Theorem lgseisen 15947

Description: Eisenstein's lemma, an expression for ( P /L Q ) when P , Q are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)

Hypotheses

Ref	Expression
lgseisen.1	|- ( ph -> P e. ( Prime \ { 2 } ) )
lgseisen.2	|- ( ph -> Q e. ( Prime \ { 2 } ) )
lgseisen.3	|- ( ph -> P =/= Q )

Assertion

Ref	Expression
lgseisen	|- ( ph -> ( Q /L P ) = ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) )

Distinct variable groups:   x, P   ph, x   x, Q

Proof of Theorem lgseisen

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lgseisen.2	. . . . 5 |- ( ph -> Q e. ( Prime \ { 2 } ) )
2	1	eldifad 3222	. . . 4 |- ( ph -> Q e. Prime )
3		prmz 12808	. . . 4 |- ( Q e. Prime -> Q e. ZZ )
4	2, 3	syl 14	. . 3 |- ( ph -> Q e. ZZ )
5		lgseisen.1	. . 3 |- ( ph -> P e. ( Prime \ { 2 } ) )
6		lgsval3 15891	. . 3 |- ( ( Q e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( Q /L P ) = ( ( ( ( Q ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
7	4, 5, 6	syl2anc 411	. 2 |- ( ph -> ( Q /L P ) = ( ( ( ( Q ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
8	1	gausslemma2dlem0a 15922	. . . . . . 7 |- ( ph -> Q e. NN )
9		oddprm 12957	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
10	5, 9	syl 14	. . . . . . . 8 |- ( ph -> ( ( P - 1 ) / 2 ) e. NN )
11	10	nnnn0d 9553	. . . . . . 7 |- ( ph -> ( ( P - 1 ) / 2 ) e. NN0 )
12	8, 11	nnexpcld 11057	. . . . . 6 |- ( ph -> ( Q ^ ( ( P - 1 ) / 2 ) ) e. NN )
13		nnq 9965	. . . . . 6 |- ( ( Q ^ ( ( P - 1 ) / 2 ) ) e. NN -> ( Q ^ ( ( P - 1 ) / 2 ) ) e. QQ )
14	12, 13	syl 14	. . . . 5 |- ( ph -> ( Q ^ ( ( P - 1 ) / 2 ) ) e. QQ )
15		1zzd 9604	. . . . . . . 8 |- ( ph -> 1 e. ZZ )
16	15	znegcld 9702	. . . . . . 7 |- ( ph -> -u 1 e. ZZ )
17		zq 9958	. . . . . . 7 |- ( -u 1 e. ZZ -> -u 1 e. QQ )
18	16, 17	syl 14	. . . . . 6 |- ( ph -> -u 1 e. QQ )
19		neg1ne0 9344	. . . . . . 7 |- -u 1 =/= 0
20	19	a1i 9	. . . . . 6 |- ( ph -> -u 1 =/= 0 )
21	10	nnzd 9699	. . . . . . . 8 |- ( ph -> ( ( P - 1 ) / 2 ) e. ZZ )
22	15, 21	fzfigd 10793	. . . . . . 7 |- ( ph -> ( 1 ... ( ( P - 1 ) / 2 ) ) e. Fin )
23	5	gausslemma2dlem0a 15922	. . . . . . . . . 10 |- ( ph -> P e. NN )
24		znq 9956	. . . . . . . . . 10 |- ( ( Q e. ZZ /\ P e. NN ) -> ( Q / P ) e. QQ )
25	4, 23, 24	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( Q / P ) e. QQ )
26		2z 9605	. . . . . . . . . . . 12 |- 2 e. ZZ
27	26	a1i 9	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ) -> 2 e. ZZ )
28		elfznn 10388	. . . . . . . . . . . . 13 |- ( x e. ( 1 ... ( ( P - 1 ) / 2 ) ) -> x e. NN )
29	28	adantl 277	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ) -> x e. NN )
30	29	nnzd 9699	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ) -> x e. ZZ )
31	27, 30	zmulcld 9706	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ) -> ( 2 x. x ) e. ZZ )
32		zq 9958	. . . . . . . . . 10 |- ( ( 2 x. x ) e. ZZ -> ( 2 x. x ) e. QQ )
33	31, 32	syl 14	. . . . . . . . 9 |- ( ( ph /\ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ) -> ( 2 x. x ) e. QQ )
34		qmulcl 9969	. . . . . . . . 9 |- ( ( ( Q / P ) e. QQ /\ ( 2 x. x ) e. QQ ) -> ( ( Q / P ) x. ( 2 x. x ) ) e. QQ )
35	25, 33, 34	syl2an2r 599	. . . . . . . 8 |- ( ( ph /\ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ) -> ( ( Q / P ) x. ( 2 x. x ) ) e. QQ )
36	35	flqcld 10637	. . . . . . 7 |- ( ( ph /\ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ) -> ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) e. ZZ )
37	22, 36	fsumzcl 12088	. . . . . 6 |- ( ph -> sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) e. ZZ )
38		qexpclz 10922	. . . . . 6 |- ( ( -u 1 e. QQ /\ -u 1 =/= 0 /\ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) e. ZZ ) -> ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) e. QQ )
39	18, 20, 37, 38	syl3anc 1274	. . . . 5 |- ( ph -> ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) e. QQ )
40		1z 9603	. . . . . 6 |- 1 e. ZZ
41		zq 9958	. . . . . 6 |- ( 1 e. ZZ -> 1 e. QQ )
42	40, 41	mp1i 10	. . . . 5 |- ( ph -> 1 e. QQ )
43		nnq 9965	. . . . . 6 |- ( P e. NN -> P e. QQ )
44	23, 43	syl 14	. . . . 5 |- ( ph -> P e. QQ )
45	23	nngt0d 9281	. . . . 5 |- ( ph -> 0 < P )
46		lgseisen.3	. . . . . 6 |- ( ph -> P =/= Q )
47		eqid 2232	. . . . . 6 |- ( ( Q x. ( 2 x. x ) ) mod P ) = ( ( Q x. ( 2 x. x ) ) mod P )
