Theorem sincosq3sgn 8623

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

Assertion

Ref	Expression
sincosq3sgn	|- (A e. (pi(,)(3 x. (pi / 2))) -> ((sin` A) < 0 /\ (cos` A) < 0))

Proof of Theorem sincosq3sgn

Step	Hyp	Ref	Expression
1		pire 8596	. . 3 |- pi e. RR
2		3re 5928	. . . 4 |- 3 e. RR
3		2re 5926	. . . . 5 |- 2 e. RR
4		2ne0 5937	. . . . 5 |- 2 =/= 0
5	1, 3, 4	redivcl 5754	. . . 4 |- (pi / 2) e. RR
6	2, 5	remulcl 5307	. . 3 |- (3 x. (pi / 2)) e. RR
7		elioo2t 6316	. . . 4 |- ((pi e. RR* /\ (3 x. (pi / 2)) e. RR*) -> (A e. (pi(,)(3 x. (pi / 2))) <-> (A e. RR /\ pi < A /\ A < (3 x. (pi / 2)))))
8		rexrt 5471	. . . 4 |- (pi e. RR -> pi e. RR*)
9		rexrt 5471	. . . 4 |- ((3 x. (pi / 2)) e. RR -> (3 x. (pi / 2)) e. RR*)
10	7, 8, 9	syl2an 454	. . 3 |- ((pi e. RR /\ (3 x. (pi / 2)) e. RR) -> (A e. (pi(,)(3 x. (pi / 2))) <-> (A e. RR /\ pi < A /\ A < (3 x. (pi / 2)))))
11	1, 6, 10	mp2an 695	. 2 |- (A e. (pi(,)(3 x. (pi / 2))) <-> (A e. RR /\ pi < A /\ A < (3 x. (pi / 2))))
12		ltaddsubt 5605	. . . . . . . . 9 |- (((pi / 2) e. RR /\ (pi / 2) e. RR /\ A e. RR) -> (((pi / 2) + (pi / 2)) < A <-> (pi / 2) < (A - (pi / 2))))
13	5, 5, 12	mp3an12 903	. . . . . . . 8 |- (A e. RR -> (((pi / 2) + (pi / 2)) < A <-> (pi / 2) < (A - (pi / 2))))
14	1	recn 5286	. . . . . . . . . 10 |- pi e. CC
15		2halvest 5986	. . . . . . . . . 10 |- (pi e. CC -> ((pi / 2) + (pi / 2)) = pi)
16	14, 15	ax-mp 7	. . . . . . . . 9 |- ((pi / 2) + (pi / 2)) = pi
17	16	breq1i 2616	. . . . . . . 8 |- (((pi / 2) + (pi / 2)) < A <-> pi < A)
18	13, 17	syl5bbr 532	. . . . . . 7 |- (A e. RR -> (pi < A <-> (pi / 2) < (A - (pi / 2))))
19		ltsubaddt 5601	. . . . . . . . 9 |- ((A e. RR /\ (pi / 2) e. RR /\ pi e. RR) -> ((A - (pi / 2)) < pi <-> A < (pi + (pi / 2))))
20	5, 1, 19	mp3an23 905	. . . . . . . 8 |- (A e. RR -> ((A - (pi / 2)) < pi <-> A < (pi + (pi / 2))))
21		df-3 5918	. . . . . . . . . . 11 |- 3 = (2 + 1)
22	21	opreq1i 3956	. . . . . . . . . 10 |- (3 x. (pi / 2)) = ((2 + 1) x. (pi / 2))
23		2cn 5927	. . . . . . . . . . 11 |- 2 e. CC
24		ax1cn 5241	. . . . . . . . . . 11 |- 1 e. CC
25	5	recn 5286	. . . . . . . . . . 11 |- (pi / 2) e. CC
26	23, 24, 25	adddir 5299	. . . . . . . . . 10 |- ((2 + 1) x. (pi / 2)) = ((2 x. (pi / 2)) + (1 x. (pi / 2)))
27	23, 14, 4	divcan2 5685	. . . . . . . . . . 11 |- (2 x. (pi / 2)) = pi
28	25	mulid2 5305	. . . . . . . . . . 11 |- (1 x. (pi / 2)) = (pi / 2)
29	27, 28	opreq12i 3958	. . . . . . . . . 10 |- ((2 x. (pi / 2)) + (1 x. (pi / 2))) = (pi + (pi / 2))
30	22, 26, 29	3eqtrr 1492	. . . . . . . . 9 |- (pi + (pi / 2)) = (3 x. (pi / 2))
31	30	breq2i 2617	. . . . . . . 8 |- (A < (pi + (pi / 2)) <-> A < (3 x. (pi / 2)))
32	20, 31	syl6rbb 535	. . . . . . 7 |- (A e. RR -> (A < (3 x. (pi / 2)) <-> (A - (pi / 2)) < pi))
33	18, 32	anbi12d 626	. . . . . 6 |- (A e. RR -> ((pi < A /\ A < (3 x. (pi / 2))) <-> ((pi / 2) < (A - (pi / 2)) /\ (A - (pi / 2)) < pi)))
34		sincosq2sgn 8622	. . . . . . . . 9 |- ((A - (pi / 2)) e. ((pi / 2)(,)pi) -> (0 < (sin` (A - (pi / 2))) /\ (cos` (A - (pi / 2))) < 0))
