Theorem numclwwlk5 25840

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 1011	. . . . 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 1016	. . . . 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 1014	. . . . 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 25839	. . . . 5 |- ( ( <. V , E >. RegUSGrph K /\ 2 || ( K - 1 ) /\ X e. V ) -> ( ( # ` ( X F 2 ) ) mod 2 ) = 1 )
7	1, 2, 3, 6	syl3anc 1268	. . . 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 2517	. . . . 5 |- ( P = 2 -> ( P e. Prime <-> 2 e. Prime ) )
10		breq1 4405	. . . . 5 |- ( P = 2 -> ( P || ( K - 1 ) <-> 2 || ( K - 1 ) ) )
11	9, 10	3anbi23d 1342	. . . 4 |- ( P = 2 -> ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) <-> ( X e. V /\ 2 e. Prime /\ 2 || ( K - 1 ) ) ) )
12	11	anbi2d 710	. . 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 6298	. . . . . 6 |- ( P = 2 -> ( X F P ) = ( X F 2 ) )
14	13	fveq2d 5869	. . . . 5 |- ( P = 2 -> ( # ` ( X F P ) ) = ( # ` ( X F 2 ) ) )
15		id 22	. . . . 5 |- ( P = 2 -> P = 2 )
16	14, 15	oveq12d 6308	. . . 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 272	. 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 1005	. . . . . . . 8 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( <. V , E >. RegUSGrph K /\ V FriendGrph E ) )
20	19	adantr 467	. . . . . . 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 468	. . . . . 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 1055	. . . . . 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 1056	. . . . . 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 14644	. . . . . . . . . 10 |- ( ( P e. Prime /\ P =/= 2 ) -> P e. ( ZZ>= ` 3 ) )
25	24	ex 436	. . . . . . . . 9 |- ( P e. Prime -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
26	25	3ad2ant2 1030	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P =/= 2 -> P e. ( ZZ>= ` 3 ) ) )
27	26	adantl 468	. . . . . . 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 432	. . . . . 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 5864	. . . . . . . . . . . 12 |- ( u = w -> ( u ` 0 ) = ( w ` 0 ) )
30	29	eqeq1d 2453	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` 0 ) = v <-> ( w ` 0 ) = v ) )
31		fveq1 5864	. . . . . . . . . . . 12 |- ( u = w -> ( u ` ( n - 2 ) ) = ( w ` ( n - 2 ) ) )
32	31, 29	eqeq12d 2466	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` ( n - 2 ) ) = ( u ` 0 ) <-> ( w ` ( n - 2 ) ) = ( w ` 0 ) ) )
33	30, 32	anbi12d 717	. . . . . . . . . 10 |- ( u = w -> ( ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) = ( u ` 0 ) ) <-> ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) ) )
34	33	cbvrabv 3044	. . . . . . . . 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 6356	. . . . . . 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 5865	. . . . . . . . . . . 12 |- ( u = w -> ( lastS ` u ) = ( lastS ` w ) )
38	37	neeq1d 2683	. . . . . . . . . . 11 |- ( u = w -> ( ( lastS ` u ) =/= v <-> ( lastS ` w ) =/= v ) )
39	30, 38	anbi12d 717	. . . . . . . . . 10 |- ( u = w -> ( ( ( u ` 0 ) = v /\ ( lastS ` u ) =/= v ) <-> ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) ) )
40	39	cbvrabv 3044	. . . . . . . . 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 6356	. . . . . . 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 2462	. . . . . . . . . 10 |- ( z = v -> ( ( u ` 0 ) = z <-> ( u ` 0 ) = v ) )
44	43	anbi1d 711	. . . . . . . . 9 |- ( z = v -> ( ( ( u ` 0 ) = z /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) <-> ( ( u ` 0 ) = v /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) ) )
45	44	rabbidv 3036	. . . . . . . 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 5865	. . . . . . . . . 10 |- ( m = n -> ( C ` m ) = ( C ` n ) )
47		oveq1 6297	. . . . . . . . . . . . 13 |- ( m = n -> ( m - 2 ) = ( n - 2 ) )
48	47	fveq2d 5869	. . . . . . . . . . . 12 |- ( m = n -> ( u ` ( m - 2 ) ) = ( u ` ( n - 2 ) ) )
49	48	neeq1d 2683	. . . . . . . . . . 11 |- ( m = n -> ( ( u ` ( m - 2 ) ) =/= ( u ` 0 ) <-> ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) )
50	49	anbi2d 710	. . . . . . . . . 10 |- ( m = n -> ( ( ( u ` 0 ) = v /\ ( u ` ( m - 2 ) ) =/= ( u ` 0 ) ) <-> ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) ) )
51	46, 50	rabeqbidv 3040	. . . . . . . . 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 2685	. . . . . . . . . . 11 |- ( u = w -> ( ( u ` ( n - 2 ) ) =/= ( u ` 0 ) <-> ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) )
53	30, 52	anbi12d 717	. . . . . . . . . 10 |- ( u = w -> ( ( ( u ` 0 ) = v /\ ( u ` ( n - 2 ) ) =/= ( u ` 0 ) ) <-> ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) ) )
54	53	cbvrabv 3044	. . . . . . . . 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 2501	. . . . . . . 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 6371	. . . . . . 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 25837	. . . . . 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 1270	. . . . 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 6305	. . . 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 25657	. . . . . . . . . . . . 13 |- ( <. V , E >. RegUSGrph K -> ( V USGrph E /\ K e. NN0 /\ A. v e. V ( ( V VDeg E ) ` v ) = K ) )
61		nn0z 10960	. . . . . . . . . . . . . 14 |- ( K e. NN0 -> K e. ZZ )
62	61	3ad2ant2 1030	. . . . . . . . . . . . 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 11040	. . . . . . . . . . 11 |- ( <. V , E >. RegUSGrph K -> K e. RR )
65		peano2rem 9941	. . . . . . . . . . 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 1029	. . . . . . . . 9 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( K - 1 ) e. RR )
68	67	adantr 467	. . . . . . . 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 25659	. . . . . . . . . . . . . . 15 |- ( <. V , E >. RegUSGrph K -> V USGrph E )
70		usgrav 25065	. . . . . . . . . . . . . . . 16 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
