Theorem wilthlem3 24796

Description: Lemma for wilth 24797. Here we round out the argument of wilthlem2 24795 with the final step of the induction. The induction argument shows that every subset of 1 ... ( P - 1 ) that is closed under inverse and contains P - 1 multiplies to -u 1 mod P, and clearly 1 ... ( P - 1 ) itself is such a set. Thus, the product of all the elements is -u 1, and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)

Hypotheses

Ref	Expression
wilthlem.t	|- T = (mulGrp ` ℂfld )
wilthlem.a	|- A = { x e. ~P ( 1 ... ( P - 1 ) ) | ( ( P - 1 ) e. x /\ A. y e. x ( ( y ^ ( P - 2 ) ) mod P ) e. x ) }

Assertion

Ref	Expression
wilthlem3	|- ( P e. Prime -> P || ( ( ! ` ( P - 1 ) ) + 1 ) )

Distinct variable groups:   x, y, A   x, P, y   x, T, y

Proof of Theorem wilthlem3

Dummy variables t s k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmuz2 15408	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
2		uz2m1nn 11763	. . . . . . . 8 |- ( P e. ( ZZ>= ` 2 ) -> ( P - 1 ) e. NN )
3	1, 2	syl 17	. . . . . . 7 |- ( P e. Prime -> ( P - 1 ) e. NN )
4		nnuz 11723	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
5	3, 4	syl6eleq 2711	. . . . . 6 |- ( P e. Prime -> ( P - 1 ) e. ( ZZ>= ` 1 ) )
6		eluzfz2 12349	. . . . . 6 |- ( ( P - 1 ) e. ( ZZ>= ` 1 ) -> ( P - 1 ) e. ( 1 ... ( P - 1 ) ) )
7	5, 6	syl 17	. . . . 5 |- ( P e. Prime -> ( P - 1 ) e. ( 1 ... ( P - 1 ) ) )
8		simpl 473	. . . . . . . 8 |- ( ( P e. Prime /\ y e. ( 1 ... ( P - 1 ) ) ) -> P e. Prime )
9		elfzelz 12342	. . . . . . . . 9 |- ( y e. ( 1 ... ( P - 1 ) ) -> y e. ZZ )
10	9	adantl 482	. . . . . . . 8 |- ( ( P e. Prime /\ y e. ( 1 ... ( P - 1 ) ) ) -> y e. ZZ )
11		prmnn 15388	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
12		fzm1ndvds 15044	. . . . . . . . 9 |- ( ( P e. NN /\ y e. ( 1 ... ( P - 1 ) ) ) -> -. P || y )
13	11, 12	sylan 488	. . . . . . . 8 |- ( ( P e. Prime /\ y e. ( 1 ... ( P - 1 ) ) ) -> -. P || y )
14		eqid 2622	. . . . . . . . 9 |- ( ( y ^ ( P - 2 ) ) mod P ) = ( ( y ^ ( P - 2 ) ) mod P )
15	14	prmdiv 15490	. . . . . . . 8 |- ( ( P e. Prime /\ y e. ZZ /\ -. P || y ) -> ( ( ( y ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) /\ P || ( ( y x. ( ( y ^ ( P - 2 ) ) mod P ) ) - 1 ) ) )
16	8, 10, 13, 15	syl3anc 1326	. . . . . . 7 |- ( ( P e. Prime /\ y e. ( 1 ... ( P - 1 ) ) ) -> ( ( ( y ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) /\ P || ( ( y x. ( ( y ^ ( P - 2 ) ) mod P ) ) - 1 ) ) )
17	16	simpld 475	. . . . . 6 |- ( ( P e. Prime /\ y e. ( 1 ... ( P - 1 ) ) ) -> ( ( y ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) )
18	17	ralrimiva 2966	. . . . 5 |- ( P e. Prime -> A. y e. ( 1 ... ( P - 1 ) ) ( ( y ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) )
19		ovex 6678	. . . . . . 7 |- ( 1 ... ( P - 1 ) ) e. _V
20	19	pwid 4174	. . . . . 6 |- ( 1 ... ( P - 1 ) ) e. ~P ( 1 ... ( P - 1 ) )
21		eleq2 2690	. . . . . . . 8 |- ( x = ( 1 ... ( P - 1 ) ) -> ( ( P - 1 ) e. x <-> ( P - 1 ) e. ( 1 ... ( P - 1 ) ) ) )
