Theorem lgsdilem 13722

Description: Lemma for lgsdi 13732 and lgsdir 13730: 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 531	. . . . . . . . . . 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 9223	. . . . . . . . . . 11 |- 0 e. ZZ
4		simpl2 996	. . . . . . . . . . . 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 9290	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ B e. ZZ ) -> ( 0 < B <-> ( 0 <_ B /\ B =/= 0 ) ) )
7	3, 5, 6	sylancr 412	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( 0 < B <-> ( 0 <_ B /\ B =/= 0 ) ) )
8		simpl1 995	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. ZZ )
9	8	zred 9334	. . . . . . . . . . . . . . . 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 8299	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> -u A e. RR )
12	11	recnd 7948	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> -u A e. CC )
13	12	mul01d 8312	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> ( -u A x. 0 ) = 0 )
14	10	recnd 7948	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> A e. CC )
15	4	zred 9334	. . . . . . . . . . . . . . 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 7948	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> B e. CC )
18	14, 17	mulneg1d 8330	. . . . . . . . . . . 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 4002	. . . . . . . . . . 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 7921	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ A < 0 ) -> 0 e. RR )
21	9	lt0neg1d 8434	. . . . . . . . . . . . 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 8772	. . . . . . . . . . . 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 1237	. . . . . . . . . . 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 7950	. . . . . . . . . . . . 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 8434	. . . . . . . . . . 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 7920	. . . . . . . . . 10 |- 0 e. RR
31		lenlt 7995	. . . . . . . . . 10 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 <_ B <-> -. B < 0 ) )
32	30, 16, 31	sylancr 412	. . . . . . . . 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 3547	. . . . . . 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 9289	. . . . . . . . . . 11 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B < 0 )
36	3, 35	mpan2 423	. . . . . . . . . 10 |- ( B e. ZZ -> DECID B < 0 )
37		oveq2 5861	. . . . . . . . . . . . 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 9090	. . . . . . . . . . . . 13 |- ( -u 1 x. -u 1 ) = 1
39	37, 38	eqtrdi 2219	. . . . . . . . . . . 12 |- ( if ( B < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = 1 )
40		oveq2 5861	. . . . . . . . . . . . 13 |- ( if ( B < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = ( -u 1 x. 1 ) )
41		ax-1cn 7867	. . . . . . . . . . . . . 14 |- 1 e. CC
42	41	mulm1i 8322	. . . . . . . . . . . . 13 |- ( -u 1 x. 1 ) = -u 1
43	40, 42	eqtrdi 2219	. . . . . . . . . . . 12 |- ( if ( B < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = -u 1 )
44	39, 43	ifsbdc 3538	. . . . . . . . . . 11 |- (DECID B < 0 -> ( -u 1 x. if ( B < 0 , -u 1 , 1 ) ) = if ( B < 0 , 1 , -u 1 ) )
45		ifnotdc 3562	. . . . . . . . . . 11 |- (DECID B < 0 -> if ( -. B < 0 , -u 1 , 1 ) = if ( B < 0 , 1 , -u 1 ) )
46	44, 45	eqtr4d 2206	. . . . . . . . . 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 1014	. . . . . . . 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 485	. . . . . . 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 2206	. . . . . 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 3531	. . . . . . . 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 5868	. . . . . 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 2206	. . . . 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 3534	. . . . . . . 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 5868	. . . . . 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 8983	. . . . . . . . 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 411	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> DECID B < 0 )
63	59, 60, 62	ifcldcd 3561	. . . . . . 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 7938	. . . . . 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 7921	. . . . . . . . 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 530	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A =/= 0 )
69	68	neneqd 2361	. . . . . . . . . 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 9255	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A < 0 \/ A = 0 \/ 0 < A ) )
73	71, 3, 72	sylancl 411	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A < 0 \/ A = 0 \/ 0 < A ) )
74		3orass 976	. . . . . . . . . . . . . 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 724	. . . . . . . . . . . 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 1344	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A = 0 \/ 0 < A ) )
78	77	orcomd 724	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( 0 < A \/ A = 0 ) )
79	69, 78	ecased 1344	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> 0 < A )
80		ltmul2 8772	. . . . . . . . 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 1237	. . . . . . . 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 7948	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> A e. CC )
83	82	mul01d 8312	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. A < 0 ) -> ( A x. 0 ) = 0 )
84	83	breq2d 4001	. . . . . . . 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 3547	. . . . . 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 2208	. . . . 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 9289	. . . . . . 7 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
89	8, 3, 88	sylancl 411	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> DECID A < 0 )
90		exmiddc 831	. . . . . 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 793	. . . 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 3547	. . 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 3547	. . . 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 3547	. . . 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 5871	. . 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 2211	. 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 927	. . . . 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 3536	. . . 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 9030	. . . 4 |- ( 1 x. 1 ) = 1
107	105, 106	eqtr4di 2221	. . 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 927	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. ( N < 0 /\ A < 0 ) )
109	108	iffalsed 3536	. . . 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 927	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ -. N < 0 ) -> -. ( N < 0 /\ B < 0 ) )
111	110	iffalsed 3536	. . . 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 5871	. . 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 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 ) ) )
114		simpl3 997	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> N e. ZZ )
115		zdclt 9289	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
116	114, 3, 115	sylancl 411	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> DECID N < 0 )
117		exmiddc 831	. . 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 793	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 703  DECID wdc 829   \/ w3o 972   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   ifcif 3526   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   x. cmul 7779   < clt 7954   <_ cle 7955   -ucneg 8091   ZZcz 9212

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-inn 8879  df-n0 9136  df-z 9213

This theorem is referenced by:  lgsdir  13730  lgsdi  13732