Theorem numclwwlk5 30845

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 991	. . . . 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 996	. . . . 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 994	. . . . 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 30844	. . . . 5 |- ( ( <. V , E >. RegUSGrph K /\ 2 || ( K - 1 ) /\ X e. V ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 )
7	1, 2, 3, 6	syl3anc 1219	. . . 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 2523	. . . . 5 |- ( P = 2 -> ( P e. Prime <-> 2 e. Prime ) )
10		breq1 4395	. . . . 5 |- ( P = 2 -> ( P || ( K - 1 ) <-> 2 || ( K - 1 ) ) )
11	9, 10	3anbi23d 1293	. . . 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 6200	. . . . . 6 |- ( P = 2 -> ( X F P ) = ( X F 2 ) )
14	13	fveq2d 5795	. . . . 5 |- ( P = 2 -> ( # ` ( X F P ) ) = ( # ` ( X F 2 ) ) )
15		id 22	. . . . 5 |- ( P = 2 -> P = 2 )
16	14, 15	oveq12d 6210	. . . 4 |- ( P = 2 -> ( ( # ` ( X F P ) ) mod P ) = ( ( # ` ( X F 2 ) ) mod 2 ) )
17	16	eqeq1d 2453	. . 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 985	. . . . . . . 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 1035	. . . . . 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 1036	. . . . . 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 30389	. . . . . . . . . 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 1010	. . . . . . . 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 5790	. . . . . . . . . . . 12 |- ( u = w -> ( u ` 0 ) = ( w ` 0 ) )
30	29	eqeq1d 2453	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` 0 ) = v <-> ( w ` 0 ) = v ) )
31		fveq1 5790	. . . . . . . . . . . 12 |- ( u = w -> ( u ` ( n - 2 ) ) = ( w ` ( n - 2 ) ) )
32	31, 29	eqeq12d 2473	. . . . . . . . . . 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 3069	. . . . . . . . 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 6252	. . . . . . 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 5791	. . . . . . . . . . . 12 |- ( u = w -> ( lastS ` u ) = ( lastS ` w ) )
38	37	neeq1d 2725	. . . . . . . . . . 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 3069	. . . . . . . . 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 6252	. . . . . . 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 2466	. . . . . . . . . 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 3062	. . . . . . . 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 5791	. . . . . . . . . 10 |- ( m = n -> ( C ` m ) = ( C ` n ) )
47		oveq1 6199	. . . . . . . . . . . . 13 |- ( m = n -> ( m - 2 ) = ( n - 2 ) )
48	47	fveq2d 5795	. . . . . . . . . . . 12 |- ( m = n -> ( u ` ( m - 2 ) ) = ( u ` ( n - 2 ) ) )
49	48	neeq1d 2725	. . . . . . . . . . 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 3065	. . . . . . . . 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 2727	. . . . . . . . . . 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 3069	. . . . . . . . 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 2508	. . . . . . . 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 6267	. . . . . . 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 30842	. . . . . 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 1221	. . . . 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 6207	. . . 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 30686	. . . . . . . . . . . . 13 |- ( <. V , E >. RegUSGrph K -> ( V USGrph E /\ K e. NN0 /\ A. v e. V ( ( V VDeg E ) ` v ) = K ) )
61		nn0z 10772	. . . . . . . . . . . . . 14 |- ( K e. NN0 -> K e. ZZ )
62	61	3ad2ant2 1010	. . . . . . . . . . . . 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 10850	. . . . . . . . . . 11 |- ( <. V , E >. RegUSGrph K -> K e. RR )
65		peano2rem 9778	. . . . . . . . . . 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 1009	. . . . . . . . 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 30688	. . . . . . . . . . . . . . 15 |- ( <. V , E >. RegUSGrph K -> V USGrph E )
70		usgrav 23407	. . . . . . . . . . . . . . . 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 1007	. . . . . . . . . . 11 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( V e. Fin /\ E e. _V ) )
76		prmm2nn0 13887	. . . . . . . . . . . . 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 1008	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( X e. V /\ ( P - 2 ) e. NN0 ) )
79	4, 5	numclwwlkffin 30815	. . . . . . . . . . 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 12229	. . . . . . . . . 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 10740	. . . . . . . 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 9517	. . . . . . 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 1009	. . . . . . . 8 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. RR )
86	76	3ad2ant2 1010	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P - 2 ) e. NN0 )
87		reexpcl 11985	. . . . . . . 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 13870	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
90	89	nnrpd 11129	. . . . . . . . 9 |- ( P e. Prime -> P e. RR+ )
91	90	3ad2ant2 1010	. . . . . . . 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 1168	. . . . . 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 11849	. . . . . 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 2459	. . . . 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 1010	. . . . . . . . . . 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 10791	. . . . . . . . . . . . 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 1009	. . . . . . . . . . 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 10848	. . . . . . . . . 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 1168	. . . . . . . . 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 996	. . . . . . . . 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 30386	. . . . . . . . 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 1009	. . . . . . . . . 10 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. ZZ )
110		simp2 989	. . . . . . . . . 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 30388	. . . . . . . . 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 6210	. . . . . . 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 6207	. . . . . 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 10536	. . . . . . . . . 10 |- ( 0 + 1 ) = 1
117	116	oveq1i 6202	. . . . . . . . 9 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
118	89	nnred 10440	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR )
119		prmgt1 13886	. . . . . . . . . 10 |- ( P e. Prime -> 1 < P )
120		1mod 11843	. . . . . . . . . 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 2504	. . . . . . . 8 |- ( P e. Prime -> ( ( 0 + 1 ) mod P ) = 1 )
123	122	3ad2ant2 1010	. . . . . . 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 2492	. . . . 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 2496	. . 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 2761	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 965   = wceq 1370   e. wcel 1758   =/= wne 2644   A.wral 2795   {crab 2799   _Vcvv 3070   <.cop 3983   class class class wbr 4392   |-> cmpt 4450   ` cfv 5518  (class class class)co 6192   |-> cmpt2 6194   Fincfn 7412   RRcr 9384   0cc0 9385   1c1 9386   + caddc 9388   x. cmul 9390   < clt 9521   - cmin 9698   NNcn 10425   2c2 10474   3c3 10475   NN0cn0 10682   ZZcz 10749   ZZ>=cuz 10964   RR+crp 11094   mod cmo 11811   ^cexp 11968   #chash 12206   lastS clsw 12326   || cdivides 13639   Primecprime 13867   USGrph cusg 23401   VDeg cvdg 23700   WWalksN cwwlkn 30452   ClWWalksN cclwwlkn 30554   RegUSGrph crusgra 30680   FriendGrph cfrgra 30720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-disj 4363  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-2o 7023  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-sup 7794  df-oi 7827  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-xadd 11193  df-fz 11541  df-fzo 11652  df-fl 11745  df-mod 11812  df-seq 11910  df-exp 11969  df-hash 12207  df-word 12333  df-lsw 12334  df-concat 12335  df-s1 12336  df-substr 12337  df-s2 12579  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-clim 13070  df-sum 13268  df-dvds 13640  df-gcd 13795  df-prm 13868  df-phi 13945  df-usgra 23403  df-nbgra 23469  df-wlk 23552  df-vdgr 23701  df-wwlk 30453  df-wwlkn 30454  df-clwwlk 30556  df-clwwlkn 30557  df-rgra 30681  df-rusgra 30682  df-frgra 30721

This theorem is referenced by:  numclwwlk6  30846