48		eqid 2232	. . . . . 6 |- ( x e. ( 1 ... ( ( P - 1 ) / 2 ) ) |-> ( ( ( ( -u 1 ^ ( ( Q x. ( 2 x. x ) ) mod P ) ) x. ( ( Q x. ( 2 x. x ) ) mod P ) ) mod P ) / 2 ) ) = ( x e. ( 1 ... ( ( P - 1 ) / 2 ) ) |-> ( ( ( ( -u 1 ^ ( ( Q x. ( 2 x. x ) ) mod P ) ) x. ( ( Q x. ( 2 x. x ) ) mod P ) ) mod P ) / 2 ) )
49		eqid 2232	. . . . . 6 |- ( ( Q x. ( 2 x. y ) ) mod P ) = ( ( Q x. ( 2 x. y ) ) mod P )
50		eqid 2232	. . . . . 6 |- (ℤ/nℤ ` P ) = (ℤ/nℤ ` P )
51		eqid 2232	. . . . . 6 |- (mulGrp ` (ℤ/nℤ ` P ) ) = (mulGrp ` (ℤ/nℤ ` P ) )
52		eqid 2232	. . . . . 6 |- ( ZRHom ` (ℤ/nℤ ` P ) ) = ( ZRHom ` (ℤ/nℤ ` P ) )
53	5, 1, 46, 47, 48, 49, 50, 51, 52	lgseisenlem4 15946	. . . . 5 |- ( ph -> ( ( Q ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) mod P ) )
54	14, 39, 42, 44, 45, 53	modqadd1 10723	. . . 4 |- ( ph -> ( ( ( Q ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) mod P ) )
55		qaddcl 9967	. . . . . 6 |- ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) e. QQ /\ 1 e. QQ ) -> ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) e. QQ )
56	39, 42, 55	syl2anc 411	. . . . 5 |- ( ph -> ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) e. QQ )
57		df-neg 8447	. . . . . . 7 |- -u 1 = ( 0 - 1 )
58		neg1cn 9342	. . . . . . . . . . . 12 |- -u 1 e. CC
59		neg1ap0 9346	. . . . . . . . . . . 12 |- -u 1 # 0
60		absexpzap 11765	. . . . . . . . . . . 12 |- ( ( -u 1 e. CC /\ -u 1 # 0 /\ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) e. ZZ ) -> ( abs ` ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) ) = ( ( abs ` -u 1 ) ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) )
61	58, 59, 37, 60	mp3an12i 1378	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) ) = ( ( abs ` -u 1 ) ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) )
62		ax-1cn 8220	. . . . . . . . . . . . . . 15 |- 1 e. CC
63	62	absnegi 11832	. . . . . . . . . . . . . 14 |- ( abs ` -u 1 ) = ( abs ` 1 )
64		abs1 11757	. . . . . . . . . . . . . 14 |- ( abs ` 1 ) = 1
65	63, 64	eqtri 2253	. . . . . . . . . . . . 13 |- ( abs ` -u 1 ) = 1
66	65	oveq1i 6060	. . . . . . . . . . . 12 |- ( ( abs ` -u 1 ) ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) = ( 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) )
67		1exp 10930	. . . . . . . . . . . . 13 |- ( sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) e. ZZ -> ( 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) = 1 )
68	37, 67	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) = 1 )
69	66, 68	eqtrid 2277	. . . . . . . . . . 11 |- ( ph -> ( ( abs ` -u 1 ) ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) = 1 )
70	61, 69	eqtrd 2265	. . . . . . . . . 10 |- ( ph -> ( abs ` ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) ) = 1 )
71		1le1 8846	. . . . . . . . . 10 |- 1 <_ 1
72	70, 71	eqbrtrdi 4148	. . . . . . . . 9 |- ( ph -> ( abs ` ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) ) <_ 1 )
73		neg1rr 9343	. . . . . . . . . . . 12 |- -u 1 e. RR
74	73	a1i 9	. . . . . . . . . . 11 |- ( ph -> -u 1 e. RR )
75	59	a1i 9	. . . . . . . . . . 11 |- ( ph -> -u 1 # 0 )
76	74, 75, 37	reexpclzapd 11060	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) e. RR )
77		1re 8273	. . . . . . . . . 10 |- 1 e. RR
78		absle 11774	. . . . . . . . . 10 |- ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) e. RR /\ 1 e. RR ) -> ( ( abs ` ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) ) <_ 1 <-> ( -u 1 <_ ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) /\ ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) <_ 1 ) ) )
79	76, 77, 78	sylancl 413	. . . . . . . . 9 |- ( ph -> ( ( abs ` ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) ) <_ 1 <-> ( -u 1 <_ ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) /\ ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) <_ 1 ) ) )
80	72, 79	mpbid 147	. . . . . . . 8 |- ( ph -> ( -u 1 <_ ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) /\ ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) <_ 1 ) )
81	80	simpld 112	. . . . . . 7 |- ( ph -> -u 1 <_ ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) )
82	57, 81	eqbrtrrid 4145	. . . . . 6 |- ( ph -> ( 0 - 1 ) <_ ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) )
83		0red 8275	. . . . . . 7 |- ( ph -> 0 e. RR )
84		1red 8289	. . . . . . 7 |- ( ph -> 1 e. RR )
