MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sincosq4sgn Unicode version

Theorem sincosq4sgn 19885
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 ressxr 8892 . . . 4  |-  RR  C_  RR*
2 3re 9833 . . . . 5  |-  3  e.  RR
3 pire 19848 . . . . . 6  |-  pi  e.  RR
4 2re 9831 . . . . . 6  |-  2  e.  RR
5 2ne0 9845 . . . . . 6  |-  2  =/=  0
63, 4, 5redivcli 9543 . . . . 5  |-  ( pi 
/  2 )  e.  RR
72, 6remulcli 8867 . . . 4  |-  ( 3  x.  ( pi  / 
2 ) )  e.  RR
81, 7sselii 3190 . . 3  |-  ( 3  x.  ( pi  / 
2 ) )  e. 
RR*
94, 3remulcli 8867 . . . 4  |-  ( 2  x.  pi )  e.  RR
101, 9sselii 3190 . . 3  |-  ( 2  x.  pi )  e. 
RR*
11 elioo2 10713 . . 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 ) ) ) )
128, 10, 11mp2an 653 . 2  |-  ( A  e.  ( ( 3  x.  ( pi  / 
2 ) ) (,) ( 2  x.  pi ) )  <->  ( A  e.  RR  /\  ( 3  x.  ( pi  / 
2 ) )  < 
A  /\  A  <  ( 2  x.  pi ) ) )
13 df-3 9821 . . . . . . . . . . . 12  |-  3  =  ( 2  +  1 )
1413oveq1i 5884 . . . . . . . . . . 11  |-  ( 3  x.  ( pi  / 
2 ) )  =  ( ( 2  +  1 )  x.  (
pi  /  2 ) )
15 2cn 9832 . . . . . . . . . . . 12  |-  2  e.  CC
16 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
176recni 8865 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  CC
1815, 16, 17adddiri 8864 . . . . . . . . . . 11  |-  ( ( 2  +  1 )  x.  ( pi  / 
2 ) )  =  ( ( 2  x.  ( pi  /  2
) )  +  ( 1  x.  ( pi 
/  2 ) ) )
193recni 8865 . . . . . . . . . . . . 13  |-  pi  e.  CC
2019, 15, 5divcan2i 9519 . . . . . . . . . . . 12  |-  ( 2  x.  ( pi  / 
2 ) )  =  pi
2117mulid2i 8856 . . . . . . . . . . . 12  |-  ( 1  x.  ( pi  / 
2 ) )  =  ( pi  /  2
)
2220, 21oveq12i 5886 . . . . . . . . . . 11  |-  ( ( 2  x.  ( pi 
/  2 ) )  +  ( 1  x.  ( pi  /  2
) ) )  =  ( pi  +  ( pi  /  2 ) )
2314, 18, 223eqtrri 2321 . . . . . . . . . 10  |-  ( pi  +  ( pi  / 
2 ) )  =  ( 3  x.  (
pi  /  2 ) )
2423breq1i 4046 . . . . . . . . 9  |-  ( ( pi  +  ( pi 
/  2 ) )  <  A  <->  ( 3  x.  ( pi  / 
2 ) )  < 
A )
25 ltaddsub 9264 . . . . . . . . . 10  |-  ( ( pi  e.  RR  /\  ( pi  /  2
)  e.  RR  /\  A  e.  RR )  ->  ( ( pi  +  ( pi  /  2
) )  <  A  <->  pi 
<  ( A  -  ( pi  /  2
) ) ) )
263, 6, 25mp3an12 1267 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( pi  +  ( pi  /  2 ) )  <  A  <->  pi  <  ( A  -  ( pi 
/  2 ) ) ) )
2724, 26syl5bbr 250 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( 3  x.  (
pi  /  2 ) )  <  A  <->  pi  <  ( A  -  ( pi 
/  2 ) ) ) )
28 ltsubadd 9260 . . . . . . . . . 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 ) ) ) )
296, 7, 28mp3an23 1269 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( A  -  (
pi  /  2 ) )  <  ( 3  x.  ( pi  / 
2 ) )  <->  A  <  ( ( 3  x.  (
pi  /  2 ) )  +  ( pi 
/  2 ) ) ) )
30 df-4 9822 . . . . . . . . . . . . 13  |-  4  =  ( 3  +  1 )
3130oveq1i 5884 . . . . . . . . . . . 12  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( ( 3  +  1 )  x.  (
pi  /  2 ) )
322recni 8865 . . . . . . . . . . . . 13  |-  3  e.  CC
3332, 16, 17adddiri 8864 . . . . . . . . . . . 12  |-  ( ( 3  +  1 )  x.  ( pi  / 
2 ) )  =  ( ( 3  x.  ( pi  /  2
) )  +  ( 1  x.  ( pi 
/  2 ) ) )
3421oveq2i 5885 . . . . . . . . . . . 12  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( 1  x.  ( pi  /  2
) ) )  =  ( ( 3  x.  ( pi  /  2
) )  +  ( pi  /  2 ) )
3531, 33, 343eqtrri 2321 . . . . . . . . . . 11  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) )  =  ( 4  x.  (
pi  /  2 ) )
36 4cn 9836 . . . . . . . . . . . . 13  |-  4  e.  CC
3715, 5pm3.2i 441 . . . . . . . . . . . . 13  |-  ( 2  e.  CC  /\  2  =/=  0 )
38 div12 9462 . . . . . . . . . . . . 13  |-  ( ( 4  e.  CC  /\  pi  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 ) )  ->  ( 4  x.  ( pi  /  2
) )  =  ( pi  x.  ( 4  /  2 ) ) )
3936, 19, 37, 38mp3an 1277 . . . . . . . . . . . 12  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( pi  x.  (
4  /  2 ) )
40 4d2e2 9892 . . . . . . . . . . . . . 14  |-  ( 4  /  2 )  =  2
4140oveq2i 5885 . . . . . . . . . . . . 13  |-  ( pi  x.  ( 4  / 
2 ) )  =  ( pi  x.  2 )
4219, 15mulcomi 8859 . . . . . . . . . . . . 13  |-  ( pi  x.  2 )  =  ( 2  x.  pi )
4341, 42eqtri 2316 . . . . . . . . . . . 12  |-  ( pi  x.  ( 4  / 
2 ) )  =  ( 2  x.  pi )
4439, 43eqtri 2316 . . . . . . . . . . 11  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( 2  x.  pi )
4535, 44eqtri 2316 . . . . . . . . . 10  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) )  =  ( 2  x.  pi )
