( P = 2 -> ( X F P ) = ( X F 2 ) )
14	13	fveq2d 5883	. . . . 5 |- ( P = 2 -> ( # ` ( X F P ) ) = ( # ` ( X F 2 ) ) )
15		id 22	. . . . 5 |- ( P = 2 -> P = 2 )
16	14, 15	oveq12d 6326	. . . 4 |- ( P = 2 -> ( ( # ` ( X F P ) ) mod P ) = ( ( # ` ( X F 2 ) ) mod 2 ) )
17	16	eqeq1d 2473	. . 3 |- ( P = 2 -> ( ( ( # ` ( X F P ) ) mod P ) = 1 <-> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 ) )
18	8, 12, 17	3imtr4d 276	. 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 1027	. . . . . . . 8 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( <. V , E >. RegUSGrph K /\ V FriendGrph E ) )
20	19	adantr 472	. . . . . . 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 473	. . . . . 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 1077	. . . . . 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 1078	. . . . . 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 14723	. . . . . . . . . 10 |- ( ( P e. Prime /\ P =/= 2 ) -> P e. ( ZZ>= ` 3 ) )
25	24	ex 441	. . . . . . . . 9 |- ( P e. Prime -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
26	25	3ad2ant2 1052	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
27	26	adantl 473	. . . . . . 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 437	. . . . . 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 5878	. . . . . . . . . . . 12 |- ( u = w -> ( u ` 0 ) = ( w ` 0 ) )
30	29	eqeq1d 2473	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` 0 ) = v <-> ( w ` 0 ) = v ) )
31		fveq1 5878	. . . . . . . . . . . 12 |- ( u = w -> ( u ` ( n - 2 ) ) = ( w ` ( n - 2 ) ) )
32	31, 29	eqeq12d 2486	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` ( n - 2 ) ) = ( u ` 0 ) <-> ( w ` ( n - 2 ) ) = ( w ` 0 ) ) )
33	30, 32	anbi12d 725	. . . . . . . . . 10 |- ( u = w -> ( ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) = ( u ` 0 ) ) <-> ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) ) )
34	33	cbvrabv 3030	. . . . . . . . 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 6375	. . . . . . 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 5879	. . . . . . . . . . . 12 |- ( u = w -> ( lastS ` u ) = ( lastS ` w ) )
38	37	neeq1d 2702	. . . . . . . . . . 11 |- ( u = w -> ( ( lastS ` u ) =/= v <-> ( lastS ` w ) =/= v ) )
39	30, 38	anbi12d 725	. . . . . . . . . 10 |- ( u = w -> ( ( ( u ` 0 ) = v /\ ( lastS ` u ) =/= v ) <-> ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) ) )
40	39	cbvrabv 3030	. . . . . . . . 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 6375	. . . . . . 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 2482	. . . . . . . . . 10 |- ( z = v -> ( ( u ` 0 ) = z <-> ( u ` 0 ) = v ) )
44	43	anbi1d 719	. . . . . . . . 9 |- ( z = v -> ( ( ( u ` 0 ) = z /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) <-> ( ( u ` 0 ) = v /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) ) )
45	44	rabbidv 3022	. . . . . . . 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 5879	. . . . . . . . . 10 |- ( m = n -> ( C ` m ) = ( C ` n ) )
47		oveq1 6315	. . . . . . . . . . . . 13 |- ( m = n -> ( m - 2 ) = ( n - 2 ) )
48	47	fveq2d 5883	. . . . . . . . . . . 12 |- ( m = n -> ( u ` ( m - 2 ) ) = ( u ` ( n - 2 ) ) )
49	48	neeq1d 2702	. . . . . . . . . . 11 |- ( m = n -> ( ( u ` ( m - 2 ) ) =/= ( u ` 0 ) <-> ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) )
50	49	anbi2d 718	. . . . . . . . . 10 |- ( m = n -> ( ( ( u ` 0 ) = v /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) <-> ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) ) )
51	46, 50	rabeqbidv 3026	. . . . . . . . 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 2704	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` ( n - 2 ) ) =/= ( u ` 0 ) <-> ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) )
53	30, 52	anbi12d 725	. . . . . . . . . 10 |- ( u = w -> ( ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) <-> ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) ) )
54	53	cbvrabv 3030	. . . . . . . . 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 2521	. . . . . . . 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 6390	. . . . . . 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 25916	. . . . . 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 1294	. . . . 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 6323	. . . 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 25736	. . . . . . . . . . . . 13 |- ( <. V , E >. RegUSGrph K -> ( V USGrph E /\ K e. NN0 /\ A. v e. V ( ( V VDeg E ) ` v ) = K ) )
61		nn0z 10984	. . . . . . . . . . . . . 14 |- ( K e. NN0 -> K e. ZZ )
62	61	3ad2ant2 1052	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ K e. NN0 /\ A. v e. V ( ( V VDeg E ) ` v ) = K ) -> K e. ZZ )
63	60, 62	syl 17	. . . . . . . . . . . 12 |- ( <. V , E >. RegUSGrph K -> K e. ZZ )
64	63	zred 11063	. . . . . . . . . . 11 |- ( <. V , E >. RegUSGrph K -> K e. RR )
65		peano2rem 9961	. . . . . . . . . . 11 |- ( K e. RR -> ( K - 1 ) e. RR )
66	64, 65	syl 17	. . . . . . . . . 10 |- ( <. V , E >. RegUSGrph K -> ( K - 1 ) e. RR )
67	66	3ad2ant1 1051	. . . . . . . . 9 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( K - 1 ) e. RR )
68	67	adantr 472	. . . . . . . 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 25738	. . . . . . . . . . . . . . 15 |- ( <. V , E >. RegUSGrph K -> V USGrph E )
70		usgrav 25144	. . . . . . . . . . . . . . . 16 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
71	70	simprd 470	. . . . . . . . . . . . . . 15 |- ( V USGrph E -> E e. _V )
72	69, 71	syl 17	. . . . . . . . . . . . . 14 |- ( <. V , E >. RegUSGrph K -> E e. _V )
73	72	anim1i 578	. . . . . . . . . . . . 13 |- ( ( <. V , E >. RegUSGrph K /\ V e. Fin ) -> ( E e. _V /\ V e. Fin ) )
