Theorem sincosq4sgn 15543

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 9207	. . . . 5 |- 3 e. RR
2		halfpire 15506	. . . . 5 |- ( pi / 2 ) e. RR
3	1, 2	remulcli 8183	. . . 4 |- ( 3 x. ( pi / 2 ) ) e. RR
4	3	rexri 8227	. . 3 |- ( 3 x. ( pi / 2 ) ) e. RR*
5		2re 9203	. . . . 5 |- 2 e. RR
6		pire 15500	. . . . 5 |- pi e. RR
7	5, 6	remulcli 8183	. . . 4 |- ( 2 x. pi ) e. RR
8	7	rexri 8227	. . 3 |- ( 2 x. pi ) e. RR*
9		elioo2 10146	. . 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 ) ) ) )
10	4, 8, 9	mp2an 426	. 2 |- ( A e. ( ( 3 x. ( pi / 2 ) ) (,) ( 2 x. pi ) ) <-> ( A e. RR /\ ( 3 x. ( pi / 2 ) ) < A /\ A < ( 2 x. pi ) ) )
11		df-3 9193	. . . . . . . . . . . 12 |- 3 = ( 2 + 1 )
12	11	oveq1i 6023	. . . . . . . . . . 11 |- ( 3 x. ( pi / 2 ) ) = ( ( 2 + 1 ) x. ( pi / 2 ) )
13		2cn 9204	. . . . . . . . . . . 12 |- 2 e. CC
14		ax-1cn 8115	. . . . . . . . . . . 12 |- 1 e. CC
15	2	recni 8181	. . . . . . . . . . . 12 |- ( pi / 2 ) e. CC
16	13, 14, 15	adddiri 8180	. . . . . . . . . . 11 |- ( ( 2 + 1 ) x. ( pi / 2 ) ) = ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) )
17	6	recni 8181	. . . . . . . . . . . . 13 |- pi e. CC
18		2ap0 9226	. . . . . . . . . . . . 13 |- 2 # 0
19	17, 13, 18	divcanap2i 8925	. . . . . . . . . . . 12 |- ( 2 x. ( pi / 2 ) ) = pi
20	15	mullidi 8172	. . . . . . . . . . . 12 |- ( 1 x. ( pi / 2 ) ) = ( pi / 2 )
21	19, 20	oveq12i 6025	. . . . . . . . . . 11 |- ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) ) = ( pi + ( pi / 2 ) )
22	12, 16, 21	3eqtrri 2255	. . . . . . . . . 10 |- ( pi + ( pi / 2 ) ) = ( 3 x. ( pi / 2 ) )
23	22	breq1i 4093	. . . . . . . . 9 |- ( ( pi + ( pi / 2 ) ) < A <-> ( 3 x. ( pi / 2 ) ) < A )
24		ltaddsub 8606	. . . . . . . . . 10 |- ( ( pi e. RR /\ ( pi / 2 ) e. RR /\ A e. RR ) -> ( ( pi + ( pi / 2 ) ) < A <-> pi < ( A - ( pi / 2 ) ) ) )
25	6, 2, 24	mp3an12 1361	. . . . . . . . 9 |- ( A e. RR -> ( ( pi + ( pi / 2 ) ) < A <-> pi < ( A - ( pi / 2 ) ) ) )
26	23, 25	bitr3id 194	. . . . . . . 8 |- ( A e. RR -> ( ( 3 x. ( pi / 2 ) ) < A <-> pi < ( A - ( pi / 2 ) ) ) )
27		ltsubadd 8602	. . . . . . . . . 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 ) ) ) )
28	2, 3, 27	mp3an23 1363	. . . . . . . . 9 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < ( 3 x. ( pi / 2 ) ) <-> A < ( ( 3 x. ( pi / 2 ) ) + ( pi / 2 ) ) ) )
29		df-4 9194	. . . . . . . . . . . . 13 |- 4 = ( 3 + 1 )
30	29	oveq1i 6023	. . . . . . . . . . . 12 |- ( 4 x. ( pi / 2 ) ) = ( ( 3 + 1 ) x. ( pi / 2 ) )
31	1	recni 8181	. . . . . . . . . . . . 13 |- 3 e. CC
32	31, 14, 15	adddiri 8180	. . . . . . . . . . . 12 |- ( ( 3 + 1 ) x. ( pi / 2 ) ) = ( ( 3 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) )
33	20	oveq2i 6024	. . . . . . . . . . . 12 |- ( ( 3 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) ) = ( ( 3 x. ( pi / 2 ) ) + ( pi / 2 ) )
34	30, 32, 33	3eqtrri 2255	. . . . . . . . . . 11 |- ( ( 3 x. ( pi / 2 ) ) + ( pi / 2 ) ) = ( 4 x. ( pi / 2 ) )
35		4cn 9211	. . . . . . . . . . . . 13 |- 4 e. CC
36	13, 18	pm3.2i 272	. . . . . . . . . . . . 13 |- ( 2 e. CC /\ 2 # 0 )
