Theorem lgsdilem 15143

Description: Lemma for lgsdi 15153 and lgsdir 15151: 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 9328	. . . . . . . . . . 11 |- 0 e. ZZ
4		simpl2 1003	. . . . . . . . . . . 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 9395	. . . . . . . . . . 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 1002	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. ZZ )
9	8	zred 9439	. . . . . . . . . . . . . . . 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 8399	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> -u A e. RR )
12	11	recnd 8048	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> -u A e. CC )
13	12	mul01d 8412	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( -u A x. 0 ) = 0 )
14	10	recnd 8048	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> A e. CC )
15	4	zred 9439	. . . . . . . . . . . . . . 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 8048	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B e. CC )
18	14, 17	mulneg1d 8430	. . . . . . . . . . . 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 4042	. . . . . . . . . . 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 8020	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> 0 e. RR )
21	9	lt0neg1d 8534	. . . . . . . . . . . . 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 8875	. . . . . . . . . . . 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 1253	. . . . . . . . . . 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 8050	. . . . . . . . . . . . 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 8534	. . . . . . . . . . 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 8019	. . . . . . . . . 10 |- 0 e. RR
31		lenlt 8095	. . . . . . . . . 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 3578	. . . . . . 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 9394	. . . . . . . . . . 11 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B < 0 )
36	3, 35	mpan2 425	. . . . . . . . . 10 |- ( B e. ZZ -> DECID B < 0 )
37		oveq2 5926	. . . . . . . . . . . . 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 9194	. . . . . . . . . . . . 13 |- ( -u 1 x. -u 1 ) = 1
39	37, 38	eqtrdi 2242	. . . . . . . . . . . 12 |- ( if ( B < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = 1 )
40		oveq2 5926	. . . . . . . . . . . . 13 |- ( if ( B < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = ( -u 1 x. 1 ) )
41		ax-1cn 7965	. . . . . . . . . . . . . 14 |- 1 e. CC
42	41	mulm1i 8422	. . . . . . . . . . . . 13 |- ( -u 1 x. 1 ) = -u 1
43	40, 42	eqtrdi 2242	. . . . . . . . . . . 12 |- ( if ( B < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = -u 1 )
44	39, 43	ifsbdc 3569	. . . . . . . . . . 11 |- (DECID B < 0 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = if ( B < 0 , 1 , -u 1 ) )
45		ifnotdc 3594	. . . . . . . . . . 11 |- (DECID B < 0 -> if ( -. B < 0 , -u 1 , 1 ) = if ( B < 0 , 1 , -u 1 ) )
46	44, 45	eqtr4d 2229	. . . . . . . . . 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 1021	. . . . . . . 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 2229	. . . . . 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 3562	. . . . . . . 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 5933	. . . . . 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 2229	. . . . 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 3565	. . . . . . . 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 5933	. . . . . 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 9087	. . . . . . . . 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 3593	. . . . . . 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 8038	. . . . . 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 8020	. . . . . . . . 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 2385	. . . . . . . . . 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 9360	. . . . . . . . . . . . . . 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 983	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . 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 1360	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A = 0 \/ 0 < A ) )
78	77	orcomd 730	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( 0 < A \/ A = 0 ) )
79	69, 78	ecased 1360	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> 0 < A )
80		ltmul2 8875	. . . . . . . . 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 1253	. . . . . . . 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 8048	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A e. CC )
83	82	mul01d 8412	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A x. 0 ) = 0 )
84	83	breq2d 4041	. . . . . . . 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 3578	. . . . . 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 2231	. . . . 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 9394	. . . . . . 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 837	. . . . . 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 799	. . . 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 3578	. . 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 3578	. . . 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 3578	. . . 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 5936	. . 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 2234	. 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 933	. . . . 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 3567	. . . 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 9134	. . . 4 |- ( 1 x. 1 ) = 1
107	105, 106	eqtr4di 2244	. . 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 933	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. ( N < 0 /\ A < 0 ) )
109	108	iffalsed 3567	. . . 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 933	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. ( N < 0 /\ B < 0 ) )
111	110	iffalsed 3567	. . . 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 5936	. . 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 2229	. 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 1004	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> N e. ZZ )
115		zdclt 9394	. . . 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 837	. . 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 799	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 709  DECID wdc 835   \/ w3o 979   /\ w3a 980   = wceq 1364   e. wcel 2164   =/= wne 2364   ifcif 3557   class class class wbr 4029  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   1c1 7873   x. cmul 7877   < clt 8054   <_ cle 8055   -ucneg 8191   ZZcz 9317

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-inn 8983  df-n0 9241  df-z 9318

This theorem is referenced by:  lgsdir  15151  lgsdi  15153