Theorem lgsdilem 15537

Description: Lemma for lgsdi 15547 and lgsdir 15545: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
lgsdilem	|- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) )

Proof of Theorem lgsdilem

Step	Hyp	Ref	Expression
1		simplrr 536	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B =/= 0 )
2	1	biantrud 304	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( 0 <_ B <-> ( 0 <_ B /\ B =/= 0 ) ) )
3		0z 9385	. . . . . . . . . . 11 |- 0 e. ZZ
4		simpl2 1004	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> B e. ZZ )
5	4	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B e. ZZ )
6		zltlen 9453	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ B e. ZZ ) -> ( 0 < B <-> ( 0 <_ B /\ B =/= 0 ) ) )
7	3, 5, 6	sylancr 414	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( 0 < B <-> ( 0 <_ B /\ B =/= 0 ) ) )
8		simpl1 1003	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. ZZ )
9	8	zred 9497	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. RR )
10	9	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> A e. RR )
11	10	renegcld 8454	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> -u A e. RR )
12	11	recnd 8103	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> -u A e. CC )
13	12	mul01d 8467	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( -u A x. 0 ) = 0 )
14	10	recnd 8103	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> A e. CC )
15	4	zred 9497	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> B e. RR )
16	15	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B e. RR )
17	16	recnd 8103	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B e. CC )
18	14, 17	mulneg1d 8485	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( -u A x. B ) = -u ( A x. B ) )
19	13, 18	breq12d 4058	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( ( -u A x. 0 ) < ( -u A x. B ) <-> 0 < -u ( A x. B ) ) )
20		0red 8075	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> 0 e. RR )
21	9	lt0neg1d 8590	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A < 0 <-> 0 < -u A ) )
22	21	biimpa 296	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> 0 < -u A )
23		ltmul2 8931	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ B e. RR /\ ( -u A e. RR /\ 0 < -u A ) ) -> ( 0 < B <-> ( -u A x. 0 ) < ( -u A x. B ) ) )
24	20, 16, 11, 22, 23	syl112anc 1254	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( 0 < B <-> ( -u A x. 0 ) < ( -u A x. B ) ) )
25	9, 15	remulcld 8105	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A x. B ) e. RR )
26	25	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( A x. B ) e. RR )
27	26	lt0neg1d 8590	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( ( A x. B ) < 0 <-> 0 < -u ( A x. B ) ) )
28	19, 24, 27	3bitr4d 220	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( 0 < B <-> ( A x. B ) < 0 ) )
29	2, 7, 28	3bitr2rd 217	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( ( A x. B ) < 0 <-> 0 <_ B ) )
30		0re 8074	. . . . . . . . . 10 |- 0 e. RR
31		lenlt 8150	. . . . . . . . . 10 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 <_ B <-> -. B < 0 ) )
32	30, 16, 31	sylancr 414	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( 0 <_ B <-> -. B < 0 ) )
33	29, 32	bitrd 188	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( ( A x. B ) < 0 <-> -. B < 0 ) )
34	33	ifbid 3592	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> if ( ( A x. B ) < 0 , -u 1 , 1 ) = if ( -. B < 0 , -u 1 , 1 ) )
35		zdclt 9452	. . . . . . . . . . 11 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B < 0 )
36	3, 35	mpan2 425	. . . . . . . . . 10 |- ( B e. ZZ -> DECID B < 0 )
37		oveq2 5954	. . . . . . . . . . . . 13 |- ( if ( B < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = ( -u 1 x. -u 1 ) )
38		neg1mulneg1e1 9251	. . . . . . . . . . . . 13 |- ( -u 1 x. -u 1 ) = 1
39	37, 38	eqtrdi 2254	. . . . . . . . . . . 12 |- ( if ( B < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = 1 )
40		oveq2 5954	. . . . . . . . . . . . 13 |- ( if ( B < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = ( -u 1 x. 1 ) )
41		ax-1cn 8020	. . . . . . . . . . . . . 14 |- 1 e. CC
42	41	mulm1i 8477	. . . . . . . . . . . . 13 |- ( -u 1 x. 1 ) = -u 1
