Theorem sincosq3sgn 15415

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 15373	. . 3 |- pi e. RR
2		3re 9145	. . . 4 |- 3 e. RR
3		halfpire 15379	. . . 4 |- ( pi / 2 ) e. RR
4	2, 3	remulcli 8121	. . 3 |- ( 3 x. ( pi / 2 ) ) e. RR
5		rexr 8153	. . . 4 |- ( pi e. RR -> pi e. RR* )
6		rexr 8153	. . . 4 |- ( ( 3 x. ( pi / 2 ) ) e. RR -> ( 3 x. ( pi / 2 ) ) e. RR* )
7		elioo2 10078	. . . 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 15382	. . . . . . . . 9 |- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
11	10	breq1i 4066	. . . . . . . 8 |- ( ( ( pi / 2 ) + ( pi / 2 ) ) < A <-> pi < A )
12		ltaddsub 8544	. . . . . . . . 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 8540	. . . . . . . . 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 9131	. . . . . . . . . . 11 |- 3 = ( 2 + 1 )
18	17	oveq1i 5977	. . . . . . . . . 10 |- ( 3 x. ( pi / 2 ) ) = ( ( 2 + 1 ) x. ( pi / 2 ) )
19		2cn 9142	. . . . . . . . . . 11 |- 2 e. CC
20		ax-1cn 8053	. . . . . . . . . . 11 |- 1 e. CC
21	3	recni 8119	. . . . . . . . . . 11 |- ( pi / 2 ) e. CC
22	19, 20, 21	adddiri 8118	. . . . . . . . . 10 |- ( ( 2 + 1 ) x. ( pi / 2 ) ) = ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) )
23	1	recni 8119	. . . . . . . . . . . 12 |- pi e. CC
24		2ap0 9164	. . . . . . . . . . . 12 |- 2 # 0
25	23, 19, 24	divcanap2i 8863	. . . . . . . . . . 11 |- ( 2 x. ( pi / 2 ) ) = pi
26	21	mullidi 8110	. . . . . . . . . . 11 |- ( 1 x. ( pi / 2 ) ) = ( pi / 2 )
27	25, 26	oveq12i 5979	. . . . . . . . . 10 |- ( ( 2 x. ( pi / 2 ) ) + ( 1 x. ( pi / 2 ) ) ) = ( pi + ( pi / 2 ) )
28	18, 22, 27	3eqtrri 2233	. . . . . . . . 9 |- ( pi + ( pi / 2 ) ) = ( 3 x. ( pi / 2 ) )
29	28	breq2i 4067	. . . . . . . 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 8371	. . . . . . . . 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 15414	. . . . . . . . 9 |- ( ( A - ( pi / 2 ) ) e. ( ( pi / 2 ) (,) pi ) -> ( 0 < ( sin ` ( A - ( pi / 2 ) ) ) /\ ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
35		rexr 8153	. . . . . . . . . . 11 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
36		elioo2 10078	. . . . . . . . . . 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 12149	. . . . . . 7 |- ( A e. RR -> ( sin ` ( A - ( pi / 2 ) ) ) e. RR )
45	44	lt0neg2d 8624	. . . . . 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 8093	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
49		pncan3 8315	. . . . . . . . 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 5603	. . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( sin ` A ) )
52	33	recnd 8136	. . . . . . . 8 |- ( A e. RR -> ( A - ( pi / 2 ) ) e. CC )
53		sinhalfpip 15407	. . . . . . . 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 2242	. . . . . 6 |- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( pi / 2 ) ) ) )
56	55	breq1d 4069	. . . . 5 |- ( A e. RR -> ( ( sin ` A ) < 0 <-> ( cos ` ( A - ( pi / 2 ) ) ) < 0 ) )
57	50	fveq2d 5603	. . . . . . 7 |- ( A e. RR -> ( cos ` ( ( pi / 2 ) + ( A - ( pi / 2 ) ) ) ) = ( cos ` A ) )
58		coshalfpip 15409	. . . . . . . 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 2242	. . . . . 6 |- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( pi / 2 ) ) ) )
61	60	breq1d 4069	. . . . 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 2178   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   RR*cxr 8141   < clt 8142   - cmin 8278   -ucneg 8279   / cdiv 8780   2c2 9122   3c3 9123   (,)cioo 10045   sincsin 12070   cosccos 12071   picpi 12073

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080  ax-pre-suploc 8081  ax-addf 8082  ax-mulf 8083

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-disj 4036  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-oadd 6529  df-er 6643  df-map 6760  df-pm 6761  df-en 6851  df-dom 6852  df-fin 6853  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-9 9137  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-xneg 9929  df-xadd 9930  df-ioo 10049  df-ioc 10050  df-ico 10051  df-icc 10052  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-exp 10721  df-fac 10908  df-bc 10930  df-ihash 10958  df-shft 11241  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-sumdc 11780  df-ef 12074  df-sin 12076  df-cos 12077  df-pi 12079  df-rest 13188  df-topgen 13207  df-psmet 14420  df-xmet 14421  df-met 14422  df-bl 14423  df-mopn 14424  df-top 14585  df-topon 14598  df-bases 14630  df-ntr 14683  df-cn 14775  df-cnp 14776  df-tx 14840  df-cncf 15158  df-limced 15243  df-dvap 15244

This theorem is referenced by:  sincosq4sgn  15416