Theorem sincosq4sgn 10056

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 7165	. . . 4 |- 3 e. RR
2		pire 10026	. . . . 5 |- pi e. RR
3		2re 7163	. . . . 5 |- 2 e. RR
4		2ne0 7174	. . . . 5 |- 2 =/= 0
5	2, 3, 4	redivcli 6976	. . . 4 |- (pi / 2) e. RR
6	1, 5	remulcli 6488	. . 3 |- (3 x. (pi / 2)) e. RR
7	3, 2	remulcli 6488	. . 3 |- (2 x. pi) e. RR
8		elioo2 7546	. . . 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		rexr 6668	. . . 4 |- ((3 x. (pi / 2)) e. RR -> (3 x. (pi / 2)) e. RR*)
10		rexr 6668	. . . 4 |- ((2 x. pi) e. RR -> (2 x. pi) e. RR*)
11	8, 9, 10	syl2an 503	. . 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 761	. 2 |- (A e. ((3 x. (pi / 2))(,)(2 x. pi)) <-> (A e. RR /\ (3 x. (pi / 2)) < A /\ A < (2 x. pi)))
13		ltaddsub 6814	. . . . . . . . . 10 |- ((pi e. RR /\ (pi / 2) e. RR /\ A e. RR) -> ((pi + (pi / 2)) < A <-> pi < (A - (pi / 2))))
14	2, 5, 13	mp3an12 1181	. . . . . . . . 9 |- (A e. RR -> ((pi + (pi / 2)) < A <-> pi < (A - (pi / 2))))
15		df-3 7155	. . . . . . . . . . . 12 |- 3 = (2 + 1)
16	15	opreq1i 4892	. . . . . . . . . . 11 |- (3 x. (pi / 2)) = ((2 + 1) x. (pi / 2))
17		2cn 7164	. . . . . . . . . . . 12 |- 2 e. CC
18		ax1cn 6422	. . . . . . . . . . . 12 |- 1 e. CC
19	5	recni 6467	. . . . . . . . . . . 12 |- (pi / 2) e. CC
20	17, 18, 19	adddiri 6480	. . . . . . . . . . 11 |- ((2 + 1) x. (pi / 2)) = ((2 x. (pi / 2)) + (1 x. (pi / 2)))
21	2	recni 6467	. . . . . . . . . . . . 13 |- pi e. CC
22	21, 17, 4	divcan2i 6905	. . . . . . . . . . . 12 |- (2 x. (pi / 2)) = pi
23	19	mulid2i 6486	. . . . . . . . . . . 12 |- (1 x. (pi / 2)) = (pi / 2)
24	22, 23	opreq12i 4894	. . . . . . . . . . 11 |- ((2 x. (pi / 2)) + (1 x. (pi / 2))) = (pi + (pi / 2))
25	16, 20, 24	3eqtrri 1913	. . . . . . . . . 10 |- (pi + (pi / 2)) = (3 x. (pi / 2))
26	25	breq1i 3345	. . . . . . . . 9 |- ((pi + (pi / 2)) < A <-> (3 x. (pi / 2)) < A)
27	14, 26	syl5bbr 593	. . . . . . . 8 |- (A e. RR -> ((3 x. (pi / 2)) < A <-> pi < (A - (pi / 2))))
28		ltsubadd 6810	. . . . . . . . . 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 1183	. . . . . . . . 9 |- (A e. RR -> ((A - (pi / 2)) < (3 x. (pi / 2)) <-> A < ((3 x. (pi / 2)) + (pi / 2))))
30		df-4 7156	. . . . . . . . . . . . 13 |- 4 = (3 + 1)
31	30	opreq1i 4892	. . . . . . . . . . . 12 |- (4 x. (pi / 2)) = ((3 + 1) x. (pi / 2))
32	1	recni 6467	. . . . . . . . . . . . 13 |- 3 e. CC
33	32, 18, 19	adddiri 6480	. . . . . . . . . . . 12 |- ((3 + 1) x. (pi / 2)) = ((3 x. (pi / 2)) + (1 x. (pi / 2)))
34	23	opreq2i 4893	. . . . . . . . . . . 12 |- ((3 x. (pi / 2)) + (1 x. (pi / 2))) = ((3 x. (pi / 2)) + (pi / 2))
35	31, 33, 34	3eqtrri 1913	. . . . . . . . . . 11 |- ((3 x. (pi / 2)) + (pi / 2)) = (4 x. (pi / 2))
