Theorem sgnmul 30604

Description: Signum of a product. (Contributed by Thierry Arnoux, 2-Oct-2018.)

Assertion

Ref	Expression
sgnmul	|- ( ( A e. RR /\ B e. RR ) -> (sgn ` ( A x. B ) ) = ( (sgn ` A ) x. (sgn ` B ) ) )

Proof of Theorem sgnmul

Step	Hyp	Ref	Expression
1		remulcl 10021	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
2	1	rexrd 10089	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR* )
3		eqeq1 2626	. 2 |- ( (sgn ` ( A x. B ) ) = 0 -> ( (sgn ` ( A x. B ) ) = ( (sgn ` A ) x. (sgn ` B ) ) <-> 0 = ( (sgn ` A ) x. (sgn ` B ) ) ) )
4		eqeq1 2626	. 2 |- ( (sgn ` ( A x. B ) ) = 1 -> ( (sgn ` ( A x. B ) ) = ( (sgn ` A ) x. (sgn ` B ) ) <-> 1 = ( (sgn ` A ) x. (sgn ` B ) ) ) )
5		eqeq1 2626	. 2 |- ( (sgn ` ( A x. B ) ) = -u 1 -> ( (sgn ` ( A x. B ) ) = ( (sgn ` A ) x. (sgn ` B ) ) <-> -u 1 = ( (sgn ` A ) x. (sgn ` B ) ) ) )
6		fveq2 6191	. . . . . . 7 |- ( A = 0 -> (sgn ` A ) = (sgn ` 0 ) )
7		sgn0 13829	. . . . . . 7 |- (sgn ` 0 ) = 0
8	6, 7	syl6eq 2672	. . . . . 6 |- ( A = 0 -> (sgn ` A ) = 0 )
9	8	oveq1d 6665	. . . . 5 |- ( A = 0 -> ( (sgn ` A ) x. (sgn ` B ) ) = ( 0 x. (sgn ` B ) ) )
10	9	adantl 482	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) = 0 ) /\ A = 0 ) -> ( (sgn ` A ) x. (sgn ` B ) ) = ( 0 x. (sgn ` B ) ) )
11		sgnclre 30601	. . . . . . 7 |- ( B e. RR -> (sgn ` B ) e. RR )
12	11	recnd 10068	. . . . . 6 |- ( B e. RR -> (sgn ` B ) e. CC )
13	12	mul02d 10234	. . . . 5 |- ( B e. RR -> ( 0 x. (sgn ` B ) ) = 0 )
14	13	ad3antlr 767	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) = 0 ) /\ A = 0 ) -> ( 0 x. (sgn ` B ) ) = 0 )
15	10, 14	eqtr2d 2657	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) = 0 ) /\ A = 0 ) -> 0 = ( (sgn ` A ) x. (sgn ` B ) ) )
16		fveq2 6191	. . . . . . 7 |- ( B = 0 -> (sgn ` B ) = (sgn ` 0 ) )
17	16, 7	syl6eq 2672	. . . . . 6 |- ( B = 0 -> (sgn ` B ) = 0 )
18	17	oveq2d 6666	. . . . 5 |- ( B = 0 -> ( (sgn ` A ) x. (sgn ` B ) ) = ( (sgn ` A ) x. 0 ) )
19	18	adantl 482	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) = 0 ) /\ B = 0 ) -> ( (sgn ` A ) x. (sgn ` B ) ) = ( (sgn ` A ) x. 0 ) )
20		sgnclre 30601	. . . . . . 7 |- ( A e. RR -> (sgn ` A ) e. RR )
21	20	recnd 10068	. . . . . 6 |- ( A e. RR -> (sgn ` A ) e. CC )
22	21	mul01d 10235	. . . . 5 |- ( A e. RR -> ( (sgn ` A ) x. 0 ) = 0 )
23	22	ad3antrrr 766	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) = 0 ) /\ B = 0 ) -> ( (sgn ` A ) x. 0 ) = 0 )
24	19, 23	eqtr2d 2657	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) = 0 ) /\ B = 0 ) -> 0 = ( (sgn ` A ) x. (sgn ` B ) ) )
25		simpl 473	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
26	25	recnd 10068	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
27		simpr 477	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
28	27	recnd 10068	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
29	26, 28	mul0ord 10677	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) ) )
30	29	biimpa 501	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) = 0 ) -> ( A = 0 \/ B = 0 ) )
31	15, 24, 30	mpjaodan 827	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) = 0 ) -> 0 = ( (sgn ` A ) x. (sgn ` B ) ) )
32		simpll 790	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) -> A e. RR )
33	32	rexrd 10089	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) -> A e. RR* )
34		oveq1 6657	. . . 4 |- ( (sgn ` A ) = 0 -> ( (sgn ` A ) x. (sgn ` B ) ) = ( 0 x. (sgn ` B ) ) )
35	34	eqeq2d 2632	. . 3 |- ( (sgn ` A ) = 0 -> ( 1 = ( (sgn ` A ) x. (sgn ` B ) ) <-> 1 = ( 0 x. (sgn ` B ) ) ) )
36		oveq1 6657	. . . 4 |- ( (sgn ` A ) = 1 -> ( (sgn ` A ) x. (sgn ` B ) ) = ( 1 x. (sgn ` B ) ) )
37	36	eqeq2d 2632	. . 3 |- ( (sgn ` A ) = 1 -> ( 1 = ( (sgn ` A ) x. (sgn ` B ) ) <-> 1 = ( 1 x. (sgn ` B ) ) ) )
38		oveq1 6657	. . . 4 |- ( (sgn ` A ) = -u 1 -> ( (sgn ` A ) x. (sgn ` B ) ) = ( -u 1 x. (sgn ` B ) ) )
39	38	eqeq2d 2632	. . 3 |- ( (sgn ` A ) = -u 1 -> ( 1 = ( (sgn ` A ) x. (sgn ` B ) ) <-> 1 = ( -u 1 x. (sgn ` B ) ) ) )
40		simpr 477	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A = 0 ) -> A = 0 )
41	26	adantr 481	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) -> A e. CC )