74	73	ancomd 458	. . . . . . . . . . . 12 |- ( ( <. V , E >. RegUSGrph K /\ V e. Fin ) -> ( V e. Fin /\ E e. _V ) )
75	74	3adant2 1049	. . . . . . . . . . 11 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( V e. Fin /\ E e. _V ) )
76		prmm2nn0 14724	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P - 2 ) e. NN0 )
77	76	anim2i 579	. . . . . . . . . . . 12 |- ( ( X e. V /\ P e. Prime ) -> ( X e. V /\ ( P - 2 ) e. NN0 ) )
78	77	3adant3 1050	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( X e. V /\ ( P - 2 ) e. NN0 ) )
79	4, 5	numclwwlkffin 25889	. . . . . . . . . . 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 485	. . . . . . . . . 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 12576	. . . . . . . . . 10 |- ( ( X F ( P - 2 ) ) e. Fin -> ( # ` ( X F ( P - 2 ) ) ) e. NN0 )
82	80, 81	syl 17	. . . . . . . . 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 10950	. . . . . . . 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 9689	. . . . . . 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 1051	. . . . . . . 8 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. RR )
86	76	3ad2ant2 1052	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P - 2 ) e. NN0 )
87		reexpcl 12327	. . . . . . . 8 |- ( ( K e. RR /\ ( P - 2 ) e. NN0 ) -> ( K ^ ( P - 2 ) ) e. RR )
88	85, 86, 87	syl2an 485	. . . . . . 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 14704	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
90	89	nnrpd 11362	. . . . . . . . 9 |- ( P e. Prime -> P e. RR+ )
91	90	3ad2ant2 1052	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. RR+ )
92	91	adantl 473	. . . . . . 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 1210	. . . . . 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 473	. . . . 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 12168	. . . . . 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 2477	. . . . 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 17	. . . 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 1052	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. NN )
99	98	adantl 473	. . . . . . . . . 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 11004	. . . . . . . . . . . . 13 |- ( K e. ZZ -> ( K - 1 ) e. ZZ )
101	63, 100	syl 17	. . . . . . . . . . . 12 |- ( <. V , E >. RegUSGrph K -> ( K - 1 ) e. ZZ )
102	101	3ad2ant1 1051	. . . . . . . . . . 11 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( K - 1 ) e. ZZ )
103	102	adantr 472	. . . . . . . . . 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 11061	. . . . . . . . . 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 1210	. . . . . . . . 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 1038	. . . . . . . . 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 14441	. . . . . . . . 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 61	. . . . . . . 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 1051	. . . . . . . . . 10 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. ZZ )
110		simp2 1031	. . . . . . . . . 10 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. Prime )
111	109, 110	anim12ci 577	. . . . . . . . 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 14833	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. ZZ ) -> ( P || ( K - 1 ) -> ( ( K ^ ( P - 2 ) ) mod P ) = 1 ) )
113	111, 106, 112	sylc 61	. . . . . . . 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 6326	. . . . . . 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 6323	. . . . . 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 10743	. . . . . . . . . 10 |- ( 0 + 1 ) = 1
117	116	oveq1i 6318	. . . . . . . . 9 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
118	89	nnred 10646	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR )
119		prmgt1 14722	. . . . . . . . . 10 |- ( P e. Prime -> 1 < P )
120		1mod 12162	. . . . . . . . . 10 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
121	118, 119, 120	syl2anc 673	. . . . . . . . 9 |- ( P e. Prime -> ( 1 mod P ) = 1 )
122	117, 121	syl5eq 2517	. . . . . . . 8 |- ( P e. Prime -> ( ( 0 + 1 ) mod P ) = 1 )
123	122	3ad2ant2 1052	. . . . . . 7 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( ( 0 + 1 ) mod P ) = 1 )
124	123	adantl 473	. . . . . 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 2505	. . . . 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 473	. . . 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 2509	. . 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 441	. 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 2726	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 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   {crab 2760   _Vcvv 3031   <.cop 3965   class class class wbr 4395   |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   Fincfn 7587   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   < clt 9693   - cmin 9880   NNcn 10631   2c2 10681   3c3 10682   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   mod cmo 12129   ^cexp 12310   #chash 12553   lastS clsw 12704   || cdvds 14382   Primecprime 14701   USGrph cusg 25136   WWalksN cwwlkn 25485   ClWWalksN cclwwlkn 25556   VDeg cvdg 25700   RegUSGrph crusgra 25730   FriendGrph cfrgra 25795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-xadd 11433  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-s2 13003  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793  df-usgra 25139  df-nbgra 25227  df-wlk 25315  df-wwlk 25486  df-wwlkn 25487  df-clwwlk 25558  df-clwwlkn 25559  df-vdgr 25701  df-rgra 25731  df-rusgra 25732  df-frgra 25796

This theorem is referenced by:  numclwwlk6  25920