Theorem sincosq2sgn 15911

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

Assertion

Ref	Expression
sincosq2sgn	|- ( A e. ( ( pi / 2 ) (,) pi ) -> ( 0 < ( sin ` A ) /\ ( cos ` A ) < 0 ) )

Proof of Theorem sincosq2sgn

Step	Hyp	Ref	Expression
1		halfpire 15876	. . 3 |- ( pi / 2 ) e. RR
2		pire 15870	. . 3 |- pi e. RR
3		rexr 8365	. . . 4 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
4		rexr 8365	. . . 4 |- ( pi e. RR -> pi e. RR* )
5		elioo2 10306	. . . 4 |- ( ( ( pi / 2 ) e. RR* /\ pi e. RR* ) -> ( A e. ( ( pi / 2 ) (,) pi ) <-> ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) ) )
6	3, 4, 5	syl2an 289	. . 3 |- ( ( ( pi / 2 ) e. RR /\ pi e. RR ) -> ( A e. ( ( pi / 2 ) (,) pi ) <-> ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) ) )
7	1, 2, 6	mp2an 430	. 2 |- ( A e. ( ( pi / 2 ) (,) pi ) <-> ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) )
8		resubcl 8584	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A - ( pi / 2 ) ) e. RR )
9	1, 8	mpan2 429	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. RR )
10		0xr 8366	. . . . . . . . . 10 |- 0 e. RR*
11	1	rexri 8377	. . . . . . . . . 10 |- ( pi / 2 ) e. RR*
12		elioo2 10306	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ ( pi / 2 ) e. RR* ) -> ( ( A - ( pi / 2 ) ) e. ( 0 (,) ( pi / 2 ) ) <-> ( ( A - ( pi / 2 ) ) e. RR /\ 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) ) )
13	10, 11, 12	mp2an 430	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( 0 (,) ( pi / 2 ) ) <-> ( ( A - ( pi / 2 ) ) e. RR /\ 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) )
14		sincosq1sgn 15910	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( 0 (,) ( pi / 2 ) ) -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) )
15	13, 14	sylbir 135	. . . . . . . 8 |- ( ( ( A - ( pi / 2 ) ) e. RR /\ 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) )
16	9, 15	syl3an1 1311	. . . . . . 7 |- ( ( A e. RR /\ 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) )
17	16	3expib 1237	. . . . . 6 |- ( A e. RR -> ( ( 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) ) )
18		0re 8320	. . . . . . . . 9 |- 0 e. RR
19		ltsub13 8765	. . . . . . . . 9 |- ( ( 0 e. RR /\ A e. RR /\ ( pi / 2 ) e. RR ) -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < ( A - 0 ) ) )
20	18, 1, 19	mp3an13 1369	. . . . . . . 8 |- ( A e. RR -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < ( A - 0 ) ) )
21		recn 8306	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
22	21	subid1d 8620	. . . . . . . . 9 |- ( A e. RR -> ( A - 0 ) = A )
23	22	breq2d 4140	. . . . . . . 8 |- ( A e. RR -> ( ( pi / 2 ) < ( A - 0 ) <-> ( pi / 2 ) < A ) )
24	20, 23	bitrd 188	. . . . . . 7 |- ( A e. RR -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < A ) )
25		ltsubadd 8754	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR /\ ( pi / 2 ) e. RR ) -> ( ( A - ( pi / 2 ) ) < ( pi / 2 ) <-> A < ( ( pi / 2 ) + ( pi / 2 ) ) ) )
26	1, 1, 25	mp3an23 1370	. . . . . . . 8 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < ( pi / 2 ) <-> A < ( ( pi / 2 ) + ( pi / 2 ) ) ) )
27		pidiv2halves 15879	. . . . . . . . 9 |- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
28	27	breq2i 4136	. . . . . . . 8 |- ( A < ( ( pi / 2 ) + ( pi / 2 ) ) <-> A < pi )
29	26, 28	bitrdi 196	. . . . . . 7 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < ( pi / 2 ) <-> A < pi ) )
30	24, 29	anbi12d 477	. . . . . 6 |- ( A e. RR -> ( ( 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) <-> ( ( pi / 2 ) < A /\ A < pi ) ) )
31	9	resincld 12473	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
32	31	lt0neg2d 8838	. . . . . . 7 |- ( A e. RR -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