42	28	adantr 481	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) -> B e. CC )
43		simpr 477	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) -> 0 < ( A x. B ) )
44	43	gt0ne0d 10592	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) -> ( A x. B ) =/= 0 )
45	41, 42, 44	mulne0bad 10682	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) -> A =/= 0 )
46	45	neneqd 2799	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) -> -. A = 0 )
47	46	adantr 481	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A = 0 ) -> -. A = 0 )
48	40, 47	pm2.21dd 186	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A = 0 ) -> 1 = ( 0 x. (sgn ` B ) ) )
49	27	ad2antrr 762	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> B e. RR )
50	49	rexrd 10089	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> B e. RR* )
51		simpll 790	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> ( A e. RR /\ B e. RR ) )
52		0red 10041	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> 0 e. RR )
53		simplll 798	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> A e. RR )
54		simpr 477	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> 0 < A )
55	52, 53, 54	ltled 10185	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> 0 <_ A )
56		simplr 792	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> 0 < ( A x. B ) )
57		prodgt0 10868	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < B )
58	51, 55, 56, 57	syl12anc 1324	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> 0 < B )
59		sgnp 13830	. . . . . 6 |- ( ( B e. RR* /\ 0 < B ) -> (sgn ` B ) = 1 )
60	50, 58, 59	syl2anc 693	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> (sgn ` B ) = 1 )
61	60	oveq2d 6666	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> ( 1 x. (sgn ` B ) ) = ( 1 x. 1 ) )
62		1t1e1 11175	. . . 4 |- ( 1 x. 1 ) = 1
63	61, 62	syl6req 2673	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ 0 < A ) -> 1 = ( 1 x. (sgn ` B ) ) )
64	27	ad2antrr 762	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> B e. RR )
65	64	rexrd 10089	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> B e. RR* )
66		simplll 798	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> A e. RR )
67	66	renegcld 10457	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> -u A e. RR )
68	64	renegcld 10457	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> -u B e. RR )
69		0red 10041	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> 0 e. RR )
70		simpr 477	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> A < 0 )
71	25	lt0neg1d 10597	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> 0 < -u A ) )
72	71	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> ( A < 0 <-> 0 < -u A ) )
73	70, 72	mpbid 222	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> 0 < -u A )
74	69, 67, 73	ltled 10185	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> 0 <_ -u A )
75		simplr 792	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> 0 < ( A x. B ) )
76	26	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> A e. CC )
77	28	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> B e. CC )
78	76, 77	mul2negd 10485	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> ( -u A x. -u B ) = ( A x. B ) )
79	75, 78	breqtrrd 4681	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> 0 < ( -u A x. -u B ) )
80		prodgt0 10868	. . . . . . . 8 |- ( ( ( -u A e. RR /\ -u B e. RR ) /\ ( 0 <_ -u A /\ 0 < ( -u A x. -u B ) ) ) -> 0 < -u B )
81	67, 68, 74, 79, 80	syl22anc 1327	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> 0 < -u B )
82	27	lt0neg1d 10597	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( B < 0 <-> 0 < -u B ) )
83	82	ad2antrr 762	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> ( B < 0 <-> 0 < -u B ) )
84	81, 83	mpbird 247	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> B < 0 )
85		sgnn 13834	. . . . . 6 |- ( ( B e. RR* /\ B < 0 ) -> (sgn ` B ) = -u 1 )
86	65, 84, 85	syl2anc 693	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> (sgn ` B ) = -u 1 )
