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Theorem efeq1 20388
Description: A complex number whose exponential is one is an integer multiple of  2 pi _i. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.)
Assertion
Ref Expression
efeq1  |-  ( A  e.  CC  ->  (
( exp `  A
)  =  1  <->  ( A  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ ) )

Proof of Theorem efeq1
StepHypRef Expression
1 halfcl 10153 . . . 4  |-  ( A  e.  CC  ->  ( A  /  2 )  e.  CC )
2 ax-icn 9009 . . . . 5  |-  _i  e.  CC
3 ine0 9429 . . . . 5  |-  _i  =/=  0
4 divcl 9644 . . . . 5  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( A  /  2
)  /  _i )  e.  CC )
52, 3, 4mp3an23 1271 . . . 4  |-  ( ( A  /  2 )  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
61, 5syl 16 . . 3  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
7 sineq0 20386 . . 3  |-  ( ( ( A  /  2
)  /  _i )  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  (
( ( A  / 
2 )  /  _i )  /  pi )  e.  ZZ ) )
86, 7syl 16 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  (
( ( A  / 
2 )  /  _i )  /  pi )  e.  ZZ ) )
9 sinval 12682 . . . . . 6  |-  ( ( ( A  /  2
)  /  _i )  e.  CC  ->  ( sin `  ( ( A  /  2 )  /  _i ) )  =  ( ( ( exp `  (
_i  x.  ( ( A  /  2 )  /  _i ) ) )  -  ( exp `  ( -u _i  x.  ( ( A  /  2 )  /  _i ) ) ) )  /  ( 2  x.  _i ) ) )
106, 9syl 16 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  ( ( A  /  2 )  /  _i ) )  =  ( ( ( exp `  (
_i  x.  ( ( A  /  2 )  /  _i ) ) )  -  ( exp `  ( -u _i  x.  ( ( A  /  2 )  /  _i ) ) ) )  /  ( 2  x.  _i ) ) )
11 divcan2 9646 . . . . . . . . . 10  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  ( A  /  2 ) )
122, 3, 11mp3an23 1271 . . . . . . . . 9  |-  ( ( A  /  2 )  e.  CC  ->  (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  ( A  /  2 ) )
131, 12syl 16 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  ( A  /  2 ) )
1413fveq2d 5695 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  ( A  /  2 ) ) )
15 mulneg1 9430 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  ( ( A  / 
2 )  /  _i )  e.  CC )  ->  ( -u _i  x.  ( ( A  / 
2 )  /  _i ) )  =  -u ( _i  x.  (
( A  /  2
)  /  _i ) ) )
162, 6, 15sylancr 645 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( _i  x.  ( ( A  / 
2 )  /  _i ) ) )
1713negeqd 9260 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  /  2
) )
1816, 17eqtrd 2440 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  / 
2 ) )
1918fveq2d 5695 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  -u ( A  /  2 ) ) )
2014, 19oveq12d 6062 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  ( ( A  /  2 )  /  _i ) ) )  -  ( exp `  ( -u _i  x.  ( ( A  /  2 )  /  _i ) ) ) )  =  ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) ) )
2120oveq1d 6059 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  ( ( A  /  2 )  /  _i ) ) )  -  ( exp `  ( -u _i  x.  ( ( A  /  2 )  /  _i ) ) ) )  /  ( 2  x.  _i ) )  =  ( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( 2  x.  _i ) ) )
2210, 21eqtrd 2440 . . . 4  |-  ( A  e.  CC  ->  ( sin `  ( ( A  /  2 )  /  _i ) )  =  ( ( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) ) )
2322eqeq1d 2416 . . 3  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) )  =  0 ) )
24 efcl 12644 . . . . . 6  |-  ( ( A  /  2 )  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
251, 24syl 16 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
261negcld 9358 . . . . . 6  |-  ( A  e.  CC  ->  -u ( A  /  2 )  e.  CC )
27 efcl 12644 . . . . . 6  |-  ( -u ( A  /  2
)  e.  CC  ->  ( exp `  -u ( A  /  2 ) )  e.  CC )
2826, 27syl 16 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  -u ( A  / 
2 ) )  e.  CC )
2925, 28subcld 9371 . . . 4  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  e.  CC )
30 2cn 10030 . . . . . 6  |-  2  e.  CC
3130, 2mulcli 9055 . . . . 5  |-  ( 2  x.  _i )  e.  CC
32 2ne0 10043 . . . . . 6  |-  2  =/=  0