33	32	anbi1d 469	. . . . . 6 |- ( A e. RR -> ( ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) <-> ( -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) ) )
34	17, 30, 33	3imtr3d 202	. . . . 5 |- ( A e. RR -> ( ( ( pi / 2 ) < A /\ A < pi ) -> ( -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) ) )
35	1	recni 8332	. . . . . . . . . 10 |- ( pi / 2 ) e. CC
36		pncan3 8528	. . . . . . . . . 10 |- ( ( ( pi / 2 ) e. CC /\ A e. CC ) -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
37	35, 21, 36	sylancr 418	. . . . . . . . 9 |- ( A e. RR -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
38	37	fveq2d 5697	. . . . . . . 8 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
39	9	recnd 8348	. . . . . . . . 9 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
40		coshalfpip 15906	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. CC -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
41	39, 40	syl 14	. . . . . . . 8 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
42	38, 41	eqtr3d 2273	. . . . . . 7 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
43	42	breq1d 4138	. . . . . 6 |- ( A e. RR -> ( ( cos ` A ) < 0 <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
44	37	fveq2d 5697	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
45		sinhalfpip 15904	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. CC -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` ( A - ( pi / 2 ) ) ) )
46	39, 45	syl 14	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` ( A - ( pi / 2 ) ) ) )
47	44, 46	eqtr3d 2273	. . . . . . 7 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
48	47	breq2d 4140	. . . . . 6 |- ( A e. RR -> ( 0 < ( sin ` A ) <-> 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) )
49	43, 48	anbi12d 477	. . . . 5 |- ( A e. RR -> ( ( ( cos ` A ) < 0 /\ 0 < ( sin ` A ) ) <-> ( -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) ) )
50	34, 49	sylibrd 169	. . . 4 |- ( A e. RR -> ( ( ( pi / 2 ) < A /\ A < pi ) -> ( ( cos ` A ) < 0 /\ 0 < ( sin ` A ) ) ) )
51	50	3impib 1232	. . 3 |- ( ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) -> ( ( cos ` A ) < 0 /\ 0 < ( sin ` A ) ) )
52	51	ancomd 267	. 2 |- ( ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) -> ( 0 < ( sin ` A ) /\ ( cos ` A ) < 0 ) )
53	7, 52	sylbi 121	1 |- ( A e. ( ( pi / 2 ) (,) pi ) -> ( 0 < ( sin ` A ) /\ ( cos ` A ) < 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   CCcc 8171   RRcr 8172   0cc0 8173   + caddc 8176   RR*cxr 8353   < clt 8354   - cmin 8491   -ucneg 8492   / cdiv 8996   2c2 9338   (,)cioo 10273   sincsin 12394   cosccos 12395   picpi 12397

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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293  ax-pre-suploc 8294  ax-addf 8295  ax-mulf 8296

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 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-disj 4105  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-map 6918  df-pm 6919  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-xneg 10157  df-xadd 10158  df-ioo 10277  df-ioc 10278  df-ico 10279  df-icc 10280  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-exp 10959  df-fac 11147  df-bc 11169  df-ihash 11198  df-shft 11563  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-sumdc 12103  df-ef 12398  df-sin 12400  df-cos 12401  df-pi 12403  df-rest 13578  df-topgen 13597  df-psmet 14863  df-xmet 14864  df-met 14865  df-bl 14866  df-mopn 14867  df-top 15082  df-topon 15095  df-bases 15127  df-ntr 15180  df-cn 15272  df-cnp 15273  df-tx 15337  df-cncf 15655  df-limced 15740  df-dvap 15741

This theorem is referenced by:  sincosq3sgn  15912