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Theorem sincosq2sgn 12949
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
StepHypRef Expression
1 halfpire 12914 . . 3  |-  ( pi 
/  2 )  e.  RR
2 pire 12908 . . 3  |-  pi  e.  RR
3 rexr 7833 . . . 4  |-  ( ( pi  /  2 )  e.  RR  ->  (
pi  /  2 )  e.  RR* )
4 rexr 7833 . . . 4  |-  ( pi  e.  RR  ->  pi  e.  RR* )
5 elioo2 9732 . . . 4  |-  ( ( ( pi  /  2
)  e.  RR*  /\  pi  e.  RR* )  ->  ( A  e.  ( (
pi  /  2 ) (,) pi )  <->  ( A  e.  RR  /\  ( pi 
/  2 )  < 
A  /\  A  <  pi ) ) )
63, 4, 5syl2an 287 . . 3  |-  ( ( ( pi  /  2
)  e.  RR  /\  pi  e.  RR )  -> 
( A  e.  ( ( pi  /  2
) (,) pi )  <-> 
( A  e.  RR  /\  ( pi  /  2
)  <  A  /\  A  <  pi ) ) )
71, 2, 6mp2an 423 . 2  |-  ( A  e.  ( ( pi 
/  2 ) (,) pi )  <->  ( A  e.  RR  /\  ( pi 
/  2 )  < 
A  /\  A  <  pi ) )
8 resubcl 8048 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  e.  RR )  ->  ( A  -  ( pi  /  2
) )  e.  RR )
91, 8mpan2 422 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  RR )
10 0xr 7834 . . . . . . . . . 10  |-  0  e.  RR*
111rexri 7845 . . . . . . . . . 10  |-  ( pi 
/  2 )  e. 
RR*
12 elioo2 9732 . . . . . . . . . 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
) ) ) )
1310, 11, 12mp2an 423 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  ( 0 (,) ( pi  /  2
) )  <->  ( ( A  -  ( pi  /  2 ) )  e.  RR  /\  0  < 
( A  -  (
pi  /  2 ) )  /\  ( A  -  ( pi  / 
2 ) )  < 
( pi  /  2
) ) )
14 sincosq1sgn 12948 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  ( 0 (,) ( pi  /  2
) )  ->  (
0  <  ( sin `  ( A  -  (
pi  /  2 ) ) )  /\  0  <  ( cos `  ( A  -  ( pi  /  2 ) ) ) ) )
1513, 14sylbir 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 ) ) ) ) )
169, 15syl3an1 1250 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  ( A  -  ( pi  /  2
) )  /\  ( A  -  ( pi  /  2 ) )  < 
( pi  /  2
) )  ->  (
0  <  ( sin `  ( A  -  (
pi  /  2 ) ) )  /\  0  <  ( cos `  ( A  -  ( pi  /  2 ) ) ) ) )
17163expib 1185 . . . . . 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 7788 . . . . . . . . 9  |-  0  e.  RR
19 ltsub13 8227 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  A  e.  RR  /\  (
pi  /  2 )  e.  RR )  -> 
( 0  <  ( A  -  ( pi  /  2 ) )  <->  ( pi  /  2 )  <  ( A  -  0 ) ) )
2018, 1, 19mp3an13 1307 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  ( A  -  ( pi  / 
2 ) )  <->  ( pi  /  2 )  <  ( A  -  0 ) ) )
21 recn 7775 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
2221subid1d 8084 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  0 )  =  A )
2322breq2d 3947 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( pi  /  2
)  <  ( A  -  0 )  <->  ( pi  /  2 )  <  A
) )
2420, 23bitrd 187 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <  ( A  -  ( pi  / 
2 ) )  <->  ( pi  /  2 )  <  A
) )
25 ltsubadd 8216 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  e.  RR  /\  ( pi  /  2
)  e.  RR )  ->  ( ( A  -  ( pi  / 
2 ) )  < 
( pi  /  2
)  <->  A  <  ( ( pi  /  2 )  +  ( pi  / 
2 ) ) ) )
261, 1, 25mp3an23 1308 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( A  -  (
pi  /  2 ) )  <  ( pi 
/  2 )  <->  A  <  ( ( pi  /  2
)  +  ( pi 
/  2 ) ) ) )
27 pidiv2halves 12917 . . . . . . . . 9  |-  ( ( pi  /  2 )  +  ( pi  / 
2 ) )  =  pi
2827breq2i 3943 . . . . . . . 8  |-  ( A  <  ( ( pi 
/  2 )  +  ( pi  /  2
) )  <->  A  <  pi )
2926, 28syl6bb 195 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A  -  (
pi  /  2 ) )  <  ( pi 
/  2 )  <->  A  <  pi ) )
3024, 29anbi12d 465 . . . . . 6  |-  ( A  e.  RR  ->  (
( 0  <  ( A  -  ( pi  /  2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  ( pi 
/  2 ) )  <-> 
( ( pi  / 
2 )  <  A  /\  A  <  pi ) ) )
319resincld 11459 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( A  -  ( pi  /  2
) ) )  e.  RR )
3231lt0neg2d 8300 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <  ( sin `  ( A  -  (
pi  /  2 ) ) )  <->  -u ( sin `  ( A  -  (
pi  /  2 ) ) )  <  0
) )
3332anbi1d 461 . . . . . 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 ) ) ) ) ) )
3417, 30, 333imtr3d 201 . . . . 5  |-  ( A  e.  RR  ->  (
( ( pi  / 
2 )  <  A  /\  A  <  pi )  ->  ( -u ( sin `  ( A  -  ( pi  /  2
) ) )  <  0  /\  0  < 
( cos `  ( A  -  ( pi  /  2 ) ) ) ) ) )
351recni 7800 . . . . . . . . . 10  |-  ( pi 
/  2 )  e.  CC
36 pncan3 7992 . . . . . . . . . 10  |-  ( ( ( pi  /  2
)  e.  CC  /\  A  e.  CC )  ->  ( ( pi  / 
