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Theorem sincosq4sgn 19869
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 8876 . . . 4  |-  RR  C_  RR*
2 3re 9817 . . . . 5  |-  3  e.  RR
3 pire 19832 . . . . . 6  |-  pi  e.  RR
4 2re 9815 . . . . . 6  |-  2  e.  RR
5 2ne0 9829 . . . . . 6  |-  2  =/=  0
63, 4, 5redivcli 9527 . . . . 5  |-  ( pi 
/  2 )  e.  RR
72, 6remulcli 8851 . . . 4  |-  ( 3  x.  ( pi  / 
2 ) )  e.  RR
81, 7sselii 3177 . . 3  |-  ( 3  x.  ( pi  / 
2 ) )  e. 
RR*
94, 3remulcli 8851 . . . 4  |-  ( 2  x.  pi )  e.  RR
101, 9sselii 3177 . . 3  |-  ( 2  x.  pi )  e. 
RR*
11 elioo2 10697 . . 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 9805 . . . . . . . . . . . 12  |-  3  =  ( 2  +  1 )
1413oveq1i 5868 . . . . . . . . . . 11  |-  ( 3  x.  ( pi  / 
2 ) )  =  ( ( 2  +  1 )  x.  (
pi  /  2 ) )
15 2cn 9816 . . . . . . . . . . . 12  |-  2  e.  CC
16 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
176recni 8849 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  CC
1815, 16, 17adddiri 8848 . . . . . . . . . . 11  |-  ( ( 2  +  1 )  x.  ( pi  / 
2 ) )  =  ( ( 2  x.  ( pi  /  2
) )  +  ( 1  x.  ( pi 
/  2 ) ) )
193recni 8849 . . . . . . . . . . . . 13  |-  pi  e.  CC
2019, 15, 5divcan2i 9503 . . . . . . . . . . . 12  |-  ( 2  x.  ( pi  / 
2 ) )  =  pi
2117mulid2i 8840 . . . . . . . . . . . 12  |-  ( 1  x.  ( pi  / 
2 ) )  =  ( pi  /  2
)
2220, 21oveq12i 5870 . . . . . . . . . . 11  |-  ( ( 2  x.  ( pi 
/  2 ) )  +  ( 1  x.  ( pi  /  2
) ) )  =  ( pi  +  ( pi  /  2 ) )
2314, 18, 223eqtrri 2308 . . . . . . . . . 10  |-  ( pi  +  ( pi  / 
2 ) )  =  ( 3  x.  (
pi  /  2 ) )
2423breq1i 4030 . . . . . . . . 9  |-  ( ( pi  +  ( pi 
/  2 ) )  <  A  <->  ( 3  x.  ( pi  / 
2 ) )  < 
A )
25 ltaddsub 9248 . . . . . . . . . 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 9244 . . . . . . . . . 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 9806 . . . . . . . . . . . . 13  |-  4  =  ( 3  +  1 )
3130oveq1i 5868 . . . . . . . . . . . 12  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( ( 3  +  1 )  x.  (
pi  /  2 ) )
322recni 8849 . . . . . . . . . . . . 13  |-  3  e.  CC
3332, 16, 17adddiri 8848 . . . . . . . . . . . 12  |-  ( ( 3  +  1 )  x.  ( pi  / 
2 ) )  =  ( ( 3  x.  ( pi  /  2
) )  +  ( 1  x.  ( pi 
/  2 ) ) )
3421oveq2i 5869 . . . . . . . . . . . 12  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( 1  x.  ( pi  /  2
) ) )  =  ( ( 3  x.  ( pi  /  2
) )  +  ( pi  /  2 ) )
3531, 33, 343eqtrri 2308 . . . . . . . . . . 11  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) )  =  ( 4  x.  (
pi  /  2 ) )
36 4cn 9820 . . . . . . . . . . . . 13  |-  4  e.  CC
3715, 5pm3.2i 441 . . . . . . . . . . . . 13  |-  ( 2  e.  CC  /\  2  =/=  0 )
38 div12 9446 . . . . . . . . . . . . 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 9876 . . . . . . . . . . . . . 14  |-  ( 4  /  2 )  =  2
4140oveq2i 5869 . . . . . . . . . . . . 13  |-  ( pi  x.  ( 4  / 
2 ) )  =  ( pi  x.  2 )
4219, 15mulcomi 8843 . . . . . . . . . . . . 13  |-  ( pi  x.  2 )  =  ( 2  x.  pi )
4341, 42eqtri 2303 . . . . . . . . . . . 12  |-  ( pi  x.  ( 4  / 
2 ) )  =  ( 2  x.  pi )
4439, 43eqtri 2303 . . . . . . . . . . 11  |-  ( 4  x.  ( pi  / 
2 ) )  =  ( 2  x.  pi )
4535, 44eqtri 2303 . . . . . . . . . 10  |-  ( ( 3  x.  ( pi 
/  2 ) )  +  ( pi  / 
2 ) )  =  ( 2  x.  pi )
4645breq2i 4031 . . . . . . . . 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 9111 . . . . . . . . . 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 3177 . . . . . . . . . . 11  |-  pi  e.  RR*
52 elioo2 10697 . . . . . . . . . . 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 19868 . . . . . . . . . 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 12423 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( A  -  ( pi  /  2
) ) )  e.  RR )
6059lt0neg1d 9342 . . . . . . 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 8827 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
64 pncan3 9059 . . . . . . . . . 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 5529 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  A
) )
6750recnd 8861 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  CC )
68 coshalfpip 19862 . . . . . . . . 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 2317 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  A )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
7170breq2d 4035 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  ( cos `  A )  <->  0  <  -u ( sin `  ( A  -  ( pi  /  2 ) ) ) ) )
7265fveq2d 5529 . . . . . . . 8  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( sin `  A
) )
73 sinhalfpip 19860 . . . . . . . . 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 2317 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
7675breq1d 4033 . . . . . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   RR*cxr 8866    < clt 8867    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   3c3 9796   4c4 9797   (,)cioo 10656   sincsin 12345   cosccos 12346   picpi 12348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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