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Theorem efeq1 20436
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 10198 . . . 4  |-  ( A  e.  CC  ->  ( A  /  2 )  e.  CC )
2 ax-icn 9054 . . . . 5  |-  _i  e.  CC
3 ine0 9474 . . . . 5  |-  _i  =/=  0
4 divcl 9689 . . . . 5  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( A  /  2
)  /  _i )  e.  CC )
52, 3, 4mp3an23 1272 . . . 4  |-  ( ( A  /  2 )  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
61, 5syl 16 . . 3  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
7 sineq0 20434 . . 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 12728 . . . . . 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 9691 . . . . . . . . . 10  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  ( A  /  2 ) )
122, 3, 11mp3an23 1272 . . . . . . . . 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 5735 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  ( A  /  2 ) ) )
15 mulneg1 9475 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  ( ( A  / 
2 )  /  _i )  e.  CC )  ->  ( -u _i  x.  ( ( A  / 
2 )  /  _i ) )  =  -u ( _i  x.  (
( A  /  2
)  /  _i ) ) )
162, 6, 15sylancr 646 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( _i  x.  ( ( A  / 
2 )  /  _i ) ) )
1713negeqd 9305 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  /  2
) )
1816, 17eqtrd 2470 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  / 
2 ) )
1918fveq2d 5735 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  -u ( A  /  2 ) ) )
2014, 19oveq12d 6102 . . . . . 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 6099 . . . . 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 2470 . . . 4  |-  ( A  e.  CC  ->  ( sin `  ( ( A  /  2 )  /  _i ) )  =  ( ( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) ) )
2322eqeq1d 2446 . . 3  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) )  =  0 ) )
24 efcl 12690 . . . . . 6  |-  ( ( A  /  2 )  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
251, 24syl 16 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
261negcld 9403 . . . . . 6  |-  ( A  e.  CC  ->  -u ( A  /  2 )  e.  CC )
27 efcl 12690 . . . . . 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 9416 . . . 4  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  e.  CC )
30 2cn 10075 . . . . . 6  |-  2  e.  CC
3130, 2mulcli 9100 . . . . 5  |-  ( 2  x.  _i )  e.  CC
32 2ne0 10088 . . . . . 6  |-  2  =/=  0
3330, 2, 32, 3mulne0i 9670 . . . . 5  |-  ( 2  x.  _i )  =/=  0
34 diveq0 9693 . . . . 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 1272 . . . 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 12703 . . . . . . . 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 9834 . . . . . 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 12706 . . . . . . . . 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 644 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( ( A  /  2 )  -  -u ( A  /  2
) ) )  =  ( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )
421, 1subnegd 9423 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  ( ( A  /  2 )  +  ( A  /  2
) ) )
43 2halves 10201 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  +  ( A  /  2 ) )  =  A )
4442, 43eqtrd 2470 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  A )
4544fveq2d 5735 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( ( A  /  2 )  -  -u ( A  /  2
) ) )  =  ( exp `  A
) )
4641, 45eqtr3d 2472 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  ( exp `  A ) )
4728, 38dividd 9793 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  1 )
4846, 47oveq12d 6102 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  -  ( ( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )  =  ( ( exp `  A
)  -  1 ) )
4939, 48eqtrd 2470 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( exp `  -u ( A  / 
2 ) ) )  =  ( ( exp `  A )  -  1 ) )
5049eqeq1d 2446 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  (
( exp `  A
)  -  1 )  =  0 ) )
5129, 28, 38diveq0ad 9805 . . . 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 12690 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
53 ax-1cn 9053 . . . . 5  |-  1  e.  CC
54 subeq0 9332 . . . . 5  |-  ( ( ( exp `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( exp `  A )  -  1 )  =  0  <->  ( exp `  A )  =  1 ) )
5552, 53, 54sylancl 645 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  1 )  =  0  <->  ( exp `  A )  =  1 ) )
5650, 51, 553bitr3d 276 . . 3  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  ( exp `  A )  =  1 ) )
5723, 36, 563bitrd 272 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  ( exp `  A )  =  1 ) )
5830, 32pm3.2i 443 . . . . . 6  |-  ( 2  e.  CC  /\  2  =/=  0 )
592, 3pm3.2i 443 . . . . . 6  |-  ( _i  e.  CC  /\  _i  =/=  0 )
60 divdiv32 9727 . . . . . 6  |-  ( ( A  e.  CC  /\  ( 2  e.  CC  /\  2  =/=  0 )  /\  ( _i  e.  CC  /\  _i  =/=  0
) )  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6158, 59, 60mp3an23 1272 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6261oveq1d 6099 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( ( ( A  /  _i )  / 
2 )  /  pi ) )
63 divcl 9689 . . . . . . 7  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  ( A  /  _i )  e.  CC )
642, 3, 63mp3an23 1272 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  e.  CC )
65 pire 20377 . . . . . . . . 9  |-  pi  e.  RR
6665recni 9107 . . . . . . . 8  |-  pi  e.  CC
67 pipos 20378 . . . . . . . . 9  |-  0  <  pi
6865, 67gt0ne0ii 9568 . . . . . . . 8  |-  pi  =/=  0
6966, 68pm3.2i 443 . . . . . . 7  |-  ( pi  e.  CC  /\  pi  =/=  0 )
70 divdiv1 9730 . . . . . . 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 1272 . . . . . 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 9100 . . . . . . 7  |-  ( 2  x.  pi )  e.  CC
7430, 66, 32, 68mulne0i 9670 . . . . . . 7  |-  ( 2  x.  pi )  =/=  0
7573, 74pm3.2i 443 . . . . . 6  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
76 divdiv1 9730 . . . . . 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 1272 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  _i )  /  ( 2  x.  pi ) )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
7872, 77eqtrd 2470 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  /  _i )  /  2
)  /  pi )  =  ( A  / 
( _i  x.  (
2  x.  pi ) ) ) )
7962, 78eqtrd 2470 . . 3  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
8079eleq1d 2504 . 2  |-  ( A  e.  CC  ->  (
( ( ( A  /  2 )  /  _i )  /  pi )  e.  ZZ  <->  ( A  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ ) )
818, 57, 803bitr3d 276 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996   _ici 8997    + caddc 8998    x. cmul 9000    - cmin 9296   -ucneg 9297    / cdiv 9682   2c2 10054   ZZcz 10287   expce 12669   sincsin 12671   picpi 12674
This theorem is referenced by:  efif1olem4  20452  eflogeq  20501  root1eq1  20644  ang180lem1  20656  proot1ex  27511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678  df-pi 12680  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759
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