Theorem sincosq3sgn 15333

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 15291	. . 3 |- pi e. RR
2		3re 9112	. . . 4 |- 3 e. RR
3		halfpire 15297	. . . 4 |- ( pi / 2 ) e. RR
4	2, 3	remulcli 8088	. . 3 |- ( 3 x. ( pi / 2 ) ) e. RR
5		rexr 8120	. . . 4 |- ( pi e. RR -> pi e. RR* )
6		rexr 8120	. . . 4 |- ( ( 3 x. ( pi / 2 ) ) e. RR -> ( 3 x. ( pi / 2 ) ) e. RR* )
7		elioo2 10045	. . . 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 289	. . 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 426	. 2 |- ( A e. ( pi (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) )
10		pidiv2halves 15300	. . . . . . . . 9 |- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
11	10	breq1i 4052	. . . . . . . 8 |- ( ( ( pi / 2 ) + ( pi / 2 ) ) < A <-> pi < A )
12		ltaddsub 8511	. . . . . . . . 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 1340	. . . . . . . 8 |- ( A e. RR -> ( ( ( pi / 2 ) + ( pi / 2 ) ) < A <-> ( pi / 2 ) < ( A - ( pi / 2 ) ) ) )
14	11, 13	bitr3id 194	. . . . . . 7 |- ( A e. RR -> ( pi < A <-> ( pi / 2 ) < ( A - ( pi / 2 ) ) ) )
15		ltsubadd 8507	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR /\ pi e. RR ) -> ( ( A - ( pi / 2 ) ) < pi <-> A < ( pi + ( pi / 2 ) ) ) )
16	3, 1, 15	mp3an23 1342	. . . . . . . 8 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < pi <-> A < ( pi + ( pi / 2 ) ) ) )
17		df-3 9098	. . . . . . . . . . 11 |- 3 = ( 2 + 1 )
18	17	oveq1i 5956	. . . . . . . . . 10 |- ( 3 x. ( pi / 2 ) ) = ( ( 2 + 1 ) x. ( pi / 2 ) )
19		2cn 9109	. . . . . . . . . . 11 |- 2 e. CC
20		ax-1cn 8020	. . . . . . . . . . 11 |- 1 e. CC
21	3	recni 8086	. . . . . . . . . . 11 |- ( pi / 2 ) e. CC
22	19, 20, 21	adddiri 8085	. . . . . . . . . 10 |- ( ( 2 + 1 ) x. ( pi / 2 ) ) = ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) )
23	1	recni 8086	. . . . . . . . . . . 12 |- pi e. CC
24		2ap0 9131	. . . . . . . . . . . 12 |- 2 # 0
25	23, 19, 24	divcanap2i 8830	. . . . . . . . . . 11 |- ( 2 x. ( pi / 2 ) ) = pi
26	21	mullidi 8077	. . . . . . . . . . 11 |- ( 1 x. ( pi / 2 ) ) = ( pi / 2 )
27	25, 26	oveq12i 5958	. . . . . . . . . 10 |- ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) ) = ( pi + ( pi / 2 ) )
28	18, 22, 27	3eqtrri 2231	. . . . . . . . 9 |- ( pi + ( pi / 2 ) ) = ( 3 x. ( pi / 2 ) )
29	28	breq2i 4053	. . . . . . . 8 |- ( A < ( pi + ( pi / 2 ) ) <-> A < ( 3 x. ( pi / 2 ) ) )
30	16, 29	bitr2di 197	. . . . . . 7 |- ( A e. RR -> ( A < ( 3 x. ( pi / 2 ) ) <-> ( A - ( pi / 2 ) ) < pi ) )
31	14, 30	anbi12d 473	. . . . . 6 |- ( A e. RR -> ( ( pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) <-> ( ( pi / 2 ) < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < pi ) ) )
32		resubcl 8338	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A - ( pi / 2 ) ) e. RR )
33	3, 32	mpan2 425	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. RR )
34		sincosq2sgn 15332	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( ( pi / 2 ) (,) pi ) -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
35		rexr 8120	. . . . . . . . . . 11 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
36		elioo2 10045	. . . . . . . . . . 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 289	. . . . . . . . . 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 426	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( ( pi / 2 ) (,) pi ) <-> ( ( A - ( pi / 2 ) ) e. RR /\ ( pi / 2 ) < ( A - ( pi / 2 ) ) /\ ( A - ( pi / 2 ) ) < pi ) )
