ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sincosq4sgn Unicode version

Theorem sincosq4sgn 12932
Description: The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
Assertion
Ref Expression
sincosq4sgn  |-  ( A  e.  ( ( 3  x.  ( pi  / 
2 ) ) (,) ( 2  x.  pi ) )  ->  (
( sin `  A
)  <  0  /\  0  <  ( cos `  A
) ) )

Proof of Theorem sincosq4sgn
StepHypRef Expression
1 3re 8806 . . . . 5  |-  3  e.  RR
2 halfpire 12895 . . . . 5  |-  ( pi 
/  2 )  e.  RR
31, 2remulcli 7792 . . . 4  |-  ( 3  x.  ( pi  / 
2 ) )  e.  RR
43rexri 7835 . . 3  |-  ( 3  x.  ( pi  / 
2 ) )  e. 
RR*
5 2re 8802 . . . . 5  |-  2  e.  RR
6 pire 12889 . . . . 5  |-  pi  e.  RR
75, 6remulcli 7792 . . . 4  |-  ( 2  x.  pi )  e.  RR
87rexri 7835 . . 3  |-  ( 2  x.  pi )  e. 
RR*
9 elioo2 9716 . . 3  |-  ( ( ( 3  x.  (
pi  /  2 ) )  e.  RR*  /\  (
2  x.  pi )  e.  RR* )  ->  ( A  e.  ( (
3  x.  ( pi 
/  2 ) ) (,) ( 2  x.  pi ) )  <->  ( A  e.  RR  /\  ( 3  x.  ( pi  / 
2 ) )  < 
A  /\  A  <  ( 2  x.  pi ) ) ) )
104, 8, 9mp2an 422 . 2  |-  ( A  e.  ( ( 3  x.  ( pi  / 
2 ) ) (,) ( 2  x.  pi ) )  <->  ( A  e.  RR  /\  ( 3  x.  ( pi  / 
2 ) )  < 
A  /\  A  <  ( 2  x.  pi ) ) )
11 df-3 8792 . . . . . . . . . . . 12  |-  3  =  ( 2  +  1 )
1211oveq1i 5784 . . . . . . . . . . 11  |-  ( 3  x.  ( pi  / 
2 ) )  =  ( ( 2  +  1 )  x.  (
pi  /  2 ) )
13 2cn 8803 . . . . . . . . . . . 12  |-  2  e.  CC
14 ax-1cn 7725 . . . . . . . . . . . 12  |-  1  e.  CC
152recni 7790 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  CC
1613, 14, 15adddiri 7789 . . . . . . . . . . 11  |-  ( ( 2  +  1 )  x.  ( pi  / 
2 ) )  =  ( ( 2  x.  ( pi  /  2
) )  +  ( 1  x.  ( pi 
/  2 ) ) )
176recni 7790 . . . . . . . . . . . . 13  |-  pi  e.  CC
18 2ap0 8825 . . . . . . . . . . . . 13  |-  2 #  0
1917, 13, 18divcanap2i 8527 . . . . . . . . . . . 12  |-  ( 2  x.  ( pi  / 
2 ) )  =  pi
2015mulid2i 7781 . . . . . . . . . . . 12  |-  ( 1  x.  ( pi  / 
2 ) )  =  ( pi  /  2
)
2119, 20oveq12i 5786 . . . . . . . . . . 11  |-  ( ( 2  x.  ( pi 
/  2 ) )  +  ( 1  x.  ( pi  /  2
) ) )  =  ( pi  +  ( pi  /  2 ) )
2212, 16, 213eqtrri 2165 . . . . . . . . . 10  |-  ( pi  +  ( pi  / 
2 ) )  =  ( 3  x.  (
pi  /  2 ) )
2322breq1i 3936 . . . . . . . . 9  |-  ( ( pi  +  ( pi 
/  2 ) )  <  A  <->  ( 3  x.  ( pi  / 
2 ) )  < 
A )
24 ltaddsub 8210 . . . . . . . . . 10  |-  ( ( pi  e.  RR  /\  ( pi  /  2
)  e.  RR  /\  A  e.  RR )  ->  ( ( pi  +  ( pi  /  2
) )  <  A  <->  pi 
<  ( A  -  ( pi  /  2
) ) ) )
256, 2, 24mp3an12 1305 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( pi  +  ( pi  /  2 ) )  <  A  <->  pi  <  ( A  -  ( pi 
/  2 ) ) ) )
2623, 25bitr3id 193 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( 3  x.  (
pi  /  2 ) )  <  A  <->  pi  <  ( A  -  ( pi 
/  2 ) ) ) )
27 ltsubadd 8206 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  e.  RR  /\  ( 3  x.  (
pi  /  2 ) )  e.  RR )  ->  ( ( A  -  ( pi  / 
2 ) )  < 
( 3  x.  (
pi  /  2 ) )  <->  A  <  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) ) ) )
282, 3, 27mp3an23 1307 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( A  -  (
pi  /  2 ) )  <  ( 3  x.  ( pi  / 
2 ) )  <->  A  <  ( ( 3  x.  (
pi  /  2 ) )  +  ( pi 
/  2 ) ) ) )
29 df-4 8793 . . . . . . . . . . . . 13  |-  4  =  ( 3  +  1 )
3029oveq1i 5784 . . . . . . . . . . . 12  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( ( 3  +  1 )  x.  (
pi  /  2 ) )
311recni 7790 . . . . . . . . . . . . 13  |-  3  e.  CC
3231, 14, 15adddiri 7789 . . . . . . . . . . . 12  |-  ( ( 3  +  1 )  x.  ( pi  / 
2 ) )  =  ( ( 3  x.  ( pi  /  2
) )  +  ( 1  x.  ( pi 
/  2 ) ) )
3320oveq2i 5785 . . . . . . . . . . . 12  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( 1  x.  ( pi  /  2