22		eleq2 2690	. . . . . . . . 9 |- ( x = ( 1 ... ( P - 1 ) ) -> ( ( ( y ^ ( P - 2 ) ) mod P ) e. x <-> ( ( y ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) ) )
23	22	raleqbi1dv 3146	. . . . . . . 8 |- ( x = ( 1 ... ( P - 1 ) ) -> ( A. y e. x ( ( y ^ ( P - 2 ) ) mod P ) e. x <-> A. y e. ( 1 ... ( P - 1 ) ) ( ( y ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) ) )
24	21, 23	anbi12d 747	. . . . . . 7 |- ( x = ( 1 ... ( P - 1 ) ) -> ( ( ( P - 1 ) e. x /\ A. y e. x ( ( y ^ ( P - 2 ) ) mod P ) e. x ) <-> ( ( P - 1 ) e. ( 1 ... ( P - 1 ) ) /\ A. y e. ( 1 ... ( P - 1 ) ) ( ( y ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) ) ) )
25		wilthlem.a	. . . . . . 7 |- A = { x e. ~P ( 1 ... ( P - 1 ) ) | ( ( P - 1 ) e. x /\ A. y e. x ( ( y ^ ( P - 2 ) ) mod P ) e. x ) }
26	24, 25	elrab2 3366	. . . . . 6 |- ( ( 1 ... ( P - 1 ) ) e. A <-> ( ( 1 ... ( P - 1 ) ) e. ~P ( 1 ... ( P - 1 ) ) /\ ( ( P - 1 ) e. ( 1 ... ( P - 1 ) ) /\ A. y e. ( 1 ... ( P - 1 ) ) ( ( y ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) ) ) )
27	20, 26	mpbiran 953	. . . . 5 |- ( ( 1 ... ( P - 1 ) ) e. A <-> ( ( P - 1 ) e. ( 1 ... ( P - 1 ) ) /\ A. y e. ( 1 ... ( P - 1 ) ) ( ( y ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) ) )
28	7, 18, 27	sylanbrc 698	. . . 4 |- ( P e. Prime -> ( 1 ... ( P - 1 ) ) e. A )
29		fzfi 12771	. . . . 5 |- ( 1 ... ( P - 1 ) ) e. Fin
30		eleq1 2689	. . . . . . . 8 |- ( s = t -> ( s e. A <-> t e. A ) )
31		reseq2 5391	. . . . . . . . . . 11 |- ( s = t -> ( _I |` s ) = ( _I |` t ) )
32	31	oveq2d 6666	. . . . . . . . . 10 |- ( s = t -> ( T gsumg ( _I |` s ) ) = ( T gsumg ( _I |` t ) ) )
33	32	oveq1d 6665	. . . . . . . . 9 |- ( s = t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( ( T gsumg ( _I |` t ) ) mod P ) )
34	33	eqeq1d 2624	. . . . . . . 8 |- ( s = t -> ( ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) <-> ( ( T gsumg ( _I |` t ) ) mod P ) = ( -u 1 mod P ) ) )
35	30, 34	imbi12d 334	. . . . . . 7 |- ( s = t -> ( ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) <-> ( t e. A -> ( ( T gsumg ( _I |` t ) ) mod P ) = ( -u 1 mod P ) ) ) )
36	35	imbi2d 330	. . . . . 6 |- ( s = t -> ( ( P e. Prime -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) <-> ( P e. Prime -> ( t e. A -> ( ( T gsumg ( _I |` t ) ) mod P ) = ( -u 1 mod P ) ) ) ) )
37		eleq1 2689	. . . . . . . 8 |- ( s = ( 1 ... ( P - 1 ) ) -> ( s e. A <-> ( 1 ... ( P - 1 ) ) e. A ) )
38		reseq2 5391	. . . . . . . . . . 11 |- ( s = ( 1 ... ( P - 1 ) ) -> ( _I |` s ) = ( _I |` ( 1 ... ( P - 1 ) ) ) )
39	38	oveq2d 6666	. . . . . . . . . 10 |- ( s = ( 1 ... ( P - 1 ) ) -> ( T gsumg ( _I |` s ) ) = ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) )
40	39	oveq1d 6665	. . . . . . . . 9 |- ( s = ( 1 ... ( P - 1 ) ) -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) mod P ) )
41	40	eqeq1d 2624	. . . . . . . 8 |- ( s = ( 1 ... ( P - 1 ) ) -> ( ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) <-> ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) mod P ) = ( -u 1 mod P ) ) )
