Theorem sincosq4sgn 15853

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 9357	. . . . 5 |- 3 e. RR
2		halfpire 15816	. . . . 5 |- ( pi / 2 ) e. RR
3	1, 2	remulcli 8330	. . . 4 |- ( 3 x. ( pi / 2 ) ) e. RR
4	3	rexri 8373	. . 3 |- ( 3 x. ( pi / 2 ) ) e. RR*
5		2re 9353	. . . . 5 |- 2 e. RR
6		pire 15810	. . . . 5 |- pi e. RR
7	5, 6	remulcli 8330	. . . 4 |- ( 2 x. pi ) e. RR
8	7	rexri 8373	. . 3 |- ( 2 x. pi ) e. RR*
9		elioo2 10302	. . 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 430	. 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 9343	. . . . . . . . . . . 12 |- 3 = ( 2 + 1 )
12	11	oveq1i 6085	. . . . . . . . . . 11 |- ( 3 x. ( pi / 2 ) ) = ( ( 2 + 1 ) x. ( pi / 2 ) )
13		2cn 9354	. . . . . . . . . . . 12 |- 2 e. CC
14		ax-1cn 8262	. . . . . . . . . . . 12 |- 1 e. CC
15	2	recni 8328	. . . . . . . . . . . 12 |- ( pi / 2 ) e. CC
16	13, 14, 15	adddiri 8327	. . . . . . . . . . 11 |- ( ( 2 + 1 ) x. ( pi / 2 ) ) = ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) )
17	6	recni 8328	. . . . . . . . . . . . 13 |- pi e. CC
18		2ap0 9376	. . . . . . . . . . . . 13 |- 2 # 0
19	17, 13, 18	divcanap2i 9075	. . . . . . . . . . . 12 |- ( 2 x. ( pi / 2 ) ) = pi
20	15	mullidi 8319	. . . . . . . . . . . 12 |- ( 1 x. ( pi / 2 ) ) = ( pi / 2 )
21	19, 20	oveq12i 6087	. . . . . . . . . . 11 |- ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) ) = ( pi + ( pi / 2 ) )
22	12, 16, 21	3eqtrri 2264	. . . . . . . . . 10 |- ( pi + ( pi / 2 ) ) = ( 3 x. ( pi / 2 ) )
23	22	breq1i 4132	. . . . . . . . 9 |- ( ( pi + ( pi / 2 ) ) < A <-> ( 3 x. ( pi / 2 ) ) < A )
24		ltaddsub 8754	. . . . . . . . . 10 |- ( ( pi e. RR /\ ( pi / 2 ) e. RR /\ A e. RR ) -> ( ( pi + ( pi / 2 ) ) < A <-> pi < ( A - ( pi / 2 ) ) ) )
25	6, 2, 24	mp3an12 1368	. . . . . . . . 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 8750	. . . . . . . . . 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 1370	. . . . . . . . 9 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < ( 3 x. ( pi / 2 ) ) <-> A < ( ( 3 x. ( pi / 2 ) ) + ( pi / 2 ) ) ) )
29		df-4 9344	. . . . . . . . . . . . 13 |- 4 = ( 3 + 1 )
30	29	oveq1i 6085	. . . . . . . . . . . 12 |- ( 4 x. ( pi / 2 ) ) = ( ( 3 + 1 ) x. ( pi / 2 ) )
31	1	recni 8328	. . . . . . . . . . . . 13 |- 3 e. CC
32	31, 14, 15	adddiri 8327	. . . . . . . . . . . 12 |- ( ( 3 + 1 ) x. ( pi / 2 ) ) = ( ( 3 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) )
33	20	oveq2i 6086	. . . . . . . . . . . 12 |- ( ( 3 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) ) = ( ( 3 x. ( pi / 2 ) ) + ( pi / 2 ) )
34	30, 32, 33	3eqtrri 2264	. . . . . . . . . . 11 |- ( ( 3 x. ( pi / 2 ) ) + ( pi / 2 ) ) = ( 4 x. ( pi / 2 ) )
35		4cn 9361	. . . . . . . . . . . . 13 |- 4 e. CC
36	13, 18	pm3.2i 272	. . . . . . . . . . . . 13 |- ( 2 e. CC /\ 2 # 0 )
37		div12ap 9014	. . . . . . . . . . . . 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 1378	. . . . . . . . . . . 12 |- ( 4 x. ( pi / 2 ) ) = ( pi x. ( 4 / 2 ) )
39		4d2e2 9444	. . . . . . . . . . . . . 14 |- ( 4 / 2 ) = 2
40	39	oveq2i 6086	. . . . . . . . . . . . 13 |- ( pi x. ( 4 / 2 ) ) = ( pi x. 2 )
41	17, 13	mulcomi 8322	. . . . . . . . . . . . 13 |- ( pi x. 2 ) = ( 2 x. pi )
42	40, 41	eqtri 2259	. . . . . . . . . . . 12 |- ( pi x. ( 4 / 2 ) ) = ( 2 x. pi )
43	38, 42	eqtri 2259	. . . . . . . . . . 11 |- ( 4 x. ( pi / 2 ) ) = ( 2 x. pi )
44	34, 43	eqtri 2259	. . . . . . . . . 10 |- ( ( 3 x. ( pi / 2 ) ) + ( pi / 2 ) ) = ( 2 x. pi )