4645breq2i 4047 . . . . . . . . 9  |-  ( A  <  ( ( 3  x.  ( pi  / 
2 ) )  +  ( pi  /  2
) )  <->  A  <  ( 2  x.  pi ) )
4729, 46syl6rbb 253 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  ( 2  x.  pi )  <->  ( A  -  ( pi  / 
2 ) )  < 
( 3  x.  (
pi  /  2 ) ) ) )
4827, 47anbi12d 691 . . . . . . 7  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  <-> 
( pi  <  ( A  -  ( pi  /  2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  ( 3  x.  ( pi  / 
2 ) ) ) ) )
49 resubcl 9127 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  e.  RR )  ->  ( A  -  ( pi  /  2
) )  e.  RR )
506, 49mpan2 652 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  RR )
511, 3sselii 3190 . . . . . . . . . . 11  |-  pi  e.  RR*
52 elioo2 10713 . . . . . . . . . . 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 ) ) ) ) )
5351, 8, 52mp2an 653 . . . . . . . . . 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 ) ) ) )
54 sincosq3sgn 19884 . . . . . . . . . 10  |-  ( ( A  -  ( pi 
/  2 ) )  e.  ( pi (,) ( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 ) )
5553, 54sylbir 204 . . . . . . . . 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 ) )
5650, 55syl3an1 1215 . . . . . . . 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 ) )
57563expib 1154 . . . . . . 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 ) ) )
5848, 57sylbid 206 . . . . . 6  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  ->  ( ( sin `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
5950resincld 12439 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( A  -  ( pi  /  2
) ) )  e.  RR )
6059lt0neg1d 9358 . . . . . . 7  |-  ( A  e.  RR  ->  (
( sin `  ( A  -  ( pi  /  2 ) ) )  <  0  <->  0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) ) ) )
6160anbi1d 685 . . . . . 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 ) ) )
6258, 61sylibd 205 . . . . 5  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  ->  ( 0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
63 recn 8843 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
64 pncan3 9075 . . . . . . . . . 10  |-  ( ( ( pi  /  2
)  e.  CC  /\  A  e.  CC )  ->  ( ( pi  / 
2 )  +  ( A  -  ( pi 
/  2 ) ) )  =  A )
6517, 63, 64sylancr 644 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( pi  /  2
)  +  ( A  -  ( pi  / 
2 ) ) )  =  A )
6665fveq2d 5545 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  A
) )
6750recnd 8877 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  CC )
68 coshalfpip 19878 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
6967, 68syl 15 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
7066, 69eqtr3d 2330 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  A )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
7170breq2d 4051 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  ( cos `  A )  <->  0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) ) ) )
7265fveq2d 5545 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( sin `  A
) )
73 sinhalfpip 19876 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7467, 73syl 15 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7572, 74eqtr3d 2330 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7675breq1d 4049 . . . . . 6  |-  ( A  e.  RR  ->  (
( sin `  A
)  <  0  <->  ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0
) )
7771, 76anbi12d 691 . . . . 5  |-  ( A  e.  RR  ->  (
( 0  <  ( cos `  A )  /\  ( sin `  A )  <  0 )  <->  ( 0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
7862, 77sylibrd 225 . . . 4  |-  ( A  e.  RR  ->  (
( ( 3  x.  ( pi  /  2
) )  <  A  /\  A  <  ( 2  x.  pi ) )  ->  ( 0  < 
( cos `  A
)  /\  ( sin `  A )  <  0
) ) )
79783impib 1149 . . 3  |-  ( ( A  e.  RR  /\  ( 3  x.  (
pi  /  2 ) )  <  A  /\  A  <  ( 2  x.  pi ) )  -> 
( 0  <  ( cos `  A )  /\  ( sin `  A )  <  0 ) )
8079ancomd 438 . 2  |-  ( ( A  e.  RR  /\  ( 3  x.  (
pi  /  2 ) )  <  A  /\  A  <  ( 2  x.  pi ) )  -> 
( ( sin `  A
)  <  0  /\  0  <  ( cos `  A
) ) )
8112, 80sylbi 187 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   RR*cxr 8882    < clt 8883    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   3c3 9812   4c4 9813   (,)cioo 10672   sincsin 12361   cosccos 12362   picpi 12364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
  Copyright terms: Public domain W3C validator