Theorem lgsdilem 13528

Description: Lemma for lgsdi 13538 and lgsdir 13536: 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 526	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B =/= 0 )
2	1	biantrud 302	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( 0 <_ B <-> ( 0 <_ B /\ B =/= 0 ) ) )
3		0z 9198	. . . . . . . . . . 11 |- 0 e. ZZ
4		simpl2 991	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> B e. ZZ )
5	4	adantr 274	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B e. ZZ )
6		zltlen 9265	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ B e. ZZ ) -> ( 0 < B <-> ( 0 <_ B /\ B =/= 0 ) ) )
7	3, 5, 6	sylancr 411	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( 0 < B <-> ( 0 <_ B /\ B =/= 0 ) ) )
8		simpl1 990	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. ZZ )
9	8	zred 9309	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. RR )
10	9	adantr 274	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> A e. RR )
11	10	renegcld 8274	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> -u A e. RR )
12	11	recnd 7923	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> -u A e. CC )
13	12	mul01d 8287	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( -u A x. 0 ) = 0 )
14	10	recnd 7923	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> A e. CC )
15	4	zred 9309	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> B e. RR )
16	15	adantr 274	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B e. RR )
17	16	recnd 7923	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B e. CC )
18	14, 17	mulneg1d 8305	. . . . . . . . . . . 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 3994	. . . . . . . . . . 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 7896	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> 0 e. RR )
21	9	lt0neg1d 8409	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A < 0 <-> 0 < -u A ) )
22	21	biimpa 294	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> 0 < -u A )
23		ltmul2 8747	. . . . . . . . . . . 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 1232	. . . . . . . . . . 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 7925	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A x. B ) e. RR )
26	25	adantr 274	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( A x. B ) e. RR )
27	26	lt0neg1d 8409	. . . . . . . . . . 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 219	. . . . . . . . . 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 216	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( ( A x. B ) < 0 <-> 0 <_ B ) )
30		0re 7895	. . . . . . . . . 10 |- 0 e. RR
31		lenlt 7970	. . . . . . . . . 10 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 <_ B <-> -. B < 0 ) )
32	30, 16, 31	sylancr 411	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( 0 <_ B <-> -. B < 0 ) )
33	29, 32	bitrd 187	. . . . . . . 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 3540	. . . . . . 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 9264	. . . . . . . . . . 11 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B < 0 )
36	3, 35	mpan2 422	. . . . . . . . . 10 |- ( B e. ZZ -> DECID B < 0 )
37		oveq2 5849	. . . . . . . . . . . . 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 9065	. . . . . . . . . . . . 13 |- ( -u 1 x. -u 1 ) = 1
39	37, 38	eqtrdi 2214	. . . . . . . . . . . 12 |- ( if ( B < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = 1 )
40		oveq2 5849	. . . . . . . . . . . . 13 |- ( if ( B < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = ( -u 1 x. 1 ) )
41		ax-1cn 7842	. . . . . . . . . . . . . 14 |- 1 e. CC
42	41	mulm1i 8297	. . . . . . . . . . . . 13 |- ( -u 1 x. 1 ) = -u 1
43	40, 42	eqtrdi 2214	. . . . . . . . . . . 12 |- ( if ( B < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = -u 1 )
44	39, 43	ifsbdc 3531	. . . . . . . . . . 11 |- (DECID B < 0 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = if ( B < 0 , 1 , -u 1 ) )
45		ifnotdc 3555	. . . . . . . . . . 11 |- (DECID B < 0 -> if ( -. B < 0 , -u 1 , 1 ) = if ( B < 0 , 1 , -u 1 ) )
46	44, 45	eqtr4d 2201	. . . . . . . . . 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 1009	. . . . . . . 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 480	. . . . . . 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 2201	. . . . . 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 3524	. . . . . . . 8 |- ( A < 0 -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
52	51	adantl 275	. . . . . . 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 5856	. . . . . 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 2201	. . . . 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 3527	. . . . . . . 8 |- ( -. A < 0 -> if ( A < 0 , -u 1 , 1 ) = 1 )
56	55	adantl 275	. . . . . . 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 5856	. . . . . 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 8958	. . . . . . . . 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 274	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> B e. ZZ )
62	61, 3, 35	sylancl 410	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> DECID B < 0 )
63	59, 60, 62	ifcldcd 3554	. . . . . . 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 7913	. . . . . 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 274	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> B e. RR )
66		0red 7896	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> 0 e. RR )
67	9	adantr 274	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A e. RR )
68		simplrl 525	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A =/= 0 )
69	68	neneqd 2356	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> -. A = 0 )
70		simpr 109	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> -. A < 0 )
71	8	adantr 274	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A e. ZZ )
72		ztri3or 9230	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A < 0 \/ A = 0 \/ 0 < A ) )
73	71, 3, 72	sylancl 410	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A < 0 \/ A = 0 \/ 0 < A ) )
74		3orass 971	. . . . . . . . . . . . . 14 |- ( ( A < 0 \/ A = 0 \/ 0 < A ) <-> ( A < 0 \/ ( A = 0 \/ 0 < A ) ) )
75	73, 74	sylib 121	. . . . . . . . . . . . 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 719	. . . . . . . . . . . 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 1339	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A = 0 \/ 0 < A ) )
78	77	orcomd 719	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( 0 < A \/ A = 0 ) )
79	69, 78	ecased 1339	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> 0 < A )
80		ltmul2 8747	. . . . . . . . 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 1232	. . . . . . . 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 7923	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A e. CC )
83	82	mul01d 8287	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A x. 0 ) = 0 )
84	83	breq2d 3993	. . . . . . . 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 187	. . . . . . 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 3540	. . . . . 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 2203	. . . . 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 9264	. . . . . . 7 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
89	8, 3, 88	sylancl 410	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> DECID A < 0 )
90		exmiddc 826	. . . . . 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 788	. . . 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 274	. . 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 109	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N < 0 ) -> N < 0 )
95	94	biantrurd 303	. . . 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 3540	. . 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 303	. . . . 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 3540	. . . 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 303	. . . . 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 3540	. . . 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 5859	. . 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 2206	. 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 109	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. N < 0 )
104	103	intnanrd 922	. . . . 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 3529	. . . 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 9005	. . . 4 |- ( 1 x. 1 ) = 1
107	105, 106	eqtr4di 2216	. . 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 922	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. ( N < 0 /\ A < 0 ) )
109	108	iffalsed 3529	. . . 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 922	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. ( N < 0 /\ B < 0 ) )
111	110	iffalsed 3529	. . . 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 5859	. . 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 2201	. 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 992	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> N e. ZZ )
115		zdclt 9264	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
116	114, 3, 115	sylancl 410	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> DECID N < 0 )
117		exmiddc 826	. . 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 788	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 103   <-> wb 104   \/ wo 698  DECID wdc 824   \/ w3o 967   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   ifcif 3519   class class class wbr 3981  (class class class)co 5841   CCcc 7747   RRcr 7748   0cc0 7749   1c1 7750   x. cmul 7754   < clt 7929   <_ cle 7930   -ucneg 8066   ZZcz 9187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rab 2452  df-v 2727  df-sbc 2951  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-br 3982  df-opab 4043  df-id 4270  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-iota 5152  df-fun 5189  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-inn 8854  df-n0 9111  df-z 9188

This theorem is referenced by:  lgsdir  13536  lgsdi  13538