Theorem sincosq2sgn 15003

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 14968	. . 3 |- ( pi / 2 ) e. RR
2		pire 14962	. . 3 |- pi e. RR
3		rexr 8067	. . . 4 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
4		rexr 8067	. . . 4 |- ( pi e. RR -> pi e. RR* )
5		elioo2 9990	. . . 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 426	. 2 |- ( A e. ( ( pi / 2 ) (,) pi ) <-> ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) )
8		resubcl 8285	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A - ( pi / 2 ) ) e. RR )
9	1, 8	mpan2 425	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. RR )
10		0xr 8068	. . . . . . . . . 10 |- 0 e. RR*
11	1	rexri 8079	. . . . . . . . . 10 |- ( pi / 2 ) e. RR*
12		elioo2 9990	. . . . . . . . . 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 426	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( 0 (,) ( pi / 2 ) ) <-> ( ( A - ( pi / 2 ) ) e. RR /\ 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) )
14		sincosq1sgn 15002	. . . . . . . . 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 1282	. . . . . . 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 1208	. . . . . 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 8021	. . . . . . . . 9 |- 0 e. RR
19		ltsub13 8464	. . . . . . . . 9 |- ( ( 0 e. RR /\ A e. RR /\ ( pi / 2 ) e. RR ) -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < ( A - 0 ) ) )
20	18, 1, 19	mp3an13 1339	. . . . . . . 8 |- ( A e. RR -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < ( A - 0 ) ) )
21		recn 8007	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
22	21	subid1d 8321	. . . . . . . . 9 |- ( A e. RR -> ( A - 0 ) = A )
23	22	breq2d 4042	. . . . . . . 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 8453	. . . . . . . . 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 1340	. . . . . . . 8 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < ( pi / 2 ) <-> A < ( ( pi / 2 ) + ( pi / 2 ) ) ) )
27		pidiv2halves 14971	. . . . . . . . 9 |- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
28	27	breq2i 4038	. . . . . . . 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 473	. . . . . 6 |- ( A e. RR -> ( ( 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) <-> ( ( pi / 2 ) < A /\ A < pi ) ) )
31	9	resincld 11869	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
32	31	lt0neg2d 8537	. . . . . . 7 |- ( A e. RR -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
33	32	anbi1d 465	. . . . . 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 8033	. . . . . . . . . 10 |- ( pi / 2 ) e. CC
36		pncan3 8229	. . . . . . . . . 10 |- ( ( ( pi / 2 ) e. CC /\ A e. CC ) -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
37	35, 21, 36	sylancr 414	. . . . . . . . 9 |- ( A e. RR -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
38	37	fveq2d 5559	. . . . . . . 8 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
39	9	recnd 8050	. . . . . . . . 9 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
40		coshalfpip 14998	. . . . . . . . 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 2228	. . . . . . 7 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
43	42	breq1d 4040	. . . . . 6 |- ( A e. RR -> ( ( cos ` A ) < 0 <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
44	37	fveq2d 5559	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
45		sinhalfpip 14996	. . . . . . . . 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 2228	. . . . . . 7 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
48	47	breq2d 4042	. . . . . 6 |- ( A e. RR -> ( 0 < ( sin ` A ) <-> 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) )
49	43, 48	anbi12d 473	. . . . 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 1203	. . 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 980   = wceq 1364   e. wcel 2164   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   + caddc 7877   RR*cxr 8055   < clt 8056   - cmin 8192   -ucneg 8193   / cdiv 8693   2c2 9035   (,)cioo 9957   sincsin 11790   cosccos 11791   picpi 11793

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994  ax-pre-suploc 7995  ax-addf 7996  ax-mulf 7997

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-disj 4008  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-of 6132  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-map 6706  df-pm 6707  df-en 6797  df-dom 6798  df-fin 6799  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-xneg 9841  df-xadd 9842  df-ioo 9961  df-ioc 9962  df-ico 9963  df-icc 9964  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-fac 10800  df-bc 10822  df-ihash 10850  df-shft 10962  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500  df-ef 11794  df-sin 11796  df-cos 11797  df-pi 11799  df-rest 12855  df-topgen 12874  df-psmet 14042  df-xmet 14043  df-met 14044  df-bl 14045  df-mopn 14046  df-top 14177  df-topon 14190  df-bases 14222  df-ntr 14275  df-cn 14367  df-cnp 14368  df-tx 14432  df-cncf 14750  df-limced 14835  df-dvap 14836

This theorem is referenced by:  sincosq3sgn  15004