Theorem sincosq3sgn 14252

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 14210	. . 3 |- pi e. RR
2		3re 8993	. . . 4 |- 3 e. RR
3		halfpire 14216	. . . 4 |- ( pi / 2 ) e. RR
4	2, 3	remulcli 7971	. . 3 |- ( 3 x. ( pi / 2 ) ) e. RR
5		rexr 8003	. . . 4 |- ( pi e. RR -> pi e. RR* )
6		rexr 8003	. . . 4 |- ( ( 3 x. ( pi / 2 ) ) e. RR -> ( 3 x. ( pi / 2 ) ) e. RR* )
7		elioo2 9921	. . . 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 14219	. . . . . . . . 9 |- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
11	10	breq1i 4011	. . . . . . . 8 |- ( ( ( pi / 2 ) + ( pi / 2 ) ) < A <-> pi < A )
12		ltaddsub 8393	. . . . . . . . 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 1327	. . . . . . . 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 8389	. . . . . . . . 9 |- ( ( A e. RR /\ ( pi / 2 ) e. RR /\ pi e. RR ) -> ( ( A - ( pi / 2 ) ) < pi <-> A < ( pi + ( pi / 2 ) ) ) )
16	3, 1, 15	mp3an23 1329	. . . . . . . 8 |- ( A e. RR -> ( ( A - ( pi / 2 ) ) < pi <-> A < ( pi + ( pi / 2 ) ) ) )
17		df-3 8979	. . . . . . . . . . 11 |- 3 = ( 2 + 1 )
18	17	oveq1i 5885	. . . . . . . . . 10 |- ( 3 x. ( pi / 2 ) ) = ( ( 2 + 1 ) x. ( pi / 2 ) )
19		2cn 8990	. . . . . . . . . . 11 |- 2 e. CC
20		ax-1cn 7904	. . . . . . . . . . 11 |- 1 e. CC
21	3	recni 7969	. . . . . . . . . . 11 |- ( pi / 2 ) e. CC
22	19, 20, 21	adddiri 7968	. . . . . . . . . 10 |- ( ( 2 + 1 ) x. ( pi / 2 ) ) = ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) )
23	1	recni 7969	. . . . . . . . . . . 12 |- pi e. CC
24		2ap0 9012	. . . . . . . . . . . 12 |- 2 # 0
25	23, 19, 24	divcanap2i 8712	. . . . . . . . . . 11 |- ( 2 x. ( pi / 2 ) ) = pi
26	21	mullidi 7960	. . . . . . . . . . 11 |- ( 1 x. ( pi / 2 ) ) = ( pi / 2 )
27	25, 26	oveq12i 5887	. . . . . . . . . 10 |- ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) ) = ( pi + ( pi / 2 ) )
28	18, 22, 27	3eqtrri 2203	. . . . . . . . 9 |- ( pi + ( pi / 2 ) ) = ( 3 x. ( pi / 2 ) )
29	28	breq2i 4012	. . . . . . . 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 8221	. . . . . . . . 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 14251	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( ( pi / 2 ) (,) pi ) -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
35		rexr 8003	. . . . . . . . . . 11 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
36		elioo2 9921	. . . . . . . . . . 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 1271	. . . . . . 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 1206	. . . . . 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 11731	. . . . . . 7 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
45	44	lt0neg2d 8473	. . . . . 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 7944	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
49		pncan3 8165	. . . . . . . . 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 5520	. . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
52	33	recnd 7986	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
53		sinhalfpip 14244	. . . . . . . 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 2212	. . . . . 6 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
56	55	breq1d 4014	. . . . 5 |- ( A e. RR -> ( ( sin ` A ) < 0 <-> ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
57	50	fveq2d 5520	. . . . . . 7 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
58		coshalfpip 14246	. . . . . . . 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 2212	. . . . . 6 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
61	60	breq1d 4014	. . . . 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 1201	. 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 978   = wceq 1353   e. wcel 2148   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   CCcc 7809   RRcr 7810   0cc0 7811   1c1 7812   + caddc 7814   x. cmul 7816   RR*cxr 7991   < clt 7992   - cmin 8128   -ucneg 8129   / cdiv 8629   2c2 8970   3c3 8971   (,)cioo 9888   sincsin 11652   cosccos 11653   picpi 11655

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931  ax-pre-suploc 7932  ax-addf 7933  ax-mulf 7934

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-disj 3982  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-of 6083  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-frec 6392  df-1o 6417  df-oadd 6421  df-er 6535  df-map 6650  df-pm 6651  df-en 6741  df-dom 6742  df-fin 6743  df-sup 6983  df-inf 6984  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-5 8981  df-6 8982  df-7 8983  df-8 8984  df-9 8985  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-xneg 9772  df-xadd 9773  df-ioo 9892  df-ioc 9893  df-ico 9894  df-icc 9895  df-fz 10009  df-fzo 10143  df-seqfrec 10446  df-exp 10520  df-fac 10706  df-bc 10728  df-ihash 10756  df-shft 10824  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-clim 11287  df-sumdc 11362  df-ef 11656  df-sin 11658  df-cos 11659  df-pi 11661  df-rest 12690  df-topgen 12709  df-psmet 13450  df-xmet 13451  df-met 13452  df-bl 13453  df-mopn 13454  df-top 13501  df-topon 13514  df-bases 13546  df-ntr 13599  df-cn 13691  df-cnp 13692  df-tx 13756  df-cncf 14061  df-limced 14128  df-dvap 14129

This theorem is referenced by:  sincosq4sgn  14253