Theorem sincosq2sgn 13542

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 13507	. . 3 |- ( pi / 2 ) e. RR
2		pire 13501	. . 3 |- pi e. RR
3		rexr 7965	. . . 4 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
4		rexr 7965	. . . 4 |- ( pi e. RR -> pi e. RR* )
5		elioo2 9878	. . . 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 424	. 2 |- ( A e. ( ( pi / 2 ) (,) pi ) <-> ( A e. RR /\ ( pi / 2 ) < A /\ A < pi ) )
8		resubcl 8183	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A - ( pi / 2 ) ) e. RR )
9	1, 8	mpan2 423	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. RR )
10		0xr 7966	. . . . . . . . . 10 |- 0 e. RR*
11	1	rexri 7977	. . . . . . . . . 10 |- ( pi / 2 ) e. RR*
12		elioo2 9878	. . . . . . . . . 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 424	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( 0 (,) ( pi / 2 ) ) <-> ( ( A - ( pi / 2 ) ) e. RR /\ 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) )
14		sincosq1sgn 13541	. . . . . . . . 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 1266	. . . . . . 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 1201	. . . . . 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 7920	. . . . . . . . 9 |- 0 e. RR
19		ltsub13 8362	. . . . . . . . 9 |- ( ( 0 e. RR /\ A e. RR /\ ( pi / 2 ) e. RR ) -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < ( A - 0 ) ) )
20	18, 1, 19	mp3an13 1323	. . . . . . . 8 |- ( A e. RR -> ( 0 < ( A - ( pi / 2 ) ) <-> ( pi / 2 ) < ( A - 0 ) ) )
21		recn 7907	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
22	21	subid1d 8219	. . . . . . . . 9 |- ( A e. RR -> ( A - 0 ) = A )
23	22	breq2d 4001	. . . . . . . 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 8351	. . . . . . . . 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 1324	. . . . . . . 8 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < ( pi / 2 ) <-> A < ( ( pi / 2 ) + ( pi / 2 ) ) ) )
27		pidiv2halves 13510	. . . . . . . . 9 |- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
28	27	breq2i 3997	. . . . . . . 8 |- ( A < ( ( pi / 2 ) + ( pi / 2 ) ) <-> A < pi )
29	26, 28	bitrdi 195	. . . . . . 7 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < ( pi / 2 ) <-> A < pi ) )
30	24, 29	anbi12d 470	. . . . . 6 |- ( A e. RR -> ( ( 0 < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < ( pi / 2 ) ) <-> ( ( pi / 2 ) < A /\ A < pi ) ) )
31	9	resincld 11686	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
32	31	lt0neg2d 8435	. . . . . . 7 |- ( A e. RR -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
33	32	anbi1d 462	. . . . . 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 7932	. . . . . . . . . 10 |- ( pi / 2 ) e. CC
36		pncan3 8127	. . . . . . . . . 10 |- ( ( ( pi / 2 ) e. CC /\ A e. CC ) -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
37	35, 21, 36	sylancr 412	. . . . . . . . 9 |- ( A e. RR -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
38	37	fveq2d 5500	. . . . . . . 8 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
39	9	recnd 7948	. . . . . . . . 9 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
40		coshalfpip 13537	. . . . . . . . 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 2205	. . . . . . 7 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
43	42	breq1d 3999	. . . . . 6 |- ( A e. RR -> ( ( cos ` A ) < 0 <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
44	37	fveq2d 5500	. . . . . . . 8 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
45		sinhalfpip 13535	. . . . . . . . 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 2205	. . . . . . 7 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
48	47	breq2d 4001	. . . . . 6 |- ( A e. RR -> ( 0 < ( sin ` A ) <-> 0 < ( cos ` ( A - ( pi / 2 ) ) ) ) )
49	43, 48	anbi12d 470	. . . . 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 1196	. . 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 973   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   + caddc 7777   RR*cxr 7953   < clt 7954   - cmin 8090   -ucneg 8091   / cdiv 8589   2c2 8929   (,)cioo 9845   sincsin 11607   cosccos 11608   picpi 11610

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894  ax-pre-suploc 7895  ax-addf 7896  ax-mulf 7897

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-disj 3967  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-map 6628  df-pm 6629  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-9 8944  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-ioo 9849  df-ioc 9850  df-ico 9851  df-icc 9852  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-bc 10682  df-ihash 10710  df-shft 10779  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611  df-sin 11613  df-cos 11614  df-pi 11616  df-rest 12581  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-met 12783  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835  df-ntr 12890  df-cn 12982  df-cnp 12983  df-tx 13047  df-cncf 13352  df-limced 13419  df-dvap 13420

This theorem is referenced by:  sincosq3sgn  13543