36		4re 7166	. . . . . . . . . . . . . 14 |- 4 e. RR
37	36	recni 6467	. . . . . . . . . . . . 13 |- 4 e. CC
38	17, 4	pm3.2i 307	. . . . . . . . . . . . 13 |- (2 e. CC /\ 2 =/= 0)
39		div12 6927	. . . . . . . . . . . . 13 |- ((4 e. CC /\ pi e. CC /\ (2 e. CC /\ 2 =/= 0)) -> (4 x. (pi / 2)) = (pi x. (4 / 2)))
40	37, 21, 38, 39	mp3an 1191	. . . . . . . . . . . 12 |- (4 x. (pi / 2)) = (pi x. (4 / 2))
41		4d2e2 7211	. . . . . . . . . . . . 13 |- (4 / 2) = 2
42	41	opreq2i 4893	. . . . . . . . . . . 12 |- (pi x. (4 / 2)) = (pi x. 2)
43	21, 17	mulcomi 6476	. . . . . . . . . . . 12 |- (pi x. 2) = (2 x. pi)
44	40, 42, 43	3eqtri 1912	. . . . . . . . . . 11 |- (4 x. (pi / 2)) = (2 x. pi)
45	35, 44	eqtri 1908	. . . . . . . . . 10 |- ((3 x. (pi / 2)) + (pi / 2)) = (2 x. pi)
46	45	breq2i 3346	. . . . . . . . 9 |- (A < ((3 x. (pi / 2)) + (pi / 2)) <-> A < (2 x. pi))
47	29, 46	syl6rbb 596	. . . . . . . 8 |- (A e. RR -> (A < (2 x. pi) <-> (A - (pi / 2)) < (3 x. (pi / 2))))
48	27, 47	anbi12d 690	. . . . . . 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		elioo2 7546	. . . . . . . . . . . 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		rexr 6668	. . . . . . . . . . . 12 |- (pi e. RR -> pi e. RR*)
51	49, 50, 9	syl2an 503	. . . . . . . . . . 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 761	. . . . . . . . . 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 10055	. . . . . . . . . 10 |- ((A - (pi / 2)) e. (pi(,)(3 x. (pi / 2))) -> ((sin` (A - (pi / 2))) < 0 /\ (cos` (A - (pi / 2))) < 0))
54	52, 53	sylbir 218	. . . . . . . . 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		resubcl 6601	. . . . . . . . . 10 |- ((A e. RR /\ (pi / 2) e. RR) -> (A - (pi / 2)) e. RR)
56	5, 55	mpan2 760	. . . . . . . . 9 |- (A e. RR -> (A - (pi / 2)) e. RR)
57	54, 56	syl3an1 1130	. . . . . . . 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 1070	. . . . . . 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 220	. . . . . 6 |- (A e. RR -> (((3 x. (pi / 2)) < A /\ A < (2 x. pi)) -> ((sin` (A - (pi / 2))) < 0 /\ (cos` (A - (pi / 2))) < 0)))
60		resincl 8703	. . . . . . . 8 |- ((A - (pi / 2)) e. RR -> (sin` (A - (pi / 2))) e. RR)
61		lt0neg1 6857	. . . . . . . 8 |- ((sin` (A - (pi / 2))) e. RR -> ((sin` (A - (pi / 2))) < 0 <-> 0 < -u(sin` (A - (pi / 2)))))
62	56, 60, 61	3syl 24	. . . . . . 7 |- (A e. RR -> ((sin` (A - (pi / 2))) < 0 <-> 0 < -u(sin` (A - (pi / 2)))))
63	62	anbi1d 679	. . . . . 6 |- (A e. RR -> (((sin` (A - (pi / 2))) < 0 /\ (cos` (A - (pi / 2))) < 0) <-> (0 < -u(sin` (A - (pi / 2))) /\ (cos` (A - (pi / 2))) < 0)))
64	59, 63	sylibd 219	. . . . 5 |- (A e. RR -> (((3 x. (pi / 2)) < A /\ A < (2 x. pi)) -> (0 < -u(sin` (A - (pi / 2))) /\ (cos` (A - (pi / 2))) < 0)))