37		div12ap 8864	. . . . . . . . . . . . 13 |- ( ( 4 e. CC /\ pi e. CC /\ ( 2 e. CC /\ 2 # 0 ) ) -> ( 4 x. ( pi / 2 ) ) = ( pi x. ( 4 / 2 ) ) )
38	35, 17, 36, 37	mp3an 1371	. . . . . . . . . . . 12 |- ( 4 x. ( pi / 2 ) ) = ( pi x. ( 4 / 2 ) )
39		4d2e2 9294	. . . . . . . . . . . . . 14 |- ( 4 / 2 ) = 2
40	39	oveq2i 6024	. . . . . . . . . . . . 13 |- ( pi x. ( 4 / 2 ) ) = ( pi x. 2 )
41	17, 13	mulcomi 8175	. . . . . . . . . . . . 13 |- ( pi x. 2 ) = ( 2 x. pi )
42	40, 41	eqtri 2250	. . . . . . . . . . . 12 |- ( pi x. ( 4 / 2 ) ) = ( 2 x. pi )
43	38, 42	eqtri 2250	. . . . . . . . . . 11 |- ( 4 x. ( pi / 2 ) ) = ( 2 x. pi )
44	34, 43	eqtri 2250	. . . . . . . . . 10 |- ( ( 3 x. ( pi / 2 ) ) + ( pi / 2 ) ) = ( 2 x. pi )
45	44	breq2i 4094	. . . . . . . . 9 |- ( A < ( ( 3 x. ( pi / 2 ) ) + ( pi / 2 ) ) <-> A < ( 2 x. pi ) )
46	28, 45	bitr2di 197	. . . . . . . 8 |- ( A e. RR -> ( A < ( 2 x. pi ) <-> ( A - ( pi / 2 ) ) < ( 3 x. ( pi / 2 ) ) ) )
47	26, 46	anbi12d 473	. . . . . . 7 |- ( A e. RR -> ( ( ( 3 x. ( pi / 2 ) ) < A /\ A < ( 2 x. pi ) ) <-> ( pi < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( 3 x. ( pi / 2 ) ) ) ) )
48		resubcl 8433	. . . . . . . . . 10 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A - ( pi / 2 ) ) e. RR )
49	2, 48	mpan2 425	. . . . . . . . 9 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. RR )
50	6	rexri 8227	. . . . . . . . . . 11 |- pi e. RR*
51		elioo2 10146	. . . . . . . . . . 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	50, 4, 51	mp2an 426	. . . . . . . . . 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 15542	. . . . . . . . . 10 |- ( ( A - ( pi / 2 ) ) e. ( pi (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( sin ` ( A - ( pi / 2 ) ) ) < 0 /\ ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
54	52, 53	sylbir 135	. . . . . . . . 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	49, 54	syl3an1 1304	. . . . . . . 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 ) )
56	55	3expib 1230	. . . . . . 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 ) ) )
57	47, 56	sylbid 150	. . . . . 6 |- ( A e. RR -> ( ( ( 3 x. ( pi / 2 ) ) < A /\ A < ( 2 x. pi ) ) -> ( ( sin ` ( A - ( pi / 2 ) ) ) < 0 /\ ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) ) )
58	49	resincld 12274	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
59	58	lt0neg1d 8685	. . . . . . 7 |- ( A e. RR -> ( ( sin ` ( A - ( pi / 2 ) ) ) < 0 <-> 0 < -u ( sin ` ( A - ( pi / 2 ) ) ) ) )
60	59	anbi1d 465	. . . . . 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 ) ) )
61	57, 60	sylibd 149	. . . . 5 |- ( A e. RR -> ( ( ( 3 x. ( pi / 2 ) ) < A /\ A < ( 2 x. pi ) ) -> ( 0 < -u ( sin ` ( A - ( pi / 2 ) ) ) /\ ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) ) )
62		recn 8155	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
63		pncan3 8377	. . . . . . . . . 10 |- ( ( ( pi / 2 ) e. CC /\ A e. CC ) -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