3330, 2, 32, 3mulne0i 9625 . . . . 5  |-  ( 2  x.  _i )  =/=  0
34 diveq0 9648 . . . . 5  |-  ( ( ( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  e.  CC  /\  ( 2  x.  _i )  e.  CC  /\  (
2  x.  _i )  =/=  0 )  -> 
( ( ( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) )  =  0  <->  ( ( exp `  ( A  / 
2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  =  0 ) )
3531, 33, 34mp3an23 1271 . . . 4  |-  ( ( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  e.  CC  ->  ( ( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( 2  x.  _i ) )  =  0  <->  ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  =  0 ) )
3629, 35syl 16 . . 3  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( 2  x.  _i ) )  =  0  <->  ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  =  0 ) )
37 efne0 12657 . . . . . . . 8  |-  ( -u ( A  /  2
)  e.  CC  ->  ( exp `  -u ( A  /  2 ) )  =/=  0 )
3826, 37syl 16 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  -u ( A  / 
2 ) )  =/=  0 )
3925, 28, 28, 38divsubdird 9789 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( exp `  -u ( A  / 
2 ) ) )  =  ( ( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  -  ( ( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) ) )
40 efsub 12660 . . . . . . . . 9  |-  ( ( ( A  /  2
)  e.  CC  /\  -u ( A  /  2
)  e.  CC )  ->  ( exp `  (
( A  /  2
)  -  -u ( A  /  2 ) ) )  =  ( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )
411, 26, 40syl2anc 643 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( ( A  /  2 )  -  -u ( A  /  2
) ) )  =  ( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )
421, 1subnegd 9378 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  ( ( A  /  2 )  +  ( A  /  2
) ) )
43 2halves 10156 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  +  ( A  /  2 ) )  =  A )
4442, 43eqtrd 2440 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  A )
4544fveq2d 5695 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( ( A  /  2 )  -  -u ( A  /  2
) ) )  =  ( exp `  A
) )
4641, 45eqtr3d 2442 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  ( exp `  A ) )
4728, 38dividd 9748 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  1 )
4846, 47oveq12d 6062 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  -  ( ( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )  =  ( ( exp `  A
)  -  1 ) )
4939, 48eqtrd 2440 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( exp `  -u ( A  / 
2 ) ) )  =  ( ( exp `  A )  -  1 ) )
5049eqeq1d 2416 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  (
( exp `  A
)  -  1 )  =  0 ) )
5129, 28, 38diveq0ad 9760 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  (
( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  =  0 ) )
52 efcl 12644 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
53 ax-1cn 9008 . . . . 5  |-  1  e.  CC
54 subeq0 9287 . . . . 5  |-  ( ( ( exp `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( exp `  A )  -  1 )  =  0  <->  ( exp `  A )  =  1 ) )
5552, 53, 54sylancl 644 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  1 )  =  0  <->  ( exp `  A )  =  1 ) )
5650, 51, 553bitr3d 275 . . 3  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  ( exp `  A )  =  1 ) )
5723, 36, 563bitrd 271 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  ( exp `  A )  =  1 ) )
5830, 32pm3.2i 442 . . . . . 6  |-  ( 2  e.  CC  /\  2  =/=  0 )
592, 3pm3.2i 442 . . . . . 6  |-  ( _i  e.  CC  /\  _i  =/=  0 )
60 divdiv32 9682 . . . . . 6  |-  ( ( A  e.  CC  /\  ( 2  e.  CC  /\  2  =/=  0 )  /\  ( _i  e.  CC  /\  _i  =/=  0
) )  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6158, 59, 60mp3an23 1271 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6261oveq1d 6059 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( ( ( A  /  _i )  / 
2 )  /  pi ) )
63 divcl 9644 . . . . . . 7  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  ( A  /  _i )  e.  CC )