71	70	simprd 465	. . . . . . . . . . . . . . 15 |- ( V USGrph E -> E e. _V )
72	69, 71	syl 17	. . . . . . . . . . . . . 14 |- ( <. V , E >. RegUSGrph K -> E e. _V )
73	72	anim1i 572	. . . . . . . . . . . . 13 |- ( ( <. V , E >. RegUSGrph K /\ V e. Fin ) -> ( E e. _V /\ V e. Fin ) )
74	73	ancomd 453	. . . . . . . . . . . 12 |- ( ( <. V , E >. RegUSGrph K /\ V e. Fin ) -> ( V e. Fin /\ E e. _V ) )
75	74	3adant2 1027	. . . . . . . . . . 11 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( V e. Fin /\ E e. _V ) )
76		prmm2nn0 14645	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P - 2 ) e. NN0 )
77	76	anim2i 573	. . . . . . . . . . . 12 |- ( ( X e. V /\ P e. Prime ) -> ( X e. V /\ ( P - 2 ) e. NN0 ) )
78	77	3adant3 1028	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( X e. V /\ ( P - 2 ) e. NN0 ) )
79	4, 5	numclwwlkffin 25810	. . . . . . . . . . 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 480	. . . . . . . . . 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 12538	. . . . . . . . . 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 10926	. . . . . . . 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 9671	. . . . . . 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 1029	. . . . . . . 8 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. RR )
86	76	3ad2ant2 1030	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( P - 2 ) e. NN0 )
87		reexpcl 12289	. . . . . . . 8 |- ( ( K e. RR /\ ( P - 2 ) e. NN0 ) -> ( K ^ ( P - 2 ) ) e. RR )
88	85, 86, 87	syl2an 480	. . . . . . 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 14625	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
90	89	nnrpd 11339	. . . . . . . . 9 |- ( P e. Prime -> P e. RR+ )
91	90	3ad2ant2 1030	. . . . . . . 8 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. RR+ )
92	91	adantl 468	. . . . . . 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 1188	. . . . . 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 468	. . . . 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 12135	. . . . . 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 2457	. . . . 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 1030	. . . . . . . . . . 11 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. NN )
99	98	adantl 468	. . . . . . . . . 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 10980	. . . . . . . . . . . . 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 1029	. . . . . . . . . . 11 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> ( K - 1 ) e. ZZ )
103	102	adantr 467	. . . . . . . . . 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 11038	. . . . . . . . . 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 1188	. . . . . . . . 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 1016	. . . . . . . . 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 14363	. . . . . . . . 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 62	. . . . . . . 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 1029	. . . . . . . . . 10 |- ( ( <. V , E >. RegUSGrph K /\ V FriendGrph E /\ V e. Fin ) -> K e. ZZ )
110		simp2 1009	. . . . . . . . . 10 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> P e. Prime )
111	109, 110	anim12ci 571	. . . . . . . . 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 14754	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. ZZ ) -> ( P || ( K - 1 ) -> ( ( K ^ ( P - 2 ) ) mod P ) = 1 ) )
113	111, 106, 112	sylc 62	. . . . . . . 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 6308	. . . . . . 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 6305	. . . . . 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 10721	. . . . . . . . . 10 |- ( 0 + 1 ) = 1
117	116	oveq1i 6300	. . . . . . . . 9 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
118	89	nnred 10624	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR )
119		prmgt1 14643	. . . . . . . . . 10 |- ( P e. Prime -> 1 < P )
120		1mod 12129	. . . . . . . . . 10 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
121	118, 119, 120	syl2anc 667	. . . . . . . . 9 |- ( P e. Prime -> ( 1 mod P ) = 1 )
122	117, 121	syl5eq 2497	. . . . . . . 8 |- ( P e. Prime -> ( ( 0 + 1 ) mod P ) = 1 )
123	122	3ad2ant2 1030	. . . . . . 7 |- ( ( X e. V /\ P e. Prime /\ P || ( K - 1 ) ) -> ( ( 0 + 1 ) mod P ) = 1 )
124	123	adantl 468	. . . . . 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 2485	. . . . 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 468	. . . 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 2489	. . 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 436	. 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 2707	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 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   {crab 2741   _Vcvv 3045   <.cop 3974   class class class wbr 4402   |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   |-> cmpt2 6292   Fincfn 7569   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   < clt 9675   - cmin 9860   NNcn 10609   2c2 10659   3c3 10660   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   mod cmo 12096   ^cexp 12272   #chash 12515   lastS clsw 12657   || cdvds 14305   Primecprime 14622   USGrph cusg 25057   WWalksN cwwlkn 25406   ClWWalksN cclwwlkn 25477   VDeg cvdg 25621   RegUSGrph crusgra 25651   FriendGrph cfrgra 25716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-xadd 11410  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-s2 12944  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-dvds 14306  df-gcd 14469  df-prm 14623  df-phi 14714  df-usgra 25060  df-nbgra 25148  df-wlk 25236  df-wwlk 25407  df-wwlkn 25408  df-clwwlk 25479  df-clwwlkn 25480  df-vdgr 25622  df-rgra 25652  df-rusgra 25653  df-frgra 25717

This theorem is referenced by:  numclwwlk6  25841