) ) )  =  ( ( 3  x.  ( pi  /  2
) )  +  ( pi  /  2 ) )
3430, 32, 333eqtrri 2165 . . . . . . . . . . 11  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) )  =  ( 4  x.  (
pi  /  2 ) )
35 4cn 8810 . . . . . . . . . . . . 13  |-  4  e.  CC
3613, 18pm3.2i 270 . . . . . . . . . . . . 13  |-  ( 2  e.  CC  /\  2 #  0 )
37 div12ap 8466 . . . . . . . . . . . . 13  |-  ( ( 4  e.  CC  /\  pi  e.  CC  /\  (
2  e.  CC  /\  2 #  0 ) )  -> 
( 4  x.  (
pi  /  2 ) )  =  ( pi  x.  ( 4  / 
2 ) ) )
3835, 17, 36, 37mp3an 1315 . . . . . . . . . . . 12  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( pi  x.  (
4  /  2 ) )
39 4d2e2 8892 . . . . . . . . . . . . . 14  |-  ( 4  /  2 )  =  2
4039oveq2i 5785 . . . . . . . . . . . . 13  |-  ( pi  x.  ( 4  / 
2 ) )  =  ( pi  x.  2 )
4117, 13mulcomi 7784 . . . . . . . . . . . . 13  |-  ( pi  x.  2 )  =  ( 2  x.  pi )
4240, 41eqtri 2160 . . . . . . . . . . . 12  |-  ( pi  x.  ( 4  / 
2 ) )  =  ( 2  x.  pi )
4338, 42eqtri 2160 . . . . . . . . . . 11  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( 2  x.  pi )
4434, 43eqtri 2160 . . . . . . . . . 10  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) )  =  ( 2  x.  pi )
4544breq2i 3937 . . . . . . . . 9  |-  ( A  <  ( ( 3  x.  ( pi  / 
2 ) )  +  ( pi  /  2
) )  <->  A  <  ( 2  x.  pi ) )
4628, 45syl6rbb 196 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  ( 2  x.  pi )  <->  ( A  -  ( pi  / 
2 ) )  < 
( 3  x.  (
pi  /  2 ) ) ) )
4726, 46anbi12d 464 . . . . . . 7  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  <-> 
( pi  <  ( A  -  ( pi  /  2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  ( 3  x.  ( pi  / 
2 ) ) ) ) )
48 resubcl 8038 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  e.  RR )  ->  ( A  -  ( pi  /  2
) )  e.  RR )
492, 48mpan2 421 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  RR )
506rexri 7835 . . . . . . . . . . 11  |-  pi  e.  RR*
51 elioo2 9716 . . . . . . . . . . 11  |-  ( ( pi  e.  RR*  /\  (
3  x.  ( pi 
/  2 ) )  e.  RR* )  ->  (
( A  -  (
pi  /  2 ) )  e.  ( pi
(,) ( 3  x.  ( pi  /  2
) ) )  <->  ( ( A  -  ( pi  /  2 ) )  e.  RR  /\  pi  <  ( A  -  ( pi 
/  2 ) )  /\  ( A  -  ( pi  /  2
) )  <  (
3  x.  ( pi 
/  2 ) ) ) ) )
5250, 4, 51mp2an 422 . . . . . . . . . 10  |-  ( ( A  -  ( pi 
/  2 ) )  e.  ( pi (,) ( 3  x.  (
pi  /  2 ) ) )  <->  ( ( A  -  ( pi  /  2 ) )  e.  RR  /\  pi  <  ( A  -  ( pi 
/  2 ) )  /\  ( A  -  ( pi  /  2
) )  <  (
3  x.  ( pi 
/  2 ) ) ) )
53 sincosq3sgn 12931 . . . . . . . . . 10  |-  ( ( A  -  ( pi 
/  2 ) )  e.  ( pi (,) ( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 ) )
5452, 53sylbir 134 . . . . . . . . 9  |-  ( ( ( A  -  (
pi  /  2 ) )  e.  RR  /\  pi  <  ( A  -  ( pi  /  2
) )  /\  ( A  -  ( pi  /  2 ) )  < 
( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 ) )
5549, 54syl3an1 1249 . . . . . . . 8  |-  ( ( A  e.  RR  /\  pi  <  ( A  -  ( pi  /  2
) )  /\  ( A  -  ( pi  /  2 ) )  < 
( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 ) )
56553expib 1184 . . . . . . 7  |-  ( A  e.  RR  ->  (
( pi  <  ( A  -  ( pi  /  2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  ( 3  x.  ( pi  / 
2 ) ) )  ->  ( ( sin `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
5747, 56sylbid 149 . . . . . 6  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  ->  ( ( sin `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
5849resincld 11441 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( A  -  ( pi  /  2
) ) )  e.  RR )
5958lt0neg1d 8289 . . . . . . 7  |-  ( A  e.  RR  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  <->  0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) ) ) )
6059anbi1d 460 . . . . . 6  |-  ( A  e.  RR  ->  (
( ( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 )  <->  ( 0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
6157, 60sylibd 148 . . . . 5  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  ->  ( 0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
62 recn 7765 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
63 pncan3 7982 . . . . . . . . . 10  |-  ( ( ( pi  /  2
)  e.  CC  /\  A  e.  CC )  ->  ( ( pi  / 
2 )  +  ( A  -  ( pi 
/  2 ) ) )  =  A )
6415, 62, 63sylancr 410 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( pi  /  2
)  +  ( A  -  ( pi  / 
2 ) ) )  =  A )
6564fveq2d 5425 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  A
) )
6649recnd 7806 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  CC )
67 coshalfpip 12925 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
6866, 67syl 14 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
6965, 68eqtr3d 2174 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  A )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
7069breq2d 3941 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  ( cos `  A )  <->  0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) ) ) )
7164fveq2d 5425 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( sin `  A
) )
72 sinhalfpip 12923 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7366, 72syl 14 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7471, 73eqtr3d 2174 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7574breq1d 3939 . . . . . 6  |-  ( A  e.  RR  ->  (
( sin `  A
)  <  0  <->  ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0
) )
7670, 75anbi12d 464 . . . . 5  |-  ( A  e.  RR  ->  (
( 0  <  ( cos `  A )  /\  ( sin `  A )  <  0 )  <->  ( 0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
7761, 76sylibrd 168 . . . 4  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  ->  ( 0  < 
( cos `  A
)  /\  ( sin `  A )  <  0
) ) )
78773impib 1179 . . 3  |-  ( ( A  e.  RR  /\  ( 3  x.  (
pi  /  2 ) )  <  A  /\  A  <  ( 2  x.  pi ) )  -> 
( 0  <  ( cos `  A )  /\  ( sin `  A )  <  0 ) )
7978ancomd 265 . 2  |-  ( ( A  e.  RR  /\  ( 3  x.  (
pi  /  2 ) )  <  A  /\  A  <  ( 2  x.  pi ) )  -> 
( ( sin `  A
)  <  0  /\  0  <  ( cos `  A
) ) )
8010, 79sylbi 120 1  |-  ( A  e.  ( ( 3  x.  ( pi  / 
2 ) ) (,) ( 2  x.  pi ) )  ->  (
( sin `  A
)  <  0  /\  0  <  ( cos `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7630   RRcr 7631   0cc0 7632   1c1 7633    + caddc 7635    x. cmul 7637   RR*cxr 7811    < clt 7812    - cmin 7945   -ucneg 7946   # cap 8355    / cdiv 8444   2c2 8783   3c3 8784   4c4 8785   (,)cioo 9683   sincsin 11362   cosccos 11363   picpi 11365
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752  ax-pre-suploc 7753  ax-addf 7754  ax-mulf 7755
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-map 6544  df-pm 6545  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-inf 6872  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-5 8794  df-6 8795  df-7 8796  df-8 8797  df-9 8798  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-xneg 9571  df-xadd 9572  df-ioo 9687  df-ioc 9688  df-ico 9689  df-icc 9690  df-fz 9803  df-fzo 9932  df-seqfrec 10231  df-exp 10305  df-fac 10484  df-bc 10506  df-ihash 10534  df-shft 10599  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-clim 11060  df-sumdc 11135  df-ef 11366  df-sin 11368  df-cos 11369  df-pi 11371  df-rest 12136  df-topgen 12155  df-psmet 12170  df-xmet 12171  df-met 12172  df-bl 12173  df-mopn 12174  df-top 12179  df-topon 12192  df-bases 12224  df-ntr 12279  df-cn 12371  df-cnp 12372  df-tx 12436  df-cncf 12741  df-limced 12808  df-dvap 12809
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator