Theorem lgsquad3 16215

Description: Extend lgsquad2 16214 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 541	. . . . . . . . . 10 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> N e. NN )
2	1	nnzd 9769	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> N e. ZZ )
3		nnz 9665	. . . . . . . . . 10 |- ( M e. NN -> M e. ZZ )
4	3	ad3antrrr 496	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> M e. ZZ )
5		lgscl 16145	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N /L M ) e. ZZ )
6	2, 4, 5	syl2anc 415	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. ZZ )
7	6	zred 9770	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. RR )
8		absresq 11846	. . . . . . 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 12753	. . . . . . . . . 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 2271	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N gcd M ) = 1 )
13		lgsabs1 16170	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( abs ` ( N /L M ) ) = 1 <-> ( N gcd M ) = 1 ) )
14	2, 4, 13	syl2anc 415	. . . . . . . . 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 6100	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( abs ` ( N /L M ) ) ^ 2 ) = ( 1 ^ 2 ) )
17		sq1 11072	. . . . . . 7 |- ( 1 ^ 2 ) = 1
18	16, 17	eqtrdi 2287	. . . . . 6 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( abs ` ( N /L M ) ) ^ 2 ) = 1 )
19	6	zcnd 9771	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. CC )
20	19	sqvald 11110	. . . . . 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 2279	. . . . 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 6101	. . . 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 16145	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M /L N ) e. ZZ )
24	4, 2, 23	syl2anc 415	. . . . . 6 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( M /L N ) e. ZZ )
25	24	zcnd 9771	. . . . 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 8349	. . . 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 2274	. . 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 8343	. . 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 539	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> M e. NN )
30		simpllr 540	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> -. 2 || M )
31		simplrr 542	. . . . 5 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> -. 2 || N )
32	29, 30, 1, 31, 11	lgsquad2 16214	. . . 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 6100	. . 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 2279	. 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 9410	. . . . . 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 9414	. . . . . 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 496	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> M e. ZZ )
40		simpllr 540	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> -. 2 || M )
41		1zzd 9673	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 1 e. ZZ )
42		2prm 12907	. . . . . . . . 9 |- 2 e. Prime
43		nprmdvds1 12920	. . . . . . . . 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 12665	. . . . . . . 8 |- ( ( ( M e. ZZ /\ -. 2 || M ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( M - 1 ) )
46	39, 40, 41, 44, 45	syl22anc 1279	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 2 || ( M - 1 ) )
47		2z 9674	. . . . . . . 8 |- 2 e. ZZ
48		2ne0 9397	. . . . . . . 8 |- 2 =/= 0
49		peano2zm 9684	. . . . . . . . 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 12559	. . . . . . . 8 |- ( ( 2 e. ZZ /\ 2 =/= 0 /\ ( M - 1 ) e. ZZ ) -> ( 2 || ( M - 1 ) <-> ( ( M - 1 ) / 2 ) e. ZZ ) )
52	47, 48, 50, 51	mp3an12i 1382	. . . . . . 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 9665	. . . . . . . . . 10 |- ( N e. NN -> N e. ZZ )
55	54	adantr 276	. . . . . . . . 9 |- ( ( N e. NN /\ -. 2 || N ) -> N e. ZZ )
56	55	ad2antlr 493	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> N e. ZZ )
57		simplrr 542	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> -. 2 || N )
58		omoe 12665	. . . . . . . 8 |- ( ( ( N e. ZZ /\ -. 2 || N ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( N - 1 ) )
59	56, 57, 41, 44, 58	syl22anc 1279	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 2 || ( N - 1 ) )
60		peano2zm 9684	. . . . . . . . 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 12559	. . . . . . . 8 |- ( ( 2 e. ZZ /\ 2 =/= 0 /\ ( N - 1 ) e. ZZ ) -> ( 2 || ( N - 1 ) <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
63	47, 48, 61, 62	mp3an12i 1382	. . . . . . 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 9776	. . . . 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 11118	. . . 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 8720	. . 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 9658	. . . . . . . 8 |- ( ( M e. NN /\ N e. NN ) -> 0 e. ZZ )
70		zdceq 9722	. . . . . . . 8 |- ( ( ( N /L M ) e. ZZ /\ 0 e. ZZ ) -> DECID ( N /L M ) = 0 )
71	68, 69, 70	syl2anc 415	. . . . . . 7 |- ( ( M e. NN /\ N e. NN ) -> DECID ( N /L M ) = 0 )
72		lgsne0 16169	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N /L M ) =/= 0 <-> ( N gcd M ) = 1 ) )
73		gcdcom 12752	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) = ( M gcd N ) )
74	73	eqeq1d 2247	. . . . . . . . . . 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 2483	. . . . . . 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 513	. . . . 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 6101	. . 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 9658	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
84		zdceq 9722	. . . . . . . 8 |- ( ( ( M /L N ) e. ZZ /\ 0 e. ZZ ) -> DECID ( M /L N ) = 0 )
85	23, 83, 84	syl2anc 415	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M /L N ) = 0 )
86		lgsne0 16169	. . . . . . . . 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 2483	. . . . . . 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 513	. . . 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 2282	. 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 12746	. . . . . 6 |- ( ( M e. NN /\ N e. NN ) -> ( M gcd N ) e. NN )
95	94	ad2ant2r 513	. . . . 5 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( M gcd N ) e. NN )
96	95	nnzd 9769	. . . 4 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( M gcd N ) e. ZZ )
97		1zzd 9673	. . . 4 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> 1 e. ZZ )
98		zdceq 9722	. . . 4 |- ( ( ( M gcd N ) e. ZZ /\ 1 e. ZZ ) -> DECID ( M gcd N ) = 1 )
99	96, 97, 98	syl2anc 415	. . 3 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> DECID ( M gcd N ) = 1 )
100		exmiddc 848	. . 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 810	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 ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   x. cmul 8184   - cmin 8497   -ucneg 8498   # cap 8910   / cdiv 9003   NNcn 9305   2c2 9356   ZZcz 9646   ^cexp 10977   abscabs 11765   || cdvds 12556   gcd cgcd 12732   Primecprime 12887   /Lclgs 16128