64	15, 62, 63	sylancr 414	. . . . . . . . 9 |- ( A e. RR -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
65	64	fveq2d 5639	. . . . . . . 8 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
66	49	recnd 8198	. . . . . . . . 9 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
67		coshalfpip 15536	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. CC -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
68	66, 67	syl 14	. . . . . . . 8 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
69	65, 68	eqtr3d 2264	. . . . . . 7 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
70	69	breq2d 4098	. . . . . 6 |- ( A e. RR -> ( 0 < ( cos ` A ) <-> 0 < -u ( sin ` ( A - ( pi / 2 ) ) ) ) )
71	64	fveq2d 5639	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
72		sinhalfpip 15534	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. CC -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` ( A - ( pi / 2 ) ) ) )
73	66, 72	syl 14	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` ( A - ( pi / 2 ) ) ) )
74	71, 73	eqtr3d 2264	. . . . . . 7 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
75	74	breq1d 4096	. . . . . 6 |- ( A e. RR -> ( ( sin ` A ) < 0 <-> ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
76	70, 75	anbi12d 473	. . . . 5 |- ( A e. RR -> ( ( 0 < ( cos ` A ) /\ ( sin ` A ) < 0 ) <-> ( 0 < -u ( sin ` ( A - ( pi / 2 ) ) ) /\ ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) ) )
77	61, 76	sylibrd 169	. . . 4 |- ( A e. RR -> ( ( ( 3 x. ( pi / 2 ) ) < A /\ A < ( 2 x. pi ) ) -> ( 0 < ( cos ` A ) /\ ( sin ` A ) < 0 ) ) )
78	77	3impib 1225	. . 3 |- ( ( A e. RR /\ ( 3 x. ( pi / 2 ) ) < A /\ A < ( 2 x. pi ) ) -> ( 0 < ( cos ` A ) /\ ( sin ` A ) < 0 ) )
79	78	ancomd 267	. 2 |- ( ( A e. RR /\ ( 3 x. ( pi / 2 ) ) < A /\ A < ( 2 x. pi ) ) -> ( ( sin ` A ) < 0 /\ 0 < ( cos ` A ) ) )
80	10, 79	sylbi 121	1 |- ( A e. ( ( 3 x. ( pi / 2 ) ) (,) ( 2 x. pi ) ) -> ( ( sin ` A ) < 0 /\ 0 < ( cos ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   RR*cxr 8203   < clt 8204   - cmin 8340   -ucneg 8341   # cap 8751   / cdiv 8842   2c2 9184   3c3 9185   4c4 9186   (,)cioo 10113   sincsin 12195   cosccos 12196   picpi 12198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142  ax-pre-suploc 8143  ax-addf 8144  ax-mulf 8145

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-disj 4063  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-map 6814  df-pm 6815  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-xneg 9997  df-xadd 9998  df-ioo 10117  df-ioc 10118  df-ico 10119  df-icc 10120  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-exp 10791  df-fac 10978  df-bc 11000  df-ihash 11028  df-shft 11366  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-sumdc 11905  df-ef 12199  df-sin 12201  df-cos 12202  df-pi 12204  df-rest 13314  df-topgen 13333  df-psmet 14547  df-xmet 14548  df-met 14549  df-bl 14550  df-mopn 14551  df-top 14712  df-topon 14725  df-bases 14757  df-ntr 14810  df-cn 14902  df-cnp 14903  df-tx 14967  df-cncf 15285  df-limced 15370  df-dvap 15371

This theorem is referenced by: (None)