Theorem sincosq3sgn 13389

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

Assertion

Ref	Expression
sincosq3sgn	|- ( A e. ( pi (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) )

Proof of Theorem sincosq3sgn

Step	Hyp	Ref	Expression
1		pire 13347	. . 3 |- pi e. RR
2		3re 8931	. . . 4 |- 3 e. RR
3		halfpire 13353	. . . 4 |- ( pi / 2 ) e. RR
4	2, 3	remulcli 7913	. . 3 |- ( 3 x. ( pi / 2 ) ) e. RR
5		rexr 7944	. . . 4 |- ( pi e. RR -> pi e. RR* )
6		rexr 7944	. . . 4 |- ( ( 3 x. ( pi / 2 ) ) e. RR -> ( 3 x. ( pi / 2 ) ) e. RR* )
7		elioo2 9857	. . . 4 |- ( ( pi e. RR* /\ ( 3 x. ( pi / 2 ) ) e. RR* ) -> ( A e. ( pi (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) ) )
8	5, 6, 7	syl2an 287	. . 3 |- ( ( pi e. RR /\ ( 3 x. ( pi / 2 ) ) e. RR ) -> ( A e. ( pi (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) ) )
9	1, 4, 8	mp2an 423	. 2 |- ( A e. ( pi (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) )
10		pidiv2halves 13356	. . . . . . . . 9 |- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
11	10	breq1i 3989	. . . . . . . 8 |- ( ( ( pi / 2 ) + ( pi / 2 ) ) < A <-> pi < A )
12		ltaddsub 8334	. . . . . . . . 9 |- ( ( ( pi / 2 ) e. RR /\ ( pi / 2 ) e. RR /\ A e. RR ) -> ( ( ( pi / 2 ) + ( pi / 2 ) ) < A <-> ( pi / 2 ) < ( A - ( pi / 2 ) ) ) )
13	3, 3, 12	mp3an12 1317	. . . . . . . 8 |- ( A e. RR -> ( ( ( pi / 2 ) + ( pi / 2 ) ) < A <-> ( pi / 2 ) < ( A - ( pi / 2 ) ) ) )
14	11, 13	bitr3id 193	. . . . . . 7 |- ( A e. RR -> ( pi < A <-> ( pi / 2 ) < ( A - ( pi / 2 ) ) ) )
15		ltsubadd 8330	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR /\ pi e. RR ) -> ( ( A - ( pi / 2 ) ) < pi <-> A < ( pi + ( pi / 2 ) ) ) )
16	3, 1, 15	mp3an23 1319	. . . . . . . 8 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < pi <-> A < ( pi + ( pi / 2 ) ) ) )
17		df-3 8917	. . . . . . . . . . 11 |- 3 = ( 2 + 1 )
18	17	oveq1i 5852	. . . . . . . . . 10 |- ( 3 x. ( pi / 2 ) ) = ( ( 2 + 1 ) x. ( pi / 2 ) )
19		2cn 8928	. . . . . . . . . . 11 |- 2 e. CC
20		ax-1cn 7846	. . . . . . . . . . 11 |- 1 e. CC
21	3	recni 7911	. . . . . . . . . . 11 |- ( pi / 2 ) e. CC
22	19, 20, 21	adddiri 7910	. . . . . . . . . 10 |- ( ( 2 + 1 ) x. ( pi / 2 ) ) = ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) )
23	1	recni 7911	. . . . . . . . . . . 12 |- pi e. CC
24		2ap0 8950	. . . . . . . . . . . 12 |- 2 # 0
25	23, 19, 24	divcanap2i 8651	. . . . . . . . . . 11 |- ( 2 x. ( pi / 2 ) ) = pi
26	21	mulid2i 7902	. . . . . . . . . . 11 |- ( 1 x. ( pi / 2 ) ) = ( pi / 2 )
27	25, 26	oveq12i 5854	. . . . . . . . . 10 |- ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) ) = ( pi + ( pi / 2 ) )
28	18, 22, 27	3eqtrri 2191	. . . . . . . . 9 |- ( pi + ( pi / 2 ) ) = ( 3 x. ( pi / 2 ) )
29	28	breq2i 3990	. . . . . . . 8 |- ( A < ( pi + ( pi / 2 ) ) <-> A < ( 3 x. ( pi / 2 ) ) )
30	16, 29	bitr2di 196	. . . . . . 7 |- ( A e. RR -> ( A < ( 3 x. ( pi / 2 ) ) <-> ( A - ( pi / 2 ) ) < pi ) )
31	14, 30	anbi12d 465	. . . . . 6 |- ( A e. RR -> ( ( pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) <-> ( ( pi / 2 ) < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < pi ) ) )
32		resubcl 8162	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A - ( pi / 2 ) ) e. RR )
33	3, 32	mpan2 422	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. RR )
34		sincosq2sgn 13388	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( ( pi / 2 ) (,) pi ) -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
35		rexr 7944	. . . . . . . . . . 11 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
36		elioo2 9857	. . . . . . . . . . 11 |- ( ( ( pi / 2 ) e. RR* /\ pi e. RR* ) -> ( ( A - ( pi / 2 ) ) e. ( ( pi / 2 ) (,) pi ) <-> ( ( A - ( pi / 2 ) ) e. RR /\ ( pi / 2 ) < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < pi ) ) )
37	35, 5, 36	syl2an 287	. . . . . . . . . 10 |- ( ( ( pi / 2 ) e. RR /\ pi e. RR ) -> ( ( A - ( pi / 2 ) ) e. ( ( pi / 2 ) (,) pi ) <-> ( ( A - ( pi / 2 ) ) e. RR /\ ( pi / 2 ) < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < pi ) ) )
