Theorem lgsquad3 15883

Description: Extend lgsquad2 15882 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 537	. . . . . . . . . 10 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> N e. NN )
2	1	nnzd 9644	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> N e. ZZ )
3		nnz 9541	. . . . . . . . . 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 15813	. . . . . . . . 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 9645	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. RR )
8		absresq 11699	. . . . . . 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 12606	. . . . . . . . . 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 2264	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N gcd M ) = 1 )
13		lgsabs1 15838	. . . . . . . . . 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 6043	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( abs ` ( N /L M ) ) ^ 2 ) = ( 1 ^ 2 ) )
17		sq1 10939	. . . . . . 7 |- ( 1 ^ 2 ) = 1
18	16, 17	eqtrdi 2280	. . . . . 6 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( abs ` ( N /L M ) ) ^ 2 ) = 1 )
19	6	zcnd 9646	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. CC )
20	19	sqvald 10976	. . . . . 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 2272	. . . . 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 6044	. . . 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 15813	. . . . . . 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 9646	. . . . 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 8246	. . . 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 2267	. . 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 8239	. . 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 535	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> M e. NN )
30		simpllr 536	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> -. 2 || M )
31		simplrr 538	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> -. 2 || N )
32	29, 30, 1, 31, 11	lgsquad2 15882	. . . 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 6043	. . 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 2272	. 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 9291	. . . . . 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 9295	. . . . . 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 536	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> -. 2 || M )
41		1zzd 9549	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 1 e. ZZ )
42		2prm 12760	. . . . . . . . 9 |- 2 e. Prime
43		nprmdvds1 12773	. . . . . . . . 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 12518	. . . . . . . 8 |- ( ( ( M e. ZZ /\ -. 2 || M ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( M - 1 ) )
46	39, 40, 41, 44, 45	syl22anc 1275	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 2 || ( M - 1 ) )
47		2z 9550	. . . . . . . 8 |- 2 e. ZZ
48		2ne0 9278	. . . . . . . 8 |- 2 =/= 0
49		peano2zm 9560	. . . . . . . . 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 12412	. . . . . . . 8 |- ( ( 2 e. ZZ /\ 2 =/= 0 /\ ( M - 1 ) e. ZZ ) -> ( 2 || ( M - 1 ) <-> ( ( M - 1 ) / 2 ) e. ZZ ) )
52	47, 48, 50, 51	mp3an12i 1378	. . . . . . 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 9541	. . . . . . . . . 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 538	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> -. 2 || N )
58		omoe 12518	. . . . . . . 8 |- ( ( ( N e. ZZ /\ -. 2 || N ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( N - 1 ) )
59	56, 57, 41, 44, 58	syl22anc 1275	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 2 || ( N - 1 ) )
60		peano2zm 9560	. . . . . . . . 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 12412	. . . . . . . 8 |- ( ( 2 e. ZZ /\ 2 =/= 0 /\ ( N - 1 ) e. ZZ ) -> ( 2 || ( N - 1 ) <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
63	47, 48, 61, 62	mp3an12i 1378	. . . . . . 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 9651	. . . . 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 10984	. . . 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 8615	. . 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 9534	. . . . . . . 8 |- ( ( M e. NN /\ N e. NN ) -> 0 e. ZZ )
70		zdceq 9598	. . . . . . . 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 15837	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N /L M ) =/= 0 <-> ( N gcd M ) = 1 ) )
73		gcdcom 12605	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) = ( M gcd N ) )
74	73	eqeq1d 2240	. . . . . . . . . . 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 2466	. . . . . . 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 6044	. . 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 9534	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
84		zdceq 9598	. . . . . . . 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 15837	. . . . . . . . 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 2466	. . . . . . 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 2275	. 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 12599	. . . . . 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 9644	. . . 4 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( M gcd N ) e. ZZ )
97		1zzd 9549	. . . 4 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> 1 e. ZZ )
98		zdceq 9598	. . . 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 844	. . 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 806	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 716  DECID wdc 842   = wceq 1398   e. wcel 2202   =/= wne 2403   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   RRcr 8074   0cc0 8075   1c1 8076   x. cmul 8080   - cmin 8393   -ucneg 8394   # cap 8804   / cdiv 8895   NNcn 9186   2c2 9237   ZZcz 9522   ^cexp 10844   abscabs 11618   || cdvds 12409   gcd cgcd 12585   Primecprime 12740   /Lclgs 15796

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195  ax-addf 8197  ax-mulf 8198

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-disj 4070  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244  df-1st 6312  df-2nd 6313  df-tpos 6454  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-2o 6626  df-oadd 6629  df-er 6745  df-ec 6747  df-qs 6751  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-sup 7226  df-inf 7227  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-q 9897  df-rp 9932  df-fz 10287  df-fzo 10421  df-fl 10574  df-mod 10629  df-seqfrec 10754  df-exp 10845  df-ihash 11082  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-clim 11900  df-sumdc 11975  df-proddc 12173  df-dvds 12410  df-gcd 12586  df-prm 12741  df-phi 12844  df-pc 12919  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-iress 13151  df-plusg 13234  df-mulr 13235  df-starv 13236  df-sca 13237  df-vsca 13238  df-ip 13239  df-tset 13240  df-ple 13241  df-ds 13243  df-unif 13244  df-0g 13402  df-igsum 13403  df-topgen 13404  df-iimas 13446  df-qus 13447  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-mhm 13603  df-submnd 13604  df-grp 13647  df-minusg 13648  df-sbg 13649  df-mulg 13768  df-subg 13818  df-nsg 13819  df-eqg 13820  df-ghm 13889  df-cmn 13934  df-abl 13935  df-mgp 13996  df-rng 14008  df-ur 14035  df-srg 14039  df-ring 14073  df-cring 14074  df-oppr 14143  df-dvdsr 14164  df-unit 14165  df-invr 14197  df-dvr 14208  df-rhm 14228  df-nzr 14256  df-subrg 14295  df-domn 14335  df-idom 14336  df-lmod 14365  df-lssm 14429  df-lsp 14463  df-sra 14511  df-rgmod 14512  df-lidl 14545  df-rsp 14546  df-2idl 14576  df-bl 14622  df-mopn 14623  df-fg 14625  df-metu 14626  df-cnfld 14633  df-zring 14667  df-zrh 14690  df-zn 14692  df-lgs 15797

This theorem is referenced by: (None)