42	37, 41	imbi12d 334	. . . . . . 7 |- ( s = ( 1 ... ( P - 1 ) ) -> ( ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) <-> ( ( 1 ... ( P - 1 ) ) e. A -> ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) mod P ) = ( -u 1 mod P ) ) ) )
43	42	imbi2d 330	. . . . . 6 |- ( s = ( 1 ... ( P - 1 ) ) -> ( ( P e. Prime -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) <-> ( P e. Prime -> ( ( 1 ... ( P - 1 ) ) e. A -> ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) mod P ) = ( -u 1 mod P ) ) ) ) )
44		bi2.04 376	. . . . . . . . . . 11 |- ( ( s C. t -> ( P e. Prime -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) <-> ( P e. Prime -> ( s C. t -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) )
45		pm2.27 42	. . . . . . . . . . . 12 |- ( P e. Prime -> ( ( P e. Prime -> ( s C. t -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) -> ( s C. t -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) )
46	45	com34 91	. . . . . . . . . . 11 |- ( P e. Prime -> ( ( P e. Prime -> ( s C. t -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) -> ( s e. A -> ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) )
47	44, 46	syl5bi 232	. . . . . . . . . 10 |- ( P e. Prime -> ( ( s C. t -> ( P e. Prime -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) -> ( s e. A -> ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) )
48	47	alimdv 1845	. . . . . . . . 9 |- ( P e. Prime -> ( A. s ( s C. t -> ( P e. Prime -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) -> A. s ( s e. A -> ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) )
49		df-ral 2917	. . . . . . . . 9 |- ( A. s e. A ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) <-> A. s ( s e. A -> ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) )
50	48, 49	syl6ibr 242	. . . . . . . 8 |- ( P e. Prime -> ( A. s ( s C. t -> ( P e. Prime -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) -> A. s e. A ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) )
51		wilthlem.t	. . . . . . . . . 10 |- T = (mulGrp ` ℂfld )
52		simp1 1061	. . . . . . . . . 10 |- ( ( P e. Prime /\ A. s e. A ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) /\ t e. A ) -> P e. Prime )
53		simp3 1063	. . . . . . . . . 10 |- ( ( P e. Prime /\ A. s e. A ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) /\ t e. A ) -> t e. A )
54		simp2 1062	. . . . . . . . . 10 |- ( ( P e. Prime /\ A. s e. A ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) /\ t e. A ) -> A. s e. A ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) )
55	51, 25, 52, 53, 54	wilthlem2 24795	. . . . . . . . 9 |- ( ( P e. Prime /\ A. s e. A ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) /\ t e. A ) -> ( ( T gsumg ( _I |` t ) ) mod P ) = ( -u 1 mod P ) )
56	55	3exp 1264	. . . . . . . 8 |- ( P e. Prime -> ( A. s e. A ( s C. t -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) -> ( t e. A -> ( ( T gsumg ( _I |` t ) ) mod P ) = ( -u 1 mod P ) ) ) )
