Theorem sincosq4sgn 8707

Description: The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Assertion

Ref	Expression
sincosq4sgn	|- (A e. ((3 x. (pi / 2))(,)(2 x. pi)) -> ((sin` A) < 0 /\ 0 < (cos` A)))

Proof of Theorem sincosq4sgn

Step	Hyp	Ref	Expression
1		3re 5981	. . . 4 |- 3 e. RR
2		pire 8677	. . . . 5 |- pi e. RR
3		2re 5979	. . . . 5 |- 2 e. RR
4		2ne0 5990	. . . . 5 |- 2 =/= 0
5	2, 3, 4	redivcl 5798	. . . 4 |- (pi / 2) e. RR
6	1, 5	remulcl 5335	. . 3 |- (3 x. (pi / 2)) e. RR
7	3, 2	remulcl 5335	. . 3 |- (2 x. pi) e. RR
8		elioo2t 6379	. . . 4 |- (((3 x. (pi / 2)) e. RR* /\ (2 x. pi) e. RR*) -> (A e. ((3 x. (pi / 2))(,)(2 x. pi)) <-> (A e. RR /\ (3 x. (pi / 2)) < A /\ A < (2 x. pi))))
9		rexrt 5499	. . . 4 |- ((3 x. (pi / 2)) e. RR -> (3 x. (pi / 2)) e. RR*)
10		rexrt 5499	. . . 4 |- ((2 x. pi) e. RR -> (2 x. pi) e. RR*)
11	8, 9, 10	syl2an 454	. . 3 |- (((3 x. (pi / 2)) e. RR /\ (2 x. pi) e. RR) -> (A e. ((3 x. (pi / 2))(,)(2 x. pi)) <-> (A e. RR /\ (3 x. (pi / 2)) < A /\ A < (2 x. pi))))
12	6, 7, 11	mp2an 697	. 2 |- (A e. ((3 x. (pi / 2))(,)(2 x. pi)) <-> (A e. RR /\ (3 x. (pi / 2)) < A /\ A < (2 x. pi)))
13		ltaddsubt 5631	. . . . . . . . . 10 |- ((pi e. RR /\ (pi / 2) e. RR /\ A e. RR) -> ((pi + (pi / 2)) < A <-> pi < (A - (pi / 2))))
14	2, 5, 13	mp3an12 906	. . . . . . . . 9 |- (A e. RR -> ((pi + (pi / 2)) < A <-> pi < (A - (pi / 2))))
15		df-3 5971	. . . . . . . . . . . 12 |- 3 = (2 + 1)
16	15	opreq1i 3971	. . . . . . . . . . 11 |- (3 x. (pi / 2)) = ((2 + 1) x. (pi / 2))
17		2cn 5980	. . . . . . . . . . . 12 |- 2 e. CC
18		ax1cn 5269	. . . . . . . . . . . 12 |- 1 e. CC
19	5	recn 5314	. . . . . . . . . . . 12 |- (pi / 2) e. CC
20	17, 18, 19	adddir 5327	. . . . . . . . . . 11 |- ((2 + 1) x. (pi / 2)) = ((2 x. (pi / 2)) + (1 x. (pi / 2)))
21	2	recn 5314	. . . . . . . . . . . . 13 |- pi e. CC
22	21, 17, 4	divcan2 5716	. . . . . . . . . . . 12 |- (2 x. (pi / 2)) = pi
23	19	mulid2 5333	. . . . . . . . . . . 12 |- (1 x. (pi / 2)) = (pi / 2)
24	22, 23	opreq12i 3973	. . . . . . . . . . 11 |- ((2 x. (pi / 2)) + (1 x. (pi / 2))) = (pi + (pi / 2))
25	16, 20, 24	3eqtrr 1500	. . . . . . . . . 10 |- (pi + (pi / 2)) = (3 x. (pi / 2))
26	25	breq1i 2626	. . . . . . . . 9 |- ((pi + (pi / 2)) < A <-> (3 x. (pi / 2)) < A)
27	14, 26	syl5bbr 534	. . . . . . . 8 |- (A e. RR -> ((3 x. (pi / 2)) < A <-> pi < (A - (pi / 2))))
28		ltsubaddt 5627	. . . . . . . . . 10 |- ((A e. RR /\ (pi / 2) e. RR /\ (3 x. (pi / 2)) e. RR) -> ((A - (pi / 2)) < (3 x. (pi / 2)) <-> A < ((3 x. (pi / 2)) + (pi / 2))))
29	5, 6, 28	mp3an23 908	. . . . . . . . 9 |- (A e. RR -> ((A - (pi / 2)) < (3 x. (pi / 2)) <-> A < ((3 x. (pi / 2)) + (pi / 2))))
30		df-4 5972	. . . . . . . . . . . . 13 |- 4 = (3 + 1)
31	30	opreq1i 3971	. . . . . . . . . . . 12 |- (4 x. (pi / 2)) = ((3 + 1) x. (pi / 2))
32	1	recn 5314	. . . . . . . . . . . . 13 |- 3 e. CC
33	32, 18, 19	adddir 5327	. . . . . . . . . . . 12 |- ((3 + 1) x. (pi / 2)) = ((3 x. (pi / 2)) + (1 x. (pi / 2)))
34	23	opreq2i 3972	. . . . . . . . . . . 12 |- ((3 x. (pi / 2)) + (1 x. (pi / 2))) = ((3 x. (pi / 2)) + (pi / 2))
35	31, 33, 34	3eqtrr 1500	. . . . . . . . . . 11 |- ((3 x. (pi / 2)) + (pi / 2)) = (4 x. (pi / 2))
36		4re 5982	. . . . . . . . . . . . . 14 |- 4 e. RR