39		ancom 266	. . . . . . . . 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 200	. . . . . . . 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 1283	. . . . . . 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 1209	. . . . . 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 150	. . . . 5 |- ( A e. RR -> ( ( pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( pi / 2 ) ) ) ) ) )
44	33	resincld 12067	. . . . . . 7 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
45	44	lt0neg2d 8591	. . . . . 6 |- ( A e. RR -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
46	45	anbi2d 464	. . . . 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 149	. . . 4 |- ( A e. RR -> ( ( pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` ( A - ( pi / 2 ) ) ) < 0 /\ -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) ) )
48		recn 8060	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
49		pncan3 8282	. . . . . . . . 9 |- ( ( ( pi / 2 ) e. CC /\ A e. CC ) -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
50	21, 48, 49	sylancr 414	. . . . . . . 8 |- ( A e. RR -> ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) = A )
51	50	fveq2d 5582	. . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
52	33	recnd 8103	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
53		sinhalfpip 15325	. . . . . . . 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 2240	. . . . . 6 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
56	55	breq1d 4055	. . . . 5 |- ( A e. RR -> ( ( sin ` A ) < 0 <-> ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
57	50	fveq2d 5582	. . . . . . 7 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
58		coshalfpip 15327	. . . . . . . 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 2240	. . . . . 6 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
61	60	breq1d 4055	. . . . 5 |- ( A e. RR -> ( ( cos ` A ) < 0 <-> -u ( sin ` ( A - ( pi / 2 ) ) ) < 0 ) )
62	56, 61	anbi12d 473	. . . 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 169	. . 3 |- ( A e. RR -> ( ( pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) ) )
64	63	3impib 1204	. 2 |- ( ( A e. RR /\ pi < A /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) )
65	9, 64	sylbi 121	1 |- ( A e. ( pi (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   CCcc 7925   RRcr 7926   0cc0 7927   1c1 7928   + caddc 7930   x. cmul 7932   RR*cxr 8108   < clt 8109   - cmin 8245   -ucneg 8246   / cdiv 8747   2c2 9089   3c3 9090   (,)cioo 10012   sincsin 11988   cosccos 11989   picpi 11991

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047  ax-pre-suploc 8048  ax-addf 8049  ax-mulf 8050

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-disj 4022  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-of 6160  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-map 6739  df-pm 6740  df-en 6830  df-dom 6831  df-fin 6832  df-sup 7088  df-inf 7089  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-5 9100  df-6 9101  df-7 9102  df-8 9103  df-9 9104  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-xneg 9896  df-xadd 9897  df-ioo 10016  df-ioc 10017  df-ico 10018  df-icc 10019  df-fz 10133  df-fzo 10267  df-seqfrec 10595  df-exp 10686  df-fac 10873  df-bc 10895  df-ihash 10923  df-shft 11159  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-sumdc 11698  df-ef 11992  df-sin 11994  df-cos 11995  df-pi 11997  df-rest 13106  df-topgen 13125  df-psmet 14338  df-xmet 14339  df-met 14340  df-bl 14341  df-mopn 14342  df-top 14503  df-topon 14516  df-bases 14548  df-ntr 14601  df-cn 14693  df-cnp 14694  df-tx 14758  df-cncf 15076  df-limced 15161  df-dvap 15162

This theorem is referenced by:  sincosq4sgn  15334