Theorem lgsquad3 16186

Description: Extend lgsquad2 16185 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 9750	. . . . . . . . 9 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> N e. ZZ )
3		nnz 9646	. . . . . . . . . 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 16116	. . . . . . . . 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 9751	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. RR )
8		absresq 11827	. . . . . . 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 12734	. . . . . . . . . 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 16141	. . . . . . . . . 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 6094	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( ( abs ` ( N /L M ) ) ^ 2 ) = ( 1 ^ 2 ) )
17		sq1 11053	. . . . . . 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 9752	. . . . . . 7 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ ( M gcd N ) = 1 ) -> ( N /L M ) e. CC )
20	19	sqvald 11091	. . . . . 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 6095	. . . 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 16116	. . . . . . 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 9752	. . . . 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 8343	. . . 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 8337	. . 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 16185	. . . 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 6094	. . 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 9392	. . . . . 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 9396	. . . . . 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 9654	. . . . . . . 8 |- ( ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) /\ -. ( M gcd N ) = 1 ) -> 1 e. ZZ )
42		2prm 12888	. . . . . . . . 9 |- 2 e. Prime
43		nprmdvds1 12901	. . . . . . . . 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 12646	. . . . . . . 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 9655	. . . . . . . 8 |- 2 e. ZZ
48		2ne0 9379	. . . . . . . 8 |- 2 =/= 0
49		peano2zm 9665	. . . . . . . . 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 12540	. . . . . . . 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 9646	. . . . . . . . . 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 12646	. . . . . . . 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 9665	. . . . . . . . 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 12540	. . . . . . . 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 9757	. . . . 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 11099	. . . 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 8714	. . 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 9639	. . . . . . . 8 |- ( ( M e. NN /\ N e. NN ) -> 0 e. ZZ )
70		zdceq 9703	. . . . . . . 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 16140	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N /L M ) =/= 0 <-> ( N gcd M ) = 1 ) )
73		gcdcom 12733	. . . . . . . . . . . 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 6095	. . 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 9639	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
84		zdceq 9703	. . . . . . . 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 16140	. . . . . . . . 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 12727	. . . . . 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 9750	. . . 4 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( M gcd N ) e. ZZ )
97		1zzd 9654	. . . 4 |- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> 1 e. ZZ )
98		zdceq 9703	. . . 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 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   CCcc 8171   RRcr 8172   0cc0 8173   1c1 8174   x. cmul 8178   - cmin 8491   -ucneg 8492   # cap 8903   / cdiv 8996   NNcn 9287   2c2 9338   ZZcz 9627   ^cexp 10958   abscabs 11746   || cdvds 12537   gcd cgcd 12713   Primecprime 12868   /Lclgs 16099

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293  ax-addf 8295  ax-mulf 8296

This theorem 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 3714  df-pr 3715  df-tp 3716  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-disj 4105  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296  df-1st 6368  df-2nd 6369  df-tpos 6510  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-ec 6803  df-qs 6807  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-sumdc 12103  df-proddc 12301  df-dvds 12538  df-gcd 12714  df-prm 12869  df-phi 12972  df-pc 13047  df-struct 13337  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-starv 13429  df-sca 13430  df-vsca 13431  df-ip 13432  df-tset 13433  df-ple 13434  df-ds 13436  df-unif 13437  df-0g 13595  df-gzsum 13596  df-topgen 13597  df-iimas 13607  df-qus 13608  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-mhm 13749  df-submnd 13750  df-grp 13791  df-minusg 13792  df-sbg 13793  df-mulg 13906  df-subg 13956  df-nsg 13957  df-eqg 13958  df-ghm 14027  df-cmn 14072  df-abl 14073  df-gsumfi 14134  df-mgp 14201  df-rng 14215  df-ur 14246  df-srg 14251  df-ring 14285  df-cring 14286  df-oppr 14356  df-dvdsr 14378  df-unit 14379  df-invr 14411  df-dvr 14422  df-rhm 14442  df-nzr 14470  df-subrg 14510  df-domn 14550  df-idom 14551  df-lmod 14608  df-lssm 14673  df-lsp 14707  df-sra 14755  df-rgmod 14756  df-lidl 14789  df-rsp 14790  df-2idl 14820  df-bl 14866  df-mopn 14867  df-fg 14869  df-metu 14870  df-cnfld 14877  df-zring 14909  df-zrh 14932  df-zn 14934  df-lgs 16100

This theorem is referenced by: (None)