37	36	recn 5314	. . . . . . . . . . . . 13 |- 4 e. CC
38		div12t 5744	. . . . . . . . . . . . . 14 |- (((4 e. CC /\ pi e. CC /\ 2 e. CC) /\ 2 =/= 0) -> (4 x. (pi / 2)) = (pi x. (4 / 2)))
39	4, 38	mpan2 696	. . . . . . . . . . . . 13 |- ((4 e. CC /\ pi e. CC /\ 2 e. CC) -> (4 x. (pi / 2)) = (pi x. (4 / 2)))
40	37, 21, 17, 39	mp3an 916	. . . . . . . . . . . 12 |- (4 x. (pi / 2)) = (pi x. (4 / 2))
41		4d2e2 6027	. . . . . . . . . . . . 13 |- (4 / 2) = 2
42	41	opreq2i 3972	. . . . . . . . . . . 12 |- (pi x. (4 / 2)) = (pi x. 2)
43	21, 17	mulcom 5323	. . . . . . . . . . . 12 |- (pi x. 2) = (2 x. pi)
44	40, 42, 43	3eqtr 1499	. . . . . . . . . . 11 |- (4 x. (pi / 2)) = (2 x. pi)
45	35, 44	eqtr 1495	. . . . . . . . . 10 |- ((3 x. (pi / 2)) + (pi / 2)) = (2 x. pi)
46	45	breq2i 2627	. . . . . . . . 9 |- (A < ((3 x. (pi / 2)) + (pi / 2)) <-> A < (2 x. pi))
47	29, 46	syl6rbb 537	. . . . . . . 8 |- (A e. RR -> (A < (2 x. pi) <-> (A - (pi / 2)) < (3 x. (pi / 2))))
48	27, 47	anbi12d 628	. . . . . . 7 |- (A e. RR -> (((3 x. (pi / 2)) < A /\ A < (2 x. pi)) <-> (pi < (A - (pi / 2)) /\ (A - (pi / 2)) < (3 x. (pi / 2)))))
49		elioo2t 6379	. . . . . . . . . . . 12 |- ((pi e. RR* /\ (3 x. (pi / 2)) e. RR*) -> ((A - (pi / 2)) e. (pi(,)(3 x. (pi / 2))) <-> ((A - (pi / 2)) e. RR /\ pi < (A - (pi / 2)) /\ (A - (pi / 2)) < (3 x. (pi / 2)))))
50		rexrt 5499	. . . . . . . . . . . 12 |- (pi e. RR -> pi e. RR*)
51	49, 50, 9	syl2an 454	. . . . . . . . . . 11 |- ((pi e. RR /\ (3 x. (pi / 2)) e. RR) -> ((A - (pi / 2)) e. (pi(,)(3 x. (pi / 2))) <-> ((A - (pi / 2)) e. RR /\ pi < (A - (pi / 2)) /\ (A - (pi / 2)) < (3 x. (pi / 2)))))
52	2, 6, 51	mp2an 697	. . . . . . . . . 10 |- ((A - (pi / 2)) e. (pi(,)(3 x. (pi / 2))) <-> ((A - (pi / 2)) e. RR /\ pi < (A - (pi / 2)) /\ (A - (pi / 2)) < (3 x. (pi / 2))))
53		sincosq3sgn 8706	. . . . . . . . . 10 |- ((A - (pi / 2)) e. (pi(,)(3 x. (pi / 2))) -> ((sin` (A - (pi / 2))) < 0 /\ (cos` (A - (pi / 2))) < 0))
54	52, 53	sylbir 201	. . . . . . . . 9 |- (((A - (pi / 2)) e. RR /\ pi < (A - (pi / 2)) /\ (A - (pi / 2)) < (3 x. (pi / 2))) -> ((sin` (A - (pi / 2))) < 0 /\ (cos` (A - (pi / 2))) < 0))
55		resubclt 5438	. . . . . . . . . 10 |- ((A e. RR /\ (pi / 2) e. RR) -> (A - (pi / 2)) e. RR)
56	5, 55	mpan2 696	. . . . . . . . 9 |- (A e. RR -> (A - (pi / 2)) e. RR)
57	54, 56	syl3an1 859	. . . . . . . 8 |- ((A e. RR /\ pi < (A - (pi / 2)) /\ (A - (pi / 2)) < (3 x. (pi / 2))) -> ((sin` (A - (pi / 2))) < 0 /\ (cos` (A - (pi / 2))) < 0))
58	57	3expib 836	. . . . . . 7 |- (A e. RR -> ((pi < (A - (pi / 2)) /\ (A - (pi / 2)) < (3 x. (pi / 2))) -> ((sin` (A - (pi / 2))) < 0 /\ (cos` (A - (pi / 2))) < 0)))
59	48, 58	sylbid 203	. . . . . 6 |- (A e. RR -> (((3 x. (pi / 2)) < A /\ A < (2 x. pi)) -> ((sin` (A - (pi / 2))) < 0 /\ (cos` (A - (pi / 2))) < 0)))
60		resinclt 7438	. . . . . . . 8 |- ((A - (pi / 2)) e. RR -> (sin` (A - (pi / 2))) e. RR)
61		lt0neg1t 5668	. . . . . . . 8 |- ((sin` (A - (pi / 2))) e. RR -> ((sin` (A - (pi / 2))) < 0 <-> 0 < -u(sin` (A - (pi / 2)))))
62	56, 60, 61	3syl 20	. . . . . . 7 |- (A e. RR -> ((sin` (A - (pi / 2))) < 0 <-> 0 < -u(sin` (A - (pi / 2)))))
63	62	anbi1d 617	. . . . . 6 |- (A e. RR -> (((sin` (A - (pi / 2))) < 0 /\ (cos` (A - (pi / 2))) < 0)