Theorem sincosq2sgn 12908

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 12873	. . 3 |- ( pi / 2 ) e. RR
2		pire 12867	. . 3 |- pi e. RR
3		rexr 7811	. . . 4 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
4		rexr 7811	. . . 4 |- ( pi e. RR -> pi e. RR* )
5		elioo2 9704	. . . 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 287	. . 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 422	. 2 |- ( A e. ( ( pi / 2 ) (,) pi ) <-> ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) )
8		resubcl 8026	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A - ( pi / 2 ) ) e. RR )
9	1, 8	mpan2 421	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. RR )
10		0xr 7812	. . . . . . . . . 10 |- 0 e. RR*
11	1	rexri 7823	. . . . . . . . . 10 |- ( pi / 2 ) e. RR*
12		elioo2 9704	. . . . . . . . . 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 422	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( 0 (,) ( pi / 2 ) ) <-> ( ( A - ( pi / 2 ) ) e. RR /\ 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) )
14		sincosq1sgn 12907	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( 0 (,) ( pi / 2 ) ) -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) )
15	13, 14	sylbir 134	. . . . . . . 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 1249	. . . . . . 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 1184	. . . . . 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 7766	. . . . . . . . 9 |- 0 e. RR
19		ltsub13 8205	. . . . . . . . 9 |- ( ( 0 e. RR /\ A e. RR /\ ( pi / 2 ) e. RR ) -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < ( A - 0 ) ) )
20	18, 1, 19	mp3an13 1306	. . . . . . . 8 |- ( A e. RR -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < ( A - 0 ) ) )
21		recn 7753	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
22	21	subid1d 8062	. . . . . . . . 9 |- ( A e. RR -> ( A - 0 ) = A )
23	22	breq2d 3941	. . . . . . . 8 |- ( A e. RR -> ( ( pi / 2 ) < ( A - 0 ) <-> ( pi / 2 ) < A ) )
24	20, 23	bitrd 187	. . . . . . 7 |- ( A e. RR -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < A ) )
25		ltsubadd 8194	. . . . . . . . 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 1307	. . . . . . . 8 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < ( pi / 2 ) <-> A < ( ( pi / 2 ) + ( pi / 2 ) ) ) )
27		pidiv2halves 12876	. . . . . . . . 9 |- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
28	27	breq2i 3937	. . . . . . . 8 |- ( A < ( ( pi / 2 ) + ( pi / 2 ) ) <-> A < pi )
29	26, 28	syl6bb 195	. . . . . . 7 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < ( pi / 2 ) <-> A < pi ) )
30	24, 29	anbi12d 464	. . . . . 6 |- ( A e. RR -> ( ( 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) <-> ( ( pi / 2 ) < A /\ A < pi ) ) )
31	9	resincld 11430	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
32	31	lt0neg2d 8278	. . . . . . 7 |- ( A e. RR -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
33	32	anbi1d 460	. . . . . 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 201	. . . . 5 |- ( A e. RR -> ( ( ( pi / 2 ) < A /\ A < pi ) -> ( -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) ) )
35	1	recni 7778	. . . . . . . . . 10 |- ( pi / 2 ) e. CC
36		pncan3 7970	. . . . . . . . . 10 |- ( ( ( pi / 2 ) e. CC /\ A e. CC ) -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
37	35, 21, 36	sylancr 410	. . . . . . . . 9 |- ( A e. RR -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
38	37	fveq2d 5425	. . . . . . . 8 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
39	9	recnd 7794	. . . . . . . . 9 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
40		coshalfpip 12903	. . . . . . . . 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 2174	. . . . . . 7 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
43	42	breq1d 3939	. . . . . 6 |- ( A e. RR -> ( ( cos ` A ) < 0 <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
44	37	fveq2d 5425	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
45		sinhalfpip 12901	. . . . . . . . 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 2174	. . . . . . 7 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
48	47	breq2d 3941	. . . . . 6 |- ( A e. RR -> ( 0 < ( sin ` A ) <-> 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) )
49	43, 48	anbi12d 464	. . . . 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 168	. . . 4 |- ( A e. RR -> ( ( ( pi / 2 ) < A /\ A < pi ) -> ( ( cos ` A ) < 0 /\ 0 < ( sin ` A ) ) ) )
51	50	3impib 1179	. . 3 |- ( ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) -> ( ( cos ` A ) < 0 /\ 0 < ( sin ` A ) ) )
52	51	ancomd 265	. 2 |- ( ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) -> ( 0 < ( sin ` A ) /\ ( cos ` A ) < 0 ) )
53	7, 52	sylbi 120	1 |- ( A e. ( ( pi / 2 ) (,) pi ) -> ( 0 < ( sin ` A ) /\ ( cos ` A ) < 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   + caddc 7623   RR*cxr 7799   < clt 7800   - cmin 7933   -ucneg 7934   / cdiv 8432   2c2 8771   (,)cioo 9671   sincsin 11350   cosccos 11351   picpi 11353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740  ax-pre-suploc 7741  ax-addf 7742  ax-mulf 7743

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-map 6544  df-pm 6545  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-5 8782  df-6 8783  df-7 8784  df-8 8785  df-9 8786  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-ioo 9675  df-ioc 9676  df-ico 9677  df-icc 9678  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-bc 10494  df-ihash 10522  df-shft 10587  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123  df-ef 11354  df-sin 11356  df-cos 11357  df-pi 11359  df-rest 12122  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-met 12158  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210  df-ntr 12265  df-cn 12357  df-cnp 12358  df-tx 12422  df-cncf 12727  df-limced 12794  df-dvap 12795

This theorem is referenced by:  sincosq3sgn  12909