642, 3, 63mp3an23 1271 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  e.  CC )
65 pire 20329 . . . . . . . . 9  |-  pi  e.  RR
6665recni 9062 . . . . . . . 8  |-  pi  e.  CC
67 pipos 20330 . . . . . . . . 9  |-  0  <  pi
6865, 67gt0ne0ii 9523 . . . . . . . 8  |-  pi  =/=  0
6966, 68pm3.2i 442 . . . . . . 7  |-  ( pi  e.  CC  /\  pi  =/=  0 )
70 divdiv1 9685 . . . . . . 7  |-  ( ( ( A  /  _i )  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 )  /\  ( pi  e.  CC  /\  pi  =/=  0 ) )  ->  ( (
( A  /  _i )  /  2 )  /  pi )  =  (
( A  /  _i )  /  ( 2  x.  pi ) ) )
7158, 69, 70mp3an23 1271 . . . . . 6  |-  ( ( A  /  _i )  e.  CC  ->  (
( ( A  /  _i )  /  2
)  /  pi )  =  ( ( A  /  _i )  / 
( 2  x.  pi ) ) )
7264, 71syl 16 . . . . 5  |-  ( A  e.  CC  ->  (
( ( A  /  _i )  /  2
)  /  pi )  =  ( ( A  /  _i )  / 
( 2  x.  pi ) ) )
7330, 66mulcli 9055 . . . . . . 7  |-  ( 2  x.  pi )  e.  CC
7430, 66, 32, 68mulne0i 9625 . . . . . . 7  |-  ( 2  x.  pi )  =/=  0
7573, 74pm3.2i 442 . . . . . 6  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
76 divdiv1 9685 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  e.  CC  /\  _i  =/=  0 )  /\  ( ( 2  x.  pi )  e.  CC  /\  ( 2  x.  pi )  =/=  0 ) )  -> 
( ( A  /  _i )  /  (
2  x.  pi ) )  =  ( A  /  ( _i  x.  ( 2  x.  pi ) ) ) )
7759, 75, 76mp3an23 1271 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  _i )  /  ( 2  x.  pi ) )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
7872, 77eqtrd 2440 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  /  _i )  /  2
)  /  pi )  =  ( A  / 
( _i  x.  (
2  x.  pi ) ) ) )
7962, 78eqtrd 2440 . . 3  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
8079eleq1d 2474 . 2  |-  ( A  e.  CC  ->  (
( ( ( A  /  2 )  /  _i )  /  pi )  e.  ZZ  <->  ( A  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ ) )
818, 57, 803bitr3d 275 1  |-  ( A  e.  CC  ->  (
( exp `  A
)  =  1  <->  ( A  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   ` cfv 5417  (class class class)co 6044   CCcc 8948   0cc0 8950   1c1 8951   _ici 8952    + caddc 8953    x. cmul 8955    - cmin 9251   -ucneg 9252    / cdiv 9637   2c2 10009   ZZcz 10242   expce 12623   sincsin 12625   picpi 12628
This theorem is referenced by:  efif1olem4  20404  eflogeq  20453  root1eq1  20596  ang180lem1  20608  proot1ex  27392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-addf 9029  ax-mulf 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-map 6983  df-pm 6984  df-ixp 7027  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-fi 7378  df-sup 7408  df-oi 7439  df-card 7786  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-ioo 10880  df-ioc 10881  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-fl 11161  df-mod 11210  df-seq 11283  df-exp 11342  df-fac 11526  df-bc 11553  df-hash 11578  df-shft 11841  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-limsup 12224  df-clim 12241  df-rlim 12242  df-sum 12439  df-ef 12629  df-sin 12631  df-cos 12632  df-pi 12634  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-starv 13503  df-sca 13504  df-vsca 13505  df-tset 13507  df-ple 13508  df-ds 13510  df-unif 13511  df-hom 13512  df-cco 13513  df-rest 13609  df-topn 13610  df-topgen 13626  df-pt 13627  df-prds 13630  df-xrs 13685  df-0g 13686  df-gsum 13687  df-qtop 13692  df-imas 13693  df-xps 13695  df-mre 13770  df-mrc 13771  df-acs 13773  df-mnd 14649  df-submnd 14698  df-mulg 14774  df-cntz 15075  df-cmn 15373  df-psmet 16653  df-xmet 16654  df-met 16655  df-bl 16656  df-mopn 16657  df-fbas 16658  df-fg 16659  df-cnfld 16663  df-top 16922  df-bases 16924  df-topon 16925  df-topsp 16926  df-cld 17042  df-ntr 17043  df-cls 17044  df-nei 17121  df-lp 17159  df-perf 17160  df-cn 17249  df-cnp 17250  df-haus 17337  df-tx 17551  df-hmeo 17744  df-fil 17835  df-fm 17927  df-flim 17928  df-flf 17929  df-xms 18307  df-ms 18308  df-tms 18309  df-cncf 18865  df-limc 19710  df-dv 19711
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