57	50, 56	syldc 48	. . . . . . 7 |- ( A. s ( s C. t -> ( P e. Prime -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) -> ( P e. Prime -> ( t e. A -> ( ( T gsumg ( _I |` t ) ) mod P ) = ( -u 1 mod P ) ) ) )
58	57	a1i 11	. . . . . 6 |- ( t e. Fin -> ( A. s ( s C. t -> ( P e. Prime -> ( s e. A -> ( ( T gsumg ( _I |` s ) ) mod P ) = ( -u 1 mod P ) ) ) ) -> ( P e. Prime -> ( t e. A -> ( ( T gsumg ( _I |` t ) ) mod P ) = ( -u 1 mod P ) ) ) ) )
59	36, 43, 58	findcard3 8203	. . . . 5 |- ( ( 1 ... ( P - 1 ) ) e. Fin -> ( P e. Prime -> ( ( 1 ... ( P - 1 ) ) e. A -> ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) mod P ) = ( -u 1 mod P ) ) ) )
60	29, 59	ax-mp 5	. . . 4 |- ( P e. Prime -> ( ( 1 ... ( P - 1 ) ) e. A -> ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) mod P ) = ( -u 1 mod P ) ) )
61	28, 60	mpd 15	. . 3 |- ( P e. Prime -> ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) mod P ) = ( -u 1 mod P ) )
62		cnfld1 19771	. . . . . 6 |- 1 = ( 1r ` ℂfld )
63	51, 62	ringidval 18503	. . . . 5 |- 1 = ( 0g ` T )
64		cncrng 19767	. . . . . 6 |- ℂfld e. CRing
65	51	crngmgp 18555	. . . . . 6 |- (ℂfld e. CRing -> T e. CMnd )
66	64, 65	mp1i 13	. . . . 5 |- ( P e. Prime -> T e. CMnd )
67	29	a1i 11	. . . . 5 |- ( P e. Prime -> ( 1 ... ( P - 1 ) ) e. Fin )
68		zsubrg 19799	. . . . . 6 |- ZZ e. (SubRing ` ℂfld )
69	51	subrgsubm 18793	. . . . . 6 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ e. (SubMnd ` T ) )
70	68, 69	mp1i 13	. . . . 5 |- ( P e. Prime -> ZZ e. (SubMnd ` T ) )
71		f1oi 6174	. . . . . . . 8 |- ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) -1-1-onto-> ( 1 ... ( P - 1 ) )
72		f1of 6137	. . . . . . . 8 |- ( ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) -1-1-onto-> ( 1 ... ( P - 1 ) ) -> ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> ( 1 ... ( P - 1 ) ) )
73	71, 72	ax-mp 5	. . . . . . 7 |- ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> ( 1 ... ( P - 1 ) )
74	9	ssriv 3607	. . . . . . 7 |- ( 1 ... ( P - 1 ) ) C_ ZZ
75		fss 6056	. . . . . . 7 |- ( ( ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> ( 1 ... ( P - 1 ) ) /\ ( 1 ... ( P - 1 ) ) C_ ZZ ) -> ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> ZZ )
76	73, 74, 75	mp2an 708	. . . . . 6 |- ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> ZZ
77	76	a1i 11	. . . . 5 |- ( P e. Prime -> ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> ZZ )
78		1ex 10035	. . . . . . 7 |- 1 e. _V
79	78	a1i 11	. . . . . 6 |- ( P e. Prime -> 1 e. _V )
80	77, 67, 79	fdmfifsupp 8285	. . . . 5 |- ( P e. Prime -> ( _I |` ( 1 ... ( P - 1 ) ) ) finSupp 1 )
81	63, 66, 67, 70, 77, 80	gsumsubmcl 18319	. . . 4 |- ( P e. Prime -> ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) e. ZZ )
82		1z 11407	. . . . 5 |- 1 e. ZZ
83		znegcl 11412	. . . . 5 |- ( 1 e. ZZ -> -u 1 e. ZZ )
84	82, 83	mp1i 13	. . . 4 |- ( P e. Prime -> -u 1 e. ZZ )
85		moddvds 14991	. . . 4 |- ( ( P e. NN /\ ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) e. ZZ /\ -u 1 e. ZZ ) -> ( ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) mod P ) = ( -u 1 mod P ) <-> P || ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) - -u 1 ) ) )
86	11, 81, 84, 85	syl3anc 1326	. . 3 |- ( P e. Prime -> ( ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) mod P ) = ( -u 1 mod P ) <-> P || ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) - -u 1 ) ) )
87	61, 86	mpbid 222	. 2 |- ( P e. Prime -> P || ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) - -u 1 ) )
88		fcoi1 6078	. . . . . . . . . 10 |- ( ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> ( 1 ... ( P - 1 ) ) -> ( ( _I |` ( 1 ... ( P - 1 ) ) ) o. ( _I |` ( 1 ... ( P - 1 ) ) ) ) = ( _I |` ( 1 ... ( P - 1 ) ) ) )
89	73, 88	ax-mp 5	. . . . . . . . 9 |- ( ( _I |` ( 1 ... ( P - 1 ) ) ) o. ( _I |` ( 1 ... ( P - 1 ) ) ) ) = ( _I |` ( 1 ... ( P - 1 ) ) )
90	89	fveq1i 6192	. . . . . . . 8 |- ( ( ( _I |` ( 1 ... ( P - 1 ) ) ) o. ( _I |` ( 1 ... ( P - 1 ) ) ) ) ` k ) = ( ( _I |` ( 1 ... ( P - 1 ) ) ) ` k )
91		fvres 6207	. . . . . . . 8 |- ( k e. ( 1 ... ( P - 1 ) ) -> ( ( _I |` ( 1 ... ( P - 1 ) ) ) ` k ) = ( _I ` k ) )
92	90, 91	syl5eq 2668	. . . . . . 7 |- ( k e. ( 1 ... ( P - 1 ) ) -> ( ( ( _I |` ( 1 ... ( P - 1 ) ) ) o. ( _I |` ( 1 ... ( P - 1 ) ) ) ) ` k ) = ( _I ` k ) )
93	92	adantl 482	. . . . . 6 |- ( ( P e. Prime /\ k e. ( 1 ... ( P - 1 ) ) ) -> ( ( ( _I |` ( 1 ... ( P - 1 ) ) ) o. ( _I |` ( 1 ... ( P - 1 ) ) ) ) ` k ) = ( _I ` k ) )
94	5, 93	seqfveq 12825	. . . . 5 |- ( P e. Prime -> ( seq 1 ( x. , ( ( _I |` ( 1 ... ( P - 1 ) ) ) o. ( _I |` ( 1 ... ( P - 1 ) ) ) ) ) ` ( P - 1 ) ) = ( seq 1 ( x. , _I ) ` ( P - 1 ) ) )
95		cnfldbas 19750	. . . . . . 7 |- CC = ( Base ` ℂfld )
96	51, 95	mgpbas 18495	. . . . . 6 |- CC = ( Base ` T )
97		cnfldmul 19752	. . . . . . 7 |- x. = ( .r ` ℂfld )
98	51, 97	mgpplusg 18493	. . . . . 6 |- x. = ( +g ` T )
99		eqid 2622	. . . . . 6 |- (Cntz ` T ) = (Cntz ` T )
100		cnring 19768	. . . . . . 7 |- ℂfld e. Ring
101	51	ringmgp 18553	. . . . . . 7 |- (ℂfld e. Ring -> T e. Mnd )
102	100, 101	mp1i 13	. . . . . 6 |- ( P e. Prime -> T e. Mnd )
103		zsscn 11385	. . . . . . . 8 |- ZZ C_ CC
104		fss 6056	. . . . . . . 8 |- ( ( ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> ZZ /\ ZZ C_ CC ) -> ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> CC )
105	76, 103, 104	mp2an 708	. . . . . . 7 |- ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> CC
106	105	a1i 11	. . . . . 6 |- ( P e. Prime -> ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) --> CC )
107	96, 99, 66, 106	cntzcmnf 18248	. . . . . 6 |- ( P e. Prime -> ran ( _I |` ( 1 ... ( P - 1 ) ) ) C_ ( (Cntz ` T ) ` ran ( _I |` ( 1 ... ( P - 1 ) ) ) ) )
108		f1of1 6136	. . . . . . 7 |- ( ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) -1-1-onto-> ( 1 ... ( P - 1 ) ) -> ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) -1-1-> ( 1 ... ( P - 1 ) ) )
109	71, 108	mp1i 13	. . . . . 6 |- ( P e. Prime -> ( _I |` ( 1 ... ( P - 1 ) ) ) : ( 1 ... ( P - 1 ) ) -1-1-> ( 1 ... ( P - 1 ) ) )