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-tpos 6516  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-ec 6809  df-qs 6813  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-z 9647  df-dec 9780  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-fl 10707  df-mod 10762  df-seqfrec 10887  df-exp 10978  df-ihash 11217  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-clim 12047  df-sumdc 12122  df-proddc 12320  df-dvds 12557  df-gcd 12733  df-prm 12888  df-phi 12991  df-pc 13066  df-struct 13356  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-starv 13448  df-sca 13449  df-vsca 13450  df-ip 13451  df-tset 13452  df-ple 13453  df-ds 13455  df-unif 13456  df-0g 13614  df-gzsum 13615  df-topgen 13616  df-iimas 13626  df-qus 13627  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-mhm 13768  df-submnd 13769  df-grp 13810  df-minusg 13811  df-sbg 13812  df-mulg 13925  df-subg 13975  df-nsg 13976  df-eqg 13977  df-ghm 14046  df-cmn 14091  df-abl 14092  df-gsumfi 14153  df-mgp 14220  df-rng 14234  df-ur 14265  df-srg 14270  df-ring 14304  df-cring 14305  df-oppr 14375  df-dvdsr 14397  df-unit 14398  df-invr 14430  df-dvr 14441  df-rhm 14461  df-nzr 14489  df-subrg 14529  df-domn 14569  df-idom 14570  df-lmod 14627  df-lssm 14692  df-lsp 14726  df-sra 14774  df-rgmod 14775  df-lidl 14808  df-rsp 14809  df-2idl 14839  df-bl 14885  df-mopn 14886  df-fg 14888  df-metu 14889  df-cnfld 14896  df-zring 14928  df-zrh 14951  df-zn 14953  df-lgs 16129

This theorem is used by: (None)