2 )  +  ( A  -  ( pi 
/  2 ) ) )  =  A )
3735, 21, 36sylancr 411 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( pi  /  2
)  +  ( A  -  ( pi  / 
2 ) ) )  =  A )
3837fveq2d 5431 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  A
) )
399recnd 7816 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  CC )
40 coshalfpip 12944 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
4139, 40syl 14 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
4238, 41eqtr3d 2175 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  A )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
4342breq1d 3945 . . . . . 6  |-  ( A  e.  RR  ->  (
( cos `  A
)  <  0  <->  -u ( sin `  ( A  -  (
pi  /  2 ) ) )  <  0
) )
4437fveq2d 5431 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( sin `  A
) )
45 sinhalfpip 12942 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
4639, 45syl 14 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
4744, 46eqtr3d 2175 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
4847breq2d 3947 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  ( sin `  A )  <->  0  <  ( cos `  ( A  -  ( pi  / 
2 ) ) ) ) )
4943, 48anbi12d 465 . . . . 5  |-  ( A  e.  RR  ->  (
( ( cos `  A
)  <  0  /\  0  <  ( sin `  A
) )  <->  ( -u ( sin `  ( A  -  ( pi  /  2
) ) )  <  0  /\  0  < 
( cos `  ( A  -  ( pi  /  2 ) ) ) ) ) )
5034, 49sylibrd 168 . . . 4  |-  ( A  e.  RR  ->  (
( ( pi  / 
2 )  <  A  /\  A  <  pi )  ->  ( ( cos `  A )  <  0  /\  0  <  ( sin `  A ) ) ) )
51503impib 1180 . . 3  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  <  A  /\  A  <  pi )  -> 
( ( cos `  A
)  <  0  /\  0  <  ( sin `  A
) ) )
5251ancomd 265 . 2  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  <  A  /\  A  <  pi )  -> 
( 0  <  ( sin `  A )  /\  ( cos `  A )  <  0 ) )
537, 52sylbi 120 1  |-  ( A  e.  ( ( pi 
/  2 ) (,) pi )  ->  (
0  <  ( sin `  A )  /\  ( cos `  A )  <  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   class class class wbr 3935   ` cfv 5129  (class class class)co 5780   CCcc 7640   RRcr 7641   0cc0 7642    + caddc 7645   RR*cxr 7821    < clt 7822    - cmin 7955   -ucneg 7956    / cdiv 8454   2c2 8793   (,)cioo 9699   sincsin 11380   cosccos 11381   picpi 11383
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-iinf 4508  ax-cnex 7733  ax-resscn 7734  ax-1cn 7735  ax-1re 7736  ax-icn 7737  ax-addcl 7738  ax-addrcl 7739  ax-mulcl 7740  ax-mulrcl 7741  ax-addcom 7742  ax-mulcom 7743  ax-addass 7744  ax-mulass 7745  ax-distr 7746  ax-i2m1 7747  ax-0lt1 7748  ax-1rid 7749  ax-0id 7750  ax-rnegex 7751  ax-precex 7752  ax-cnre 7753  ax-pre-ltirr 7754  ax-pre-ltwlin 7755  ax-pre-lttrn 7756  ax-pre-apti 7757  ax-pre-ltadd 7758  ax-pre-mulgt0 7759  ax-pre-mulext 7760  ax-arch 7761  ax-caucvg 7762  ax-pre-suploc 7763  ax-addf 7764  ax-mulf 7765
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-if 3478  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-disj 3913  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-id 4221  df-po 4224  df-iso 4225  df-iord 4294  df-on 4296  df-ilim 4297  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-isom 5138  df-riota 5736  df-ov 5783  df-oprab 5784  df-mpo 5785  df-of 5988  df-1st 6044  df-2nd 6045  df-recs 6208  df-irdg 6273  df-frec 6294  df-1o 6319  df-oadd 6323  df-er 6435  df-map 6550  df-pm 6551  df-en 6641  df-dom 6642  df-fin 6643  df-sup 6877  df-inf 6878  df-pnf 7824  df-mnf 7825  df-xr 7826  df-ltxr 7827  df-le 7828  df-sub 7957  df-neg 7958  df-reap 8359  df-ap 8366  df-div 8455  df-inn 8743  df-2 8801  df-3 8802  df-4 8803  df-5 8804  df-6 8805  df-7 8806  df-8 8807  df-9 8808  df-n0 9000  df-z 9077  df-uz 9349  df-q 9437  df-rp 9469  df-xneg 9587  df-xadd 9588  df-ioo 9703  df-ioc 9704  df-ico 9705  df-icc 9706  df-fz 9820  df-fzo 9949  df-seqfrec 10248  df-exp 10322  df-fac 10502  df-bc 10524  df-ihash 10552  df-shft 10617  df-cj 10644  df-re 10645  df-im 10646  df-rsqrt 10800  df-abs 10801  df-clim 11078  df-sumdc 11153  df-ef 11384  df-sin 11386  df-cos 11387  df-pi 11389  df-rest 12154  df-topgen 12173  df-psmet 12188  df-xmet 12189  df-met 12190  df-bl 12191  df-mopn 12192  df-top 12197  df-topon 12210  df-bases 12242  df-ntr 12297  df-cn 12389  df-cnp 12390  df-tx 12454  df-cncf 12759  df-limced 12826  df-dvap 12827
This theorem is referenced by:  sincosq3sgn  12950
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