110		suppssdm 7308	. . . . . . . . 9 |- ( ( _I |` ( 1 ... ( P - 1 ) ) ) supp 1 ) C_ dom ( _I |` ( 1 ... ( P - 1 ) ) )
111		dmresi 5457	. . . . . . . . 9 |- dom ( _I |` ( 1 ... ( P - 1 ) ) ) = ( 1 ... ( P - 1 ) )
112	110, 111	sseqtri 3637	. . . . . . . 8 |- ( ( _I |` ( 1 ... ( P - 1 ) ) ) supp 1 ) C_ ( 1 ... ( P - 1 ) )
113		rnresi 5479	. . . . . . . 8 |- ran ( _I |` ( 1 ... ( P - 1 ) ) ) = ( 1 ... ( P - 1 ) )
114	112, 113	sseqtr4i 3638	. . . . . . 7 |- ( ( _I |` ( 1 ... ( P - 1 ) ) ) supp 1 ) C_ ran ( _I |` ( 1 ... ( P - 1 ) ) )
115	114	a1i 11	. . . . . 6 |- ( P e. Prime -> ( ( _I |` ( 1 ... ( P - 1 ) ) ) supp 1 ) C_ ran ( _I |` ( 1 ... ( P - 1 ) ) ) )
116		eqid 2622	. . . . . 6 |- ( ( ( _I |` ( 1 ... ( P - 1 ) ) ) o. ( _I |` ( 1 ... ( P - 1 ) ) ) ) supp 1 ) = ( ( ( _I |` ( 1 ... ( P - 1 ) ) ) o. ( _I |` ( 1 ... ( P - 1 ) ) ) ) supp 1 )
117	96, 63, 98, 99, 102, 67, 106, 107, 3, 109, 115, 116	gsumval3 18308	. . . . 5 |- ( P e. Prime -> ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) = ( seq 1 ( x. , ( ( _I |` ( 1 ... ( P - 1 ) ) ) o. ( _I |` ( 1 ... ( P - 1 ) ) ) ) ) ` ( P - 1 ) ) )
118		facnn 13062	. . . . . 6 |- ( ( P - 1 ) e. NN -> ( ! ` ( P - 1 ) ) = ( seq 1 ( x. , _I ) ` ( P - 1 ) ) )
119	3, 118	syl 17	. . . . 5 |- ( P e. Prime -> ( ! ` ( P - 1 ) ) = ( seq 1 ( x. , _I ) ` ( P - 1 ) ) )
120	94, 117, 119	3eqtr4d 2666	. . . 4 |- ( P e. Prime -> ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) = ( ! ` ( P - 1 ) ) )
121	120	oveq1d 6665	. . 3 |- ( P e. Prime -> ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) - -u 1 ) = ( ( ! ` ( P - 1 ) ) - -u 1 ) )
122		nnm1nn0 11334	. . . . . . 7 |- ( P e. NN -> ( P - 1 ) e. NN0 )
123	11, 122	syl 17	. . . . . 6 |- ( P e. Prime -> ( P - 1 ) e. NN0 )
124		faccl 13070	. . . . . 6 |- ( ( P - 1 ) e. NN0 -> ( ! ` ( P - 1 ) ) e. NN )
125	123, 124	syl 17	. . . . 5 |- ( P e. Prime -> ( ! ` ( P - 1 ) ) e. NN )
126	125	nncnd 11036	. . . 4 |- ( P e. Prime -> ( ! ` ( P - 1 ) ) e. CC )
127		ax-1cn 9994	. . . 4 |- 1 e. CC
128		subneg 10330	. . . 4 |- ( ( ( ! ` ( P - 1 ) ) e. CC /\ 1 e. CC ) -> ( ( ! ` ( P - 1 ) ) - -u 1 ) = ( ( ! ` ( P - 1 ) ) + 1 ) )
129	126, 127, 128	sylancl 694	. . 3 |- ( P e. Prime -> ( ( ! ` ( P - 1 ) ) - -u 1 ) = ( ( ! ` ( P - 1 ) ) + 1 ) )
130	121, 129	eqtrd 2656	. 2 |- ( P e. Prime -> ( ( T gsumg ( _I |` ( 1 ... ( P - 1 ) ) ) ) - -u 1 ) = ( ( ! ` ( P - 1 ) ) + 1 ) )
131	87, 130	breqtrd 4679	1 |- ( P e. Prime -> P || ( ( ! ` ( P - 1 ) ) + 1 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   C. wpss 3575   ~Pcpw 4158   class class class wbr 4653   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   mod cmo 12668   seqcseq 12801   ^cexp 12860   !cfa 13060   || cdvds 14983   Primecprime 15385   gsum_g cgsu 16101   Mndcmnd 17294  SubMndcsubmnd 17334  Cntzccntz 17748  CMndccmn 18193  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548  SubRingcsubrg 18776  ℂ_fldccnfld 19746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-cnfld 19747

This theorem is referenced by:  wilth  24797