Theorem lgsquad3 15728

Description: Extend lgsquad2 15727 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)

Assertion

Ref	Expression
lgsquad3	|- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( M /L N ) = ( ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) x. ( N /L M ) ) )

Proof of Theorem lgsquad3

Step	Hyp	Ref	Expression
1		simplrl 535	. . . . . . . . . 10 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> N e. NN )
2	1	nnzd 9536	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> N e. ZZ )
3		nnz 9433	. . . . . . . . . 10 |- ( M e. NN -> M e. ZZ )
4	3	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> M e. ZZ )
5		lgscl 15658	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N /L M ) e. ZZ )
6	2, 4, 5	syl2anc 411	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. ZZ )
7	6	zred 9537	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. RR )
8		absresq 11555	. . . . . . 7 |- ( ( N /L M ) e. RR -> ( ( abs ` ( N /L M ) ) ^ 2 ) = ( ( N /L M ) ^ 2 ) )
9	7, 8	syl 14	. . . . . 6 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( abs ` ( N /L M ) ) ^ 2 ) = ( ( N /L M ) ^ 2 ) )
10	2, 4	gcdcomd 12461	. . . . . . . . . 10 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N gcd M ) = ( M gcd N ) )
11		simpr 110	. . . . . . . . . 10 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( M gcd N ) = 1 )
12	10, 11	eqtrd 2242	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N gcd M ) = 1 )
13		lgsabs1 15683	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( abs ` ( N /L M ) ) = 1 <-> ( N gcd M ) = 1 ) )
14	2, 4, 13	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( abs ` ( N /L M ) ) = 1 <-> ( N gcd M ) = 1 ) )
15	12, 14	mpbird 167	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( abs ` ( N /L M ) ) = 1 )
16	15	oveq1d 5989	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( abs ` ( N /L M ) ) ^ 2 ) = ( 1 ^ 2 ) )
17		sq1 10822	. . . . . . 7 |- ( 1 ^ 2 ) = 1
18	16, 17	eqtrdi 2258	. . . . . 6 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( abs ` ( N /L M ) ) ^ 2 ) = 1 )
19	6	zcnd 9538	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. CC )
20	19	sqvald 10859	. . . . . 6 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( N /L M ) ^ 2 ) = ( ( N /L M ) x. ( N /L M ) ) )
21	9, 18, 20	3eqtr3d 2250	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> 1 = ( ( N /L M ) x. ( N /L M ) ) )
22	21	oveq2d 5990	. . . 4 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( M /L N ) x. 1 ) = ( ( M /L N ) x. ( ( N /L M ) x. ( N /L M ) ) ) )
23		lgscl 15658	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M /L N ) e. ZZ )
24	4, 2, 23	syl2anc 411	. . . . . 6 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( M /L N ) e. ZZ )
25	24	zcnd 9538	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( M /L N ) e. CC )
26	25, 19, 19	mulassd 8138	. . . 4 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( ( M /L N ) x. ( N /L M ) ) x. ( N /L M ) ) = ( ( M /L N ) x. ( ( N /L M ) x. ( N /L M ) ) ) )
27	22, 26	eqtr4d 2245	. . 3 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( M /L N ) x. 1 ) = ( ( ( M /L N ) x. ( N /L M ) ) x. ( N /L M ) ) )
28	25	mulridd 8131	. . 3 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( M /L N ) x. 1 ) = ( M /L N ) )
29		simplll 533	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> M e. NN )
30		simpllr 534	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> -. 2 || M )
31		simplrr 536	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> -. 2 || N )
32	29, 30, 1, 31, 11	lgsquad2 15727	. . . 4 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( M /L N ) x. ( N /L M ) ) = ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )
33	32	oveq1d 5989	. . 3 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( ( M /L N ) x. ( N /L M ) ) x. ( N /L M ) ) = ( ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) x. ( N /L M ) ) )
34	27, 28, 33	3eqtr3d 2250	. 2 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( M /L N ) = ( ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) x. ( N /L M ) ) )
35		neg1cn 9183	. . . . . 6 |- -u 1 e. CC
36	35	a1i 9	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> -u 1 e. CC )
37		neg1ap0 9187	. . . . . 6 |- -u 1 # 0
38	37	a1i 9	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> -u 1 # 0 )
39	3	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> M e. ZZ )
40		simpllr 534	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> -. 2 || M )
41		1zzd 9441	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 1 e. ZZ )
42		2prm 12615	. . . . . . . . 9 |- 2 e. Prime
43		nprmdvds1 12628	. . . . . . . . 9 |- ( 2 e. Prime -> -. 2 || 1 )
44	42, 43	mp1i 10	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> -. 2 || 1 )
45		omoe 12373	. . . . . . . 8 |- ( ( ( M e. ZZ /\ -. 2 || M ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( M - 1 ) )
46	39, 40, 41, 44, 45	syl22anc 1253	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 2 || ( M - 1 ) )
47		2z 9442	. . . . . . . 8 |- 2 e. ZZ
48		2ne0 9170	. . . . . . . 8 |- 2 =/= 0
49		peano2zm 9452	. . . . . . . . 9 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
50	39, 49	syl 14	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( M - 1 ) e. ZZ )
51		dvdsval2 12267	. . . . . . . 8 |- ( ( 2 e. ZZ /\ 2 =/= 0 /\ ( M - 1 ) e. ZZ ) -> ( 2 || ( M - 1 ) <-> ( ( M - 1 ) / 2 ) e. ZZ ) )
52	47, 48, 50, 51	mp3an12i 1356	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( 2 || ( M - 1 ) <-> ( ( M - 1 ) / 2 ) e. ZZ ) )
53	46, 52	mpbid 147	. . . . . 6 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( ( M - 1 ) / 2 ) e. ZZ )
54		nnz 9433	. . . . . . . . . 10 |- ( N e. NN -> N e. ZZ )