85	83, 84, 76	lesubaddd 8816	. . . . . 6 |- ( ph -> ( ( 0 - 1 ) <_ ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) <-> 0 <_ ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) ) )
86	82, 85	mpbid 147	. . . . 5 |- ( ph -> 0 <_ ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) )
87	23	nnred 9250	. . . . . . . 8 |- ( ph -> P e. RR )
88		peano2rem 8540	. . . . . . . 8 |- ( P e. RR -> ( P - 1 ) e. RR )
89	87, 88	syl 14	. . . . . . 7 |- ( ph -> ( P - 1 ) e. RR )
90	80	simprd 114	. . . . . . 7 |- ( ph -> ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) <_ 1 )
91		df-2 9296	. . . . . . . . 9 |- 2 = ( 1 + 1 )
92		eldifsni 3822	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> P =/= 2 )
93	5, 92	syl 14	. . . . . . . . . . 11 |- ( ph -> P =/= 2 )
94	23	nnzd 9699	. . . . . . . . . . . 12 |- ( ph -> P e. ZZ )
95		zapne 9652	. . . . . . . . . . . 12 |- ( ( P e. ZZ /\ 2 e. ZZ ) -> ( P # 2 <-> P =/= 2 ) )
96	94, 26, 95	sylancl 413	. . . . . . . . . . 11 |- ( ph -> ( P # 2 <-> P =/= 2 ) )
97	93, 96	mpbird 167	. . . . . . . . . 10 |- ( ph -> P # 2 )
98		2re 9307	. . . . . . . . . . . 12 |- 2 e. RR
99	98	a1i 9	. . . . . . . . . . 11 |- ( ph -> 2 e. RR )
100	5	eldifad 3222	. . . . . . . . . . . 12 |- ( ph -> P e. Prime )
101		prmuz2 12828	. . . . . . . . . . . 12 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
102		eluzle 9866	. . . . . . . . . . . 12 |- ( P e. ( ZZ>= ` 2 ) -> 2 <_ P )
103	100, 101, 102	3syl 17	. . . . . . . . . . 11 |- ( ph -> 2 <_ P )
104	99, 87, 103	leltapd 8913	. . . . . . . . . 10 |- ( ph -> ( 2 < P <-> P # 2 ) )
105	97, 104	mpbird 167	. . . . . . . . 9 |- ( ph -> 2 < P )
106	91, 105	eqbrtrrid 4145	. . . . . . . 8 |- ( ph -> ( 1 + 1 ) < P )
107	84, 84, 87	ltaddsubd 8819	. . . . . . . 8 |- ( ph -> ( ( 1 + 1 ) < P <-> 1 < ( P - 1 ) ) )
108	106, 107	mpbid 147	. . . . . . 7 |- ( ph -> 1 < ( P - 1 ) )
109	76, 84, 89, 90, 108	lelttrd 8398	. . . . . 6 |- ( ph -> ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) < ( P - 1 ) )
110	76, 84, 87	ltaddsubd 8819	. . . . . 6 |- ( ph -> ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) < P <-> ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) < ( P - 1 ) ) )
111	109, 110	mpbird 167	. . . . 5 |- ( ph -> ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) < P )
112		modqid 10711	. . . . 5 |- ( ( ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) e. QQ /\ P e. QQ ) /\ ( 0 <_ ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) /\ ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) < P ) ) -> ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) mod P ) = ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) )
113	56, 44, 86, 111, 112	syl22anc 1275	. . . 4 |- ( ph -> ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) mod P ) = ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) )
114	54, 113	eqtrd 2265	. . 3 |- ( ph -> ( ( ( Q ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) )
115	114	oveq1d 6065	. 2 |- ( ph -> ( ( ( ( Q ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) - 1 ) )
116	76	recnd 8302	. . 3 |- ( ph -> ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) e. CC )
117		pncan 8479	. . 3 |- ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) e. CC /\ 1 e. CC ) -> ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) - 1 ) = ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) )
118	116, 62, 117	sylancl 413	. 2 |- ( ph -> ( ( ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) + 1 ) - 1 ) = ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) )
119	7, 115, 118	3eqtrd 2269	1 |- ( ph -> ( Q /L P ) = ( -u 1 ^ sum_ x e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. x ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   =/= wne 2412   \ cdif 3208   {csn 3689   class class class wbr 4109   |-> cmpt 4171   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   < clt 8308   <_ cle 8309   - cmin 8444   -ucneg 8445   # cap 8855   / cdiv 8946   NNcn 9237   2c2 9288   ZZcz 9577   ZZ>=cuz 9853   QQcq 9951   ...cfz 10342   |_cfl 10628   mod cmo 10684   ^cexp 10900   abscabs 11682   sum_csu 12038   Primecprime 12804  mulGrpcmgp 14064   ZRHomczrh 14759  ℤ/nℤczn 14761   /Lclgs 15870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247  ax-addf 8249  ax-mulf 8250