38	3, 1, 37	mp2an 423	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( ( pi / 2 ) (,) pi ) <-> ( ( A - ( pi / 2 ) ) e. RR /\ ( pi / 2 ) < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < pi ) )
39		ancom 264	. . . . . . . . 9 |- ( ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) <-> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( pi / 2 ) ) ) ) )
40	34, 38, 39	3imtr3i 199	. . . . . . . 8 |- ( ( ( A - ( pi / 2 ) ) e. RR /\ ( pi / 2 ) < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < pi ) -> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( pi / 2 ) ) ) ) )
41	33, 40	syl3an1 1261	. . . . . . 7 |- ( ( A e. RR /\ ( pi / 2 ) < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < pi ) -> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( pi / 2 ) ) ) ) )
42	41	3expib 1196	. . . . . 6 |- ( A e. RR -> ( ( ( pi / 2 ) < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < pi ) -> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( pi / 2 ) ) ) ) ) )
43	31, 42	sylbid 149	. . . . 5 |- ( A e. RR -> ( ( pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( pi / 2 ) ) ) ) ) )
44	33	resincld 11664	. . . . . . 7 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
45	44	lt0neg2d 8414	. . . . . 6 |- ( A e. RR -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
46	45	anbi2d 460	. . . . 5 |- ( A e. RR -> ( ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( pi / 2 ) ) ) ) <-> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) ) )
47	43, 46	sylibd 148	. . . 4 |- ( A e. RR -> ( ( pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) ) )
48		recn 7886	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
49		pncan3 8106	. . . . . . . . 9 |- ( ( ( pi / 2 ) e. CC /\ A e. CC ) -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
50	21, 48, 49	sylancr 411	. . . . . . . 8 |- ( A e. RR -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
51	50	fveq2d 5490	. . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
52	33	recnd 7927	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
53		sinhalfpip 13381	. . . . . . . 8 |- ( ( A - ( pi / 2 ) ) e. CC -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` ( A - ( pi / 2 ) ) ) )
54	52, 53	syl 14	. . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` ( A - ( pi / 2 ) ) ) )
55	51, 54	eqtr3d 2200	. . . . . 6 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
56	55	breq1d 3992	. . . . 5 |- ( A e. RR -> ( ( sin ` A ) < 0 <-> ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
57	50	fveq2d 5490	. . . . . . 7 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
58		coshalfpip 13383	. . . . . . . 8 |- ( ( A - ( pi / 2 ) ) e. CC -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
59	52, 58	syl 14	. . . . . . 7 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
60	57, 59	eqtr3d 2200	. . . . . 6 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
61	60	breq1d 3992	. . . . 5 |- ( A e. RR -> ( ( cos ` A ) < 0 <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
62	56, 61	anbi12d 465	. . . 4 |- ( A e. RR -> ( ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) <-> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) ) )
63	47, 62	sylibrd 168	. . 3 |- ( A e. RR -> ( ( pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) ) )
64	63	3impib 1191	. 2 |- ( ( A e. RR /\ pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) )
65	9, 64	sylbi 120	1 |- ( A e. ( pi (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   RR*cxr 7932   < clt 7933   - cmin 8069   -ucneg 8070   / cdiv 8568   2c2 8908   3c3 8909   (,)cioo 9824   sincsin 11585   cosccos 11586   picpi 11588

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-pre-suploc 7874  ax-addf 7875  ax-mulf 7876

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-map 6616  df-pm 6617  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-5 8919  df-6 8920  df-7 8921  df-8 8922  df-9 8923  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ioo 9828  df-ioc 9829  df-ico 9830  df-icc 9831  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-sin 11591  df-cos 11592  df-pi 11594  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266

This theorem is referenced by:  sincosq4sgn  13390