43	40, 42	eqtrdi 2254	. . . . . . . . . . . 12 |- ( if ( B < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = -u 1 )
44	39, 43	ifsbdc 3583	. . . . . . . . . . 11 |- (DECID B < 0 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = if ( B < 0 , 1 , -u 1 ) )
45		ifnotdc 3609	. . . . . . . . . . 11 |- (DECID B < 0 -> if ( -. B < 0 , -u 1 , 1 ) = if ( B < 0 , 1 , -u 1 ) )
46	44, 45	eqtr4d 2241	. . . . . . . . . 10 |- (DECID B < 0 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = if ( -. B < 0 , -u 1 , 1 ) )
47	36, 46	syl 14	. . . . . . . . 9 |- ( B e. ZZ -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = if ( -. B < 0 , -u 1 , 1 ) )
48	47	3ad2ant2 1022	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = if ( -. B < 0 , -u 1 , 1 ) )
49	48	ad2antrr 488	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = if ( -. B < 0 , -u 1 , 1 ) )
50	34, 49	eqtr4d 2241	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> if ( ( A x. B ) < 0 , -u 1 , 1 ) = ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) )
51		iftrue 3576	. . . . . . . 8 |- ( A < 0 -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
52	51	adantl 277	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
53	52	oveq1d 5961	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( if ( A < 0 , -u 1 , 1 ) x. if ( B < 0 , -u 1 , 1 ) ) = ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) )
54	50, 53	eqtr4d 2241	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> if ( ( A x. B ) < 0 , -u 1 , 1 ) = ( if ( A < 0 , -u 1 , 1 ) x. if ( B < 0 , -u 1 , 1 ) ) )
55		iffalse 3579	. . . . . . . 8 |- ( -. A < 0 -> if ( A < 0 , -u 1 , 1 ) = 1 )
56	55	adantl 277	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = 1 )
57	56	oveq1d 5961	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( if ( A < 0 , -u 1 , 1 ) x. if ( B < 0 , -u 1 , 1 ) ) = ( 1 x. if ( B < 0 , -u 1 , 1 ) ) )
58		neg1cn 9143	. . . . . . . . 9 |- -u 1 e. CC
59	58	a1i 9	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> -u 1 e. CC )
60	41	a1i 9	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> 1 e. CC )
61	4	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> B e. ZZ )
62	61, 3, 35	sylancl 413	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> DECID B < 0 )
63	59, 60, 62	ifcldcd 3608	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> if ( B < 0 , -u 1 , 1 ) e. CC )
64	63	mulid2d 8093	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( 1 x. if ( B < 0 , -u 1 , 1 ) ) = if ( B < 0 , -u 1 , 1 ) )
65	15	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> B e. RR )
66		0red 8075	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> 0 e. RR )
67	9	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A e. RR )
68		simplrl 535	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A =/= 0 )
69	68	neneqd 2397	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> -. A = 0 )
70		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> -. A < 0 )
71	8	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A e. ZZ )
72		ztri3or 9417	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A < 0 \/ A = 0 \/ 0 < A ) )
73	71, 3, 72	sylancl 413	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A < 0 \/ A = 0 \/ 0 < A ) )
74		3orass 984	. . . . . . . . . . . . . 14 |- ( ( A < 0 \/ A = 0 \/ 0 < A ) <-> ( A < 0 \/ ( A = 0 \/ 0 < A ) ) )
75	73, 74	sylib 122	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A < 0 \/ ( A = 0 \/ 0 < A ) ) )
76	75	orcomd 731	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( ( A = 0 \/ 0 < A ) \/ A < 0 ) )
77	70, 76	ecased 1362	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A = 0 \/ 0 < A ) )
78	77	orcomd 731	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( 0 < A \/ A = 0 ) )
79	69, 78	ecased 1362	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> 0 < A )
80		ltmul2 8931	. . . . . . . . 9 |- ( ( B e. RR /\ 0 e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( B < 0 <-> ( A x. B ) < ( A x. 0 ) ) )
81	65, 66, 67, 79, 80	syl112anc 1254	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( B < 0 <-> ( A x. B ) < ( A x. 0 ) ) )
82	67	recnd 8103	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A e. CC )
83	82	mul01d 8467	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A x. 0 ) = 0 )