45	44	breq2i 4133	. . . . . . . . 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 477	. . . . . . 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 8580	. . . . . . . . . 10 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A - ( pi / 2 ) ) e. RR )
49	2, 48	mpan2 429	. . . . . . . . 9 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. RR )
50	6	rexri 8373	. . . . . . . . . . 11 |- pi e. RR*
51		elioo2 10302	. . . . . . . . . . 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 430	. . . . . . . . . 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 15852	. . . . . . . . . 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 1311	. . . . . . . 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 1237	. . . . . . 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 12468	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
59	58	lt0neg1d 8833	. . . . . . 7 |- ( A e. RR -> ( ( sin ` ( A - ( pi / 2 ) ) ) < 0 <-> 0 < -u ( sin ` ( A - ( pi / 2 ) ) ) ) )
60	59	anbi1d 469	. . . . . 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 8302	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
63		pncan3 8524	. . . . . . . . . 10 |- ( ( ( pi / 2 ) e. CC /\ A e. CC ) -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
64	15, 62, 63	sylancr 418	. . . . . . . . 9 |- ( A e. RR -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
65	64	fveq2d 5694	. . . . . . . 8 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
66	49	recnd 8344	. . . . . . . . 9 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
67		coshalfpip 15846	. . . . . . . . 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 2273	. . . . . . 7 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
70	69	breq2d 4137	. . . . . 6 |- ( A e. RR -> ( 0 < ( cos ` A ) <-> 0 < -u ( sin ` ( A - ( pi / 2 ) ) ) ) )
71	64	fveq2d 5694	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
72		sinhalfpip 15844	. . . . . . . . 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 2273	. . . . . . 7 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
75	74	breq1d 4135	. . . . . 6 |- ( A e. RR -> ( ( sin ` A ) < 0 <-> ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
76	70, 75	anbi12d 477	. . . . 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 1232	. . 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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   RR*cxr 8349   < clt 8350   - cmin 8487   -ucneg 8488   # cap 8899   / cdiv 8992   2c2 9334   3c3 9335   4c4 9336   (,)cioo 10269   sincsin 12389   cosccos 12390   picpi 12392

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289  ax-pre-suploc 8290  ax-addf 8291  ax-mulf 8292

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-disj 4102  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-map 6914  df-pm 6915  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-xneg 10153  df-xadd 10154  df-ioo 10273  df-ioc 10274  df-ico 10275  df-icc 10276  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-fac 11142  df-bc 11164  df-ihash 11193  df-shft 11558  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098  df-ef 12393  df-sin 12395  df-cos 12396  df-pi 12398  df-rest 13572  df-topgen 13591  df-psmet 14852  df-xmet 14853  df-met 14854  df-bl 14855  df-mopn 14856  df-top 15022  df-topon 15035  df-bases 15067  df-ntr 15120  df-cn 15212  df-cnp 15213  df-tx 15277  df-cncf 15595  df-limced 15680  df-dvap 15681

This theorem is referenced by: (None)