87	86	oveq2d 6666	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> ( -u 1 x. (sgn ` B ) ) = ( -u 1 x. -u 1 ) )
88		neg1mulneg1e1 11245	. . . 4 |- ( -u 1 x. -u 1 ) = 1
89	87, 88	syl6req 2673	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) /\ A < 0 ) -> 1 = ( -u 1 x. (sgn ` B ) ) )
90	33, 35, 37, 39, 48, 63, 89	sgn3da 30603	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < ( A x. B ) ) -> 1 = ( (sgn ` A ) x. (sgn ` B ) ) )
91		simpll 790	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) -> A e. RR )
92	91	rexrd 10089	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) -> A e. RR* )
93	34	eqeq2d 2632	. . 3 |- ( (sgn ` A ) = 0 -> ( -u 1 = ( (sgn ` A ) x. (sgn ` B ) ) <-> -u 1 = ( 0 x. (sgn ` B ) ) ) )
94	36	eqeq2d 2632	. . 3 |- ( (sgn ` A ) = 1 -> ( -u 1 = ( (sgn ` A ) x. (sgn ` B ) ) <-> -u 1 = ( 1 x. (sgn ` B ) ) ) )
95	38	eqeq2d 2632	. . 3 |- ( (sgn ` A ) = -u 1 -> ( -u 1 = ( (sgn ` A ) x. (sgn ` B ) ) <-> -u 1 = ( -u 1 x. (sgn ` B ) ) ) )
96		simpr 477	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A = 0 ) -> A = 0 )
97	26	ad2antrr 762	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A = 0 ) -> A e. CC )
98	28	ad2antrr 762	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A = 0 ) -> B e. CC )
99		simplr 792	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A = 0 ) -> ( A x. B ) < 0 )
100	99	lt0ne0d 10593	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A = 0 ) -> ( A x. B ) =/= 0 )
101	97, 98, 100	mulne0bad 10682	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A = 0 ) -> A =/= 0 )
102	101	neneqd 2799	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A = 0 ) -> -. A = 0 )
103	96, 102	pm2.21dd 186	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A = 0 ) -> -u 1 = ( 0 x. (sgn ` B ) ) )
104	27	ad2antrr 762	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ 0 < A ) -> B e. RR )
105	104	rexrd 10089	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ 0 < A ) -> B e. RR* )
106		simplr 792	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) -> B e. RR )
107	26, 28	mulcomd 10061	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) = ( B x. A ) )
108	107	breq1d 4663	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. B ) < 0 <-> ( B x. A ) < 0 ) )
109	108	biimpa 501	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) -> ( B x. A ) < 0 )
110	106, 91, 109	mul2lt0rgt0 11933	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ 0 < A ) -> B < 0 )
111	105, 110, 85	syl2anc 693	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ 0 < A ) -> (sgn ` B ) = -u 1 )
112	111	oveq2d 6666	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ 0 < A ) -> ( 1 x. (sgn ` B ) ) = ( 1 x. -u 1 ) )
113		neg1cn 11124	. . . . 5 |- -u 1 e. CC
114	113	mulid2i 10043	. . . 4 |- ( 1 x. -u 1 ) = -u 1
115	112, 114	syl6req 2673	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ 0 < A ) -> -u 1 = ( 1 x. (sgn ` B ) ) )
116	106	adantr 481	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A < 0 ) -> B e. RR )
117	116	rexrd 10089	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A < 0 ) -> B e. RR* )
118	106, 91, 109	mul2lt0rlt0 11932	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A < 0 ) -> 0 < B )
119	117, 118, 59	syl2anc 693	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A < 0 ) -> (sgn ` B ) = 1 )
120	119	oveq2d 6666	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A < 0 ) -> ( -u 1 x. (sgn ` B ) ) = ( -u 1 x. 1 ) )
121	113	mulid1i 10042	. . . 4 |- ( -u 1 x. 1 ) = -u 1
122	120, 121	syl6req 2673	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) /\ A < 0 ) -> -u 1 = ( -u 1 x. (sgn ` B ) ) )
123	92, 93, 94, 95, 103, 115, 122	sgn3da 30603	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A x. B ) < 0 ) -> -u 1 = ( (sgn ` A ) x. (sgn ` B ) ) )
124	2, 3, 4, 5, 31, 90, 123	sgn3da 30603	1 |- ( ( A e. RR /\ B e. RR ) -> (sgn ` ( A x. B ) ) = ( (sgn ` A ) x. (sgn ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   -ucneg 10267  sgncsgn 13826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-rp 11833  df-sgn 13827

This theorem is referenced by:  sgnmulrp2  30605  sgnmulsgn  30611  sgnmulsgp  30612