84	83	breq2d 4057	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( ( A x. B ) < ( A x. 0 ) <-> ( A x. B ) < 0 ) )
85	81, 84	bitrd 188	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( B < 0 <-> ( A x. B ) < 0 ) )
86	85	ifbid 3592	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> if ( B < 0 , -u 1 , 1 ) = if ( ( A x. B ) < 0 , -u 1 , 1 ) )
87	57, 64, 86	3eqtrrd 2243	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> if ( ( A x. B ) < 0 , -u 1 , 1 ) = ( if ( A < 0 , -u 1 , 1 ) x. if ( B < 0 , -u 1 , 1 ) ) )
88		zdclt 9452	. . . . . . 7 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
89	8, 3, 88	sylancl 413	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> DECID A < 0 )
90		exmiddc 838	. . . . . 6 |- (DECID A < 0 -> ( A < 0 \/ -. A < 0 ) )
91	89, 90	syl 14	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A < 0 \/ -. A < 0 ) )
92	54, 87, 91	mpjaodan 800	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> if ( ( A x. B ) < 0 , -u 1 , 1 ) = ( if ( A < 0 , -u 1 , 1 ) x. if ( B < 0 , -u 1 , 1 ) ) )
93	92	adantr 276	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> if ( ( A x. B ) < 0 , -u 1 , 1 ) = ( if ( A < 0 , -u 1 , 1 ) x. if ( B < 0 , -u 1 , 1 ) ) )
94		simpr 110	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> N < 0 )
95	94	biantrurd 305	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> ( ( A x. B ) < 0 <-> ( N < 0 /\ ( A x. B ) < 0 ) ) )
96	95	ifbid 3592	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> if ( ( A x. B ) < 0 , -u 1 , 1 ) = if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) )
97	94	biantrurd 305	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> ( A < 0 <-> ( N < 0 /\ A < 0 ) ) )
98	97	ifbid 3592	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> if ( A < 0 , -u 1 , 1 ) = if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) )
99	94	biantrurd 305	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> ( B < 0 <-> ( N < 0 /\ B < 0 ) ) )
100	99	ifbid 3592	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> if ( B < 0 , -u 1 , 1 ) = if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) )
101	98, 100	oveq12d 5964	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> ( if ( A < 0 , -u 1 , 1 ) x. if ( B < 0 , -u 1 , 1 ) ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) )
102	93, 96, 101	3eqtr3d 2246	. 2 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) )
103		simpr 110	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. N < 0 )
104	103	intnanrd 934	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. ( N < 0 /\ ( A x. B ) < 0 ) )
105	104	iffalsed 3581	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) = 1 )
106		1t1e1 9191	. . . 4 |- ( 1 x. 1 ) = 1
107	105, 106	eqtr4di 2256	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) = ( 1 x. 1 ) )
108	103	intnanrd 934	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. ( N < 0 /\ A < 0 ) )
109	108	iffalsed 3581	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) = 1 )
110	103	intnanrd 934	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. ( N < 0 /\ B < 0 ) )
111	110	iffalsed 3581	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) = 1 )
112	109, 111	oveq12d 5964	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) = ( 1 x. 1 ) )
113	107, 112	eqtr4d 2241	. 2 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) )
114		simpl3 1005	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> N e. ZZ )
115		zdclt 9452	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
116	114, 3, 115	sylancl 413	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> DECID N < 0 )
117		exmiddc 838	. . 3 |- (DECID N < 0 -> ( N < 0 \/ -. N < 0 ) )
118	116, 117	syl 14	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( N < 0 \/ -. N < 0 ) )
119	102, 113, 118	mpjaodan 800	1 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710  DECID wdc 836   \/ w3o 980   /\ w3a 981   = wceq 1373   e. wcel 2176   =/= wne 2376   ifcif 3571   class class class wbr 4045  (class class class)co 5946   CCcc 7925   RRcr 7926   0cc0 7927   1c1 7928   x. cmul 7932   < clt 8109   <_ cle 8110   -ucneg 8246   ZZcz 9374

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-inn 9039  df-n0 9298  df-z 9375

This theorem is referenced by:  lgsdir  15545  lgsdi  15547