55	54	adantr 276	. . . . . . . . 9 |- ( ( N e. NN /\ -. 2 || N ) -> N e. ZZ )
56	55	ad2antlr 489	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> N e. ZZ )
57		simplrr 536	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> -. 2 || N )
58		omoe 12373	. . . . . . . 8 |- ( ( ( N e. ZZ /\ -. 2 || N ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( N - 1 ) )
59	56, 57, 41, 44, 58	syl22anc 1253	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 2 || ( N - 1 ) )
60		peano2zm 9452	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
61	56, 60	syl 14	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( N - 1 ) e. ZZ )
62		dvdsval2 12267	. . . . . . . 8 |- ( ( 2 e. ZZ /\ 2 =/= 0 /\ ( N - 1 ) e. ZZ ) -> ( 2 || ( N - 1 ) <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
63	47, 48, 61, 62	mp3an12i 1356	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( 2 || ( N - 1 ) <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
64	59, 63	mpbid 147	. . . . . 6 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( ( N - 1 ) / 2 ) e. ZZ )
65	53, 64	zmulcld 9543	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) e. ZZ )
66	36, 38, 65	expclzapd 10867	. . . 4 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) e. CC )
67	66	mul01d 8507	. . 3 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) x. 0 ) = 0 )
68	54, 3, 5	syl2anr 290	. . . . . . . 8 |- ( ( M e. NN /\ N e. NN ) -> ( N /L M ) e. ZZ )
69		0zd 9426	. . . . . . . 8 |- ( ( M e. NN /\ N e. NN ) -> 0 e. ZZ )
70		zdceq 9490	. . . . . . . 8 |- ( ( ( N /L M ) e. ZZ /\ 0 e. ZZ ) -> DECID ( N /L M ) = 0 )
71	68, 69, 70	syl2anc 411	. . . . . . 7 |- ( ( M e. NN /\ N e. NN ) -> DECID ( N /L M ) = 0 )
72		lgsne0 15682	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N /L M ) =/= 0 <-> ( N gcd M ) = 1 ) )
73		gcdcom 12460	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) = ( M gcd N ) )
74	73	eqeq1d 2218	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N gcd M ) = 1 <-> ( M gcd N ) = 1 ) )
75	72, 74	bitrd 188	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N /L M ) =/= 0 <-> ( M gcd N ) = 1 ) )
76	54, 3, 75	syl2anr 290	. . . . . . . . 9 |- ( ( M e. NN /\ N e. NN ) -> ( ( N /L M ) =/= 0 <-> ( M gcd N ) = 1 ) )
77	76	a1d 22	. . . . . . . 8 |- ( ( M e. NN /\ N e. NN ) -> (DECID ( N /L M ) = 0 -> ( ( N /L M ) =/= 0 <-> ( M gcd N ) = 1 ) ) )
78	77	necon1bbiddc 2443	. . . . . . 7 |- ( ( M e. NN /\ N e. NN ) -> (DECID ( N /L M ) = 0 -> ( -. ( M gcd N ) = 1 <-> ( N /L M ) = 0 ) ) )
79	71, 78	mpd 13	. . . . . 6 |- ( ( M e. NN /\ N e. NN ) -> ( -. ( M gcd N ) = 1 <-> ( N /L M ) = 0 ) )
80	79	ad2ant2r 509	. . . . 5 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( -. ( M gcd N ) = 1 <-> ( N /L M ) = 0 ) )
81	80	biimpa 296	. . . 4 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( N /L M ) = 0 )
82	81	oveq2d 5990	. . 3 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) x. ( N /L M ) ) = ( ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) x. 0 ) )
83		0zd 9426	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
84		zdceq 9490	. . . . . . . 8 |- ( ( ( M /L N ) e. ZZ /\ 0 e. ZZ ) -> DECID ( M /L N ) = 0 )
85	23, 83, 84	syl2anc 411	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M /L N ) = 0 )
86		lgsne0 15682	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M /L N ) =/= 0 <-> ( M gcd N ) = 1 ) )
87	86	a1d 22	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> (DECID ( M /L N ) = 0 -> ( ( M /L N ) =/= 0 <-> ( M gcd N ) = 1 ) ) )
88	87	necon1bbiddc 2443	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> (DECID ( M /L N ) = 0 -> ( -. ( M gcd N ) = 1 <-> ( M /L N ) = 0 ) ) )
89	85, 88	mpd 13	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M gcd N ) = 1 <-> ( M /L N ) = 0 ) )
90	3, 54, 89	syl2an 289	. . . . 5 |- ( ( M e. NN /\ N e. NN ) -> ( -. ( M gcd N ) = 1 <-> ( M /L N ) = 0 ) )
91	90	ad2ant2r 509	. . . 4 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( -. ( M gcd N ) = 1 <-> ( M /L N ) = 0 ) )
92	91	biimpa 296	. . 3 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( M /L N ) = 0 )
93	67, 82, 92	3eqtr4rd 2253	. 2 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> ( M /L N ) = ( ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) x. ( N /L M ) ) )
94		gcdnncl 12454	. . . . . 6 |- ( ( M e. NN /\ N e. NN ) -> ( M gcd N ) e. NN )
95	94	ad2ant2r 509	. . . . 5 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( M gcd N ) e. NN )
96	95	nnzd 9536	. . . 4 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( M gcd N ) e. ZZ )
97		1zzd 9441	. . . 4 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> 1 e. ZZ )
98		zdceq 9490	. . . 4 |- ( ( ( M gcd N ) e. ZZ /\ 1 e. ZZ ) -> DECID ( M gcd N ) = 1 )
99	96, 97, 98	syl2anc 411	. . 3 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> DECID ( M gcd N ) = 1 )
100		exmiddc 840	. . 3 |- (DECID ( M gcd N ) = 1 -> ( ( M gcd N ) = 1 \/ -. ( M gcd N ) = 1 ) )
101	99, 100	syl 14	. 2 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( ( M gcd N ) = 1 \/ -. ( M gcd N ) = 1 ) )
102	34, 93, 101	mpjaodan 802	1 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( M /L N ) = ( ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) x. ( N /L M ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 712  DECID wdc 838   = wceq 1375   e. wcel 2180   =/= wne 2380   class class class wbr 4062   ` cfv 5294  (class class class)co 5974   CCcc 7965   RRcr 7966   0cc0 7967   1c1 7968   x. cmul 7972   - cmin 8285   -ucneg 8286   # cap 8696   / cdiv 8787   NNcn 9078   2c2 9129   ZZcz 9414   ^cexp 10727   abscabs 11474   || cdvds 12264   gcd cgcd 12440   Primecprime 12595   /Lclgs 15641