35		elioo2t 6316	. . . . . . . . . . 11 |- (((pi / 2) e. RR* /\ pi e. RR*) -> ((A - (pi / 2)) e. ((pi / 2)(,)pi) <-> ((A - (pi / 2)) e. RR /\ (pi / 2) < (A - (pi / 2)) /\ (A - (pi / 2)) < pi)))
36		rexrt 5471	. . . . . . . . . . 11 |- ((pi / 2) e. RR -> (pi / 2) e. RR*)
37	35, 36, 8	syl2an 454	. . . . . . . . . 10 |- (((pi / 2) e. RR /\ pi e. RR) -> ((A - (pi / 2)) e. ((pi / 2)(,)pi) <-> ((A - (pi / 2)) e. RR /\ (pi / 2) < (A - (pi / 2)) /\ (A - (pi / 2)) < pi)))
38	5, 1, 37	mp2an 695	. . . . . . . . 9 |- ((A - (pi / 2)) e. ((pi / 2)(,)pi) <-> ((A - (pi / 2)) e. RR /\ (pi / 2) < (A - (pi / 2)) /\ (A - (pi / 2)) < pi))
39		ancom 435	. . . . . . . . 9 |- ((0 < (sin` (A - (pi / 2))) /\ (cos` (A - (pi / 2))) < 0) <-> ((cos` (A - (pi / 2))) < 0 /\ 0 < (sin` (A - (pi / 2)))))
40	34, 38, 39	3imtr3 218	. . . . . . . 8 |- (((A - (pi / 2)) e. RR /\ (pi / 2) < (A - (pi / 2)) /\ (A - (pi / 2)) < pi) -> ((cos` (A - (pi / 2))) < 0 /\ 0 < (sin` (A - (pi / 2)))))
41		resubclt 5410	. . . . . . . . 9 |- ((A e. RR /\ (pi / 2) e. RR) -> (A - (pi / 2)) e. RR)
42	5, 41	mpan2 694	. . . . . . . 8 |- (A e. RR -> (A - (pi / 2)) e. RR)
43	40, 42	syl3an1 857	. . . . . . 7 |- ((A e. RR /\ (pi / 2) < (A - (pi / 2)) /\ (A - (pi / 2)) < pi) -> ((cos` (A - (pi / 2))) < 0 /\ 0 < (sin` (A - (pi / 2)))))
44	43	3expib 834	. . . . . 6 |- (A e. RR -> (((pi / 2) < (A - (pi / 2)) /\ (A - (pi / 2)) < pi) -> ((cos` (A - (pi / 2))) < 0 /\ 0 < (sin` (A - (pi / 2))))))
45	33, 44	sylbid 203	. . . . 5 |- (A e. RR -> ((pi < A /\ A < (3 x. (pi / 2))) -> ((cos` (A - (pi / 2))) < 0 /\ 0 < (sin` (A - (pi / 2))))))
46		resinclt 7380	. . . . . . 7 |- ((A - (pi / 2)) e. RR -> (sin` (A - (pi / 2))) e. RR)
47		lt0neg2t 5642	. . . . . . 7 |- ((sin` (A - (pi / 2))) e. RR -> (0 < (sin` (A - (pi / 2))) <-> -u(sin` (A - (pi / 2))) < 0))
48	42, 46, 47	3syl 20	. . . . . 6 |- (A e. RR -> (0 < (sin` (A - (pi / 2))) <-> -u(sin` (A - (pi / 2))) < 0))
49	48	anbi2d 614	. . . . 5 |- (A e. RR -> (((cos` (A - (pi / 2))) < 0 /\ 0 < (sin` (A - (pi / 2)))) <-> ((cos` (A - (pi / 2))) < 0 /\ -u(sin` (A - (pi / 2))) < 0)))
50	45, 49	sylibd 202	. . . 4 |- (A e. RR -> ((pi < A /\ A < (3 x. (pi / 2))) -> ((cos` (A - (pi / 2))) < 0 /\ -u(sin` (A - (pi / 2))) < 0)))
51		recnt 5285	. . . . . . . . 9 |- (A e. RR -> A e. CC)
52		pncan3t 5349	. . . . . . . . . 10 |- (((pi / 2) e. CC /\ A e. CC) -> ((pi / 2) + (A - (pi / 2))) = A)
53	25, 52	mpan 693	. . . . . . . . 9 |- (A e. CC -> ((pi / 2) + (A - (pi / 2))) = A)
54	51, 53	syl 10	. . . . . . . 8 |- (A e. RR -> ((pi / 2) + (A - (pi / 2))) = A)
55	54	fveq2d 3713	. . . . . . 7 |- (A e. RR -> (sin` ((pi / 2) + (A - (pi / 2)))) = (sin` A))
56	42	recnd 5287	. . . . . . . 8 |- (A e. RR -> (A - (pi / 2)) e. CC)
57		sinhalfpip 8616	. . . . . . . 8 |- ((A - (pi / 2)) e. CC -> (sin` ((pi / 2) + (A - (pi / 2)))) = (cos` (A - (pi / 2))))
58	56, 57	syl 10	. . . . . . 7 |- (A e. RR -> (sin` ((pi / 2) + (A - (pi / 2)))) = (cos` (A - (pi / 2))))
59	55, 58	eqtr3d 1501	. . . . . 6 |- (A e. RR -> (sin` A) = (cos` (A - (pi / 2))