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-tp 3697  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-of 6266  df-1st 6334  df-2nd 6335  df-tpos 6476  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-2o 6648  df-oadd 6651  df-er 6767  df-ec 6769  df-qs 6773  df-map 6884  df-en 6976  df-dom 6977  df-fin 6978  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-z 9578  df-dec 9710  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039  df-proddc 12237  df-dvds 12474  df-gcd 12650  df-prm 12805  df-phi 12908  df-pc 12983  df-struct 13214  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-starv 13305  df-sca 13306  df-vsca 13307  df-ip 13308  df-tset 13309  df-ple 13310  df-ds 13312  df-unif 13313  df-0g 13471  df-igsum 13472  df-topgen 13473  df-iimas 13515  df-qus 13516  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mhm 13672  df-submnd 13673  df-grp 13716  df-minusg 13717  df-sbg 13718  df-mulg 13837  df-subg 13887  df-nsg 13888  df-eqg 13889  df-ghm 13958  df-cmn 14003  df-abl 14004  df-mgp 14065  df-rng 14077  df-ur 14104  df-srg 14108  df-ring 14142  df-cring 14143  df-oppr 14212  df-dvdsr 14233  df-unit 14234  df-invr 14266  df-dvr 14277  df-rhm 14297  df-nzr 14325  df-subrg 14364  df-domn 14404  df-idom 14405  df-lmod 14437  df-lssm 14501  df-lsp 14535  df-sra 14583  df-rgmod 14584  df-lidl 14617  df-rsp 14618  df-2idl 14648  df-bl 14694  df-mopn 14695  df-fg 14697  df-metu 14698  df-cnfld 14705  df-zring 14739  df-zrh 14762  df-zn 14764  df-lgs 15871

This theorem is referenced by:  lgsquadlem2  15951