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087  ax-addf 8089  ax-mulf 8090

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-xor 1398  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-tp 3654  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-disj 4039  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-of 6188  df-1st 6256  df-2nd 6257  df-tpos 6361  df-recs 6421  df-irdg 6486  df-frec 6507  df-1o 6532  df-2o 6533  df-oadd 6536  df-er 6650  df-ec 6652  df-qs 6656  df-map 6767  df-en 6858  df-dom 6859  df-fin 6860  df-sup 7119  df-inf 7120  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-z 9415  df-dec 9547  df-uz 9691  df-q 9783  df-rp 9818  df-fz 10173  df-fzo 10307  df-fl 10457  df-mod 10512  df-seqfrec 10637  df-exp 10728  df-ihash 10965  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476  df-clim 11756  df-sumdc 11831  df-proddc 12028  df-dvds 12265  df-gcd 12441  df-prm 12596  df-phi 12699  df-pc 12774  df-struct 13000  df-ndx 13001  df-slot 13002  df-base 13004  df-sets 13005  df-iress 13006  df-plusg 13089  df-mulr 13090  df-starv 13091  df-sca 13092  df-vsca 13093  df-ip 13094  df-tset 13095  df-ple 13096  df-ds 13098  df-unif 13099  df-0g 13257  df-igsum 13258  df-topgen 13259  df-iimas 13301  df-qus 13302  df-mgm 13355  df-sgrp 13401  df-mnd 13416  df-mhm 13458  df-submnd 13459  df-grp 13502  df-minusg 13503  df-sbg 13504  df-mulg 13623  df-subg 13673  df-nsg 13674  df-eqg 13675  df-ghm 13744  df-cmn 13789  df-abl 13790  df-mgp 13850  df-rng 13862  df-ur 13889  df-srg 13893  df-ring 13927  df-cring 13928  df-oppr 13997  df-dvdsr 14018  df-unit 14019  df-invr 14050  df-dvr 14061  df-rhm 14081  df-nzr 14109  df-subrg 14148  df-domn 14188  df-idom 14189  df-lmod 14218  df-lssm 14282  df-lsp 14316  df-sra 14364  df-rgmod 14365  df-lidl 14398  df-rsp 14399  df-2idl 14429  df-bl 14475  df-mopn 14476  df-fg 14478  df-metu 14479  df-cnfld 14486  df-zring 14520  df-zrh 14543  df-zn 14545  df-lgs 15642

This theorem is referenced by: (None)