65		recn 6466	. . . . . . . . . 10 |- (A e. RR -> A e. CC)
66		pncan3 6534	. . . . . . . . . . 11 |- (((pi / 2) e. CC /\ A e. CC) -> ((pi / 2) + (A - (pi / 2))) = A)
67	19, 66	mpan 759	. . . . . . . . . 10 |- (A e. CC -> ((pi / 2) + (A - (pi / 2))) = A)
68	65, 67	syl 12	. . . . . . . . 9 |- (A e. RR -> ((pi / 2) + (A - (pi / 2))) = A)
69	68	fveq2d 4685	. . . . . . . 8 |- (A e. RR -> (cos` ((pi / 2) + (A - (pi / 2)))) = (cos` A))
70	56	recnd 6468	. . . . . . . . 9 |- (A e. RR -> (A - (pi / 2)) e. CC)
71		coshalfpip 10050	. . . . . . . . 9 |- ((A - (pi / 2)) e. CC -> (cos` ((pi / 2) + (A - (pi / 2)))) = -u(sin` (A - (pi / 2))))
72	70, 71	syl 12	. . . . . . . 8 |- (A e. RR -> (cos` ((pi / 2) + (A - (pi / 2)))) = -u(sin` (A - (pi / 2))))
73	69, 72	eqtr3d 1927	. . . . . . 7 |- (A e. RR -> (cos` A) = -u(sin` (A - (pi / 2))))
74	73	breq2d 3350	. . . . . 6 |- (A e. RR -> (0 < (cos` A) <-> 0 < -u(sin` (A - (pi / 2)))))
75	68	fveq2d 4685	. . . . . . . 8 |- (A e. RR -> (sin` ((pi / 2) + (A - (pi / 2)))) = (sin` A))
76		sinhalfpip 10048	. . . . . . . . 9 |- ((A - (pi / 2)) e. CC -> (sin` ((pi / 2) + (A - (pi / 2)))) = (cos` (A - (pi / 2))))
77	70, 76	syl 12	. . . . . . . 8 |- (A e. RR -> (sin` ((pi / 2) + (A - (pi / 2)))) = (cos` (A - (pi / 2))))
78	75, 77	eqtr3d 1927	. . . . . . 7 |- (A e. RR -> (sin` A) = (cos` (A - (pi / 2))))
79	78	breq1d 3348	. . . . . 6 |- (A e. RR -> ((sin` A) < 0 <-> (cos` (A - (pi / 2))) < 0))
80	74, 79	anbi12d 690	. . . . 5 |- (A e. RR -> ((0 < (cos` A) /\ (sin` A) < 0) <-> (0 < -u(sin` (A - (pi / 2))) /\ (cos` (A - (pi / 2))) < 0)))
81	64, 80	sylibrd 221	. . . 4 |- (A e. RR -> (((3 x. (pi / 2)) < A /\ A < (2 x. pi)) -> (0 < (cos` A) /\ (sin` A) < 0)))
82	81	3impib 1065	. . 3 |- ((A e. RR /\ (3 x. (pi / 2)) < A /\ A < (2 x. pi)) -> (0 < (cos` A) /\ (sin` A) < 0))
83		ancom 482	. . 3 |- ((0 < (cos` A) /\ (sin` A) < 0) <-> ((sin` A) < 0 /\ 0 < (cos` A)))
84	82, 83	sylib 215	. 2 |- ((A e. RR /\ (3 x. (pi / 2)) < A /\ A < (2 x. pi)) -> ((sin` A) < 0 /\ 0 < (cos` A)))
85	12, 84	sylbi 216	1 |- (A e. ((3 x. (pi / 2))(,)(2 x. pi)) -> ((sin` A) < 0 /\ 0 < (cos` A)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  RR*cxr 6652   < clt 6653  2c2 7145  3c3 7146  4c4 7147  (,)cioo 7524  sincsin 8557  cosccos 8558  picpi 8559

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-5 7157  df-6 7158  df-7 7159  df-8 7160  df-9 7161  df-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-ioc 7529  df-ico 7530  df-icc 7531  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-bc 8209  df-clim 8235  df-sum 8240  df-cncf 8525  df-ef 8560  df-sin 8562  df-cos 8563  df-pi 8564  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200

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