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Theorem efeq1 21997
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 10562 . . . 4  |-  ( A  e.  CC  ->  ( A  /  2 )  e.  CC )
2 ax-icn 9353 . . . . 5  |-  _i  e.  CC
3 ine0 9792 . . . . 5  |-  _i  =/=  0
4 divcl 10012 . . . . 5  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( A  /  2
)  /  _i )  e.  CC )
52, 3, 4mp3an23 1306 . . . 4  |-  ( ( A  /  2 )  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
61, 5syl 16 . . 3  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
7 sineq0 21995 . . 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 13418 . . . . . 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 10014 . . . . . . . . . 10  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  ( A  /  2 ) )
122, 3, 11mp3an23 1306 . . . . . . . . 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 5707 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  ( A  /  2 ) ) )
15 mulneg1 9793 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  ( ( A  / 
2 )  /  _i )  e.  CC )  ->  ( -u _i  x.  ( ( A  / 
2 )  /  _i ) )  =  -u ( _i  x.  (
( A  /  2
)  /  _i ) ) )
162, 6, 15sylancr 663 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( _i  x.  ( ( A  / 
2 )  /  _i ) ) )
1713negeqd 9616 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  /  2
) )
1816, 17eqtrd 2475 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  / 
2 ) )
1918fveq2d 5707 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  -u ( A  /  2 ) ) )
2014, 19oveq12d 6121 . . . . . 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 6118 . . . . 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 2475 . . . 4  |-  ( A  e.  CC  ->  ( sin `  ( ( A  /  2 )  /  _i ) )  =  ( ( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) ) )
2322eqeq1d 2451 . . 3  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) )  =  0 ) )
24 efcl 13380 . . . . . 6  |-  ( ( A  /  2 )  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
251, 24syl 16 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
261negcld 9718 . . . . . 6  |-  ( A  e.  CC  ->  -u ( A  /  2 )  e.  CC )
27 efcl 13380 . . . . . 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 9731 . . . 4  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  e.  CC )
30 2cn 10404 . . . . . 6  |-  2  e.  CC
3130, 2mulcli 9403 . . . . 5  |-  ( 2  x.  _i )  e.  CC
32 2ne0 10426 . . . . . 6  |-  2  =/=  0
3330, 2, 32, 3mulne0i 9991 . . . . 5  |-  ( 2  x.  _i )  =/=  0
34 diveq0 10016 . . . . 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 1306 . . . 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 13393 . . . . . . . 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 10158 . . . . . 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 13396 . . . . . . . . 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 661 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( ( A  /  2 )  -  -u ( A  /  2
) ) )  =  ( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )
421, 1subnegd 9738 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  ( ( A  /  2 )  +  ( A  /  2
) ) )
43 2halves 10565 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  +  ( A  /  2 ) )  =  A )
4442, 43eqtrd 2475 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  A )
4544fveq2d 5707 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( ( A  /  2 )  -  -u ( A  /  2
) ) )  =  ( exp `  A
) )
4641, 45eqtr3d 2477 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  ( exp `  A ) )
4728, 38dividd 10117 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  1 )
4846, 47oveq12d 6121 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  -  ( ( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )  =  ( ( exp `  A
)  -  1 ) )
4939, 48eqtrd 2475 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( exp `  -u ( A  / 
2 ) ) )  =  ( ( exp `  A )  -  1 ) )
5049eqeq1d 2451 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  (
( exp `  A
)  -  1 )  =  0 ) )
5129, 28, 38diveq0ad 10129 . . . 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 13380 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
53 ax-1cn 9352 . . . . 5  |-  1  e.  CC
54 subeq0 9647 . . . . 5  |-  ( ( ( exp `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( exp `  A )  -  1 )  =  0  <->  ( exp `  A )  =  1 ) )
5552, 53, 54sylancl 662 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  1 )  =  0  <->  ( exp `  A )  =  1 ) )
5650, 51, 553bitr3d 283 . . 3  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  ( exp `  A )  =  1 ) )
5723, 36, 563bitrd 279 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  ( exp `  A )  =  1 ) )
58 2cnne0 10548 . . . . . 6  |-  ( 2  e.  CC  /\  2  =/=  0 )
592, 3pm3.2i 455 . . . . . 6  |-  ( _i  e.  CC  /\  _i  =/=  0 )
60 divdiv32 10051 . . . . . 6  |-  ( ( A  e.  CC  /\  ( 2  e.  CC  /\  2  =/=  0 )  /\  ( _i  e.  CC  /\  _i  =/=  0
) )  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6158, 59, 60mp3an23 1306 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6261oveq1d 6118 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( ( ( A  /  _i )  / 
2 )  /  pi ) )
63 divcl 10012 . . . . . . 7  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  ( A  /  _i )  e.  CC )
642, 3, 63mp3an23 1306 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  e.  CC )
65 picn 21934 . . . . . . . 8  |-  pi  e.  CC
66 pire 21933 . . . . . . . . 9  |-  pi  e.  RR
67 pipos 21935 . . . . . . . . 9  |-  0  <  pi
6866, 67gt0ne0ii 9888 . . . . . . . 8  |-  pi  =/=  0
6965, 68pm3.2i 455 . . . . . . 7  |-  ( pi  e.  CC  /\  pi  =/=  0 )
70 divdiv1 10054 . . . . . . 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 1306 . . . . . 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, 65mulcli 9403 . . . . . . 7  |-  ( 2  x.  pi )  e.  CC
7430, 65, 32, 68mulne0i 9991 . . . . . . 7  |-  ( 2  x.  pi )  =/=  0
7573, 74pm3.2i 455 . . . . . 6  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
76 divdiv1 10054 . . . . . 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 1306 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  _i )  /  ( 2  x.  pi ) )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
7872, 77eqtrd 2475 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  /  _i )  /  2
)  /  pi )  =  ( A  / 
( _i  x.  (
2  x.  pi ) ) ) )
7962, 78eqtrd 2475 . . 3  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
8079eleq1d 2509 . 2  |-  ( A  e.  CC  ->  (
( ( ( A  /  2 )  /  _i )  /  pi )  e.  ZZ  <->  ( A  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ ) )
818, 57, 803bitr3d 283 1  |-  ( A  e.  CC  ->  (
( exp `  A
)  =  1  <->  ( A  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618   ` cfv 5430  (class class class)co 6103   CCcc 9292   0cc0 9294   1c1 9295   _ici 9296    + caddc 9297    x. cmul 9299    - cmin 9607   -ucneg 9608    / cdiv 10005   2c2 10383   ZZcz 10658   expce 13359   sincsin 13361   picpi 13364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-fi 7673  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-ioo 11316  df-ioc 11317  df-ico 11318  df-icc 11319  df-fz 11450  df-fzo 11561  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-fac 12064  df-bc 12091  df-hash 12116  df-shft 12568  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-limsup 12961  df-clim 12978  df-rlim 12979  df-sum 13176  df-ef 13365  df-sin 13367  df-cos 13368  df-pi 13370  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-rest 14373  df-topn 14374  df-0g 14392  df-gsum 14393  df-topgen 14394  df-pt 14395  df-prds 14398  df-xrs 14452  df-qtop 14457  df-imas 14458  df-xps 14460  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-submnd 15477  df-mulg 15560  df-cntz 15847  df-cmn 16291  df-psmet 17821  df-xmet 17822  df-met 17823  df-bl 17824  df-mopn 17825  df-fbas 17826  df-fg 17827  df-cnfld 17831  df-top 18515  df-bases 18517  df-topon 18518  df-topsp 18519  df-cld 18635  df-ntr 18636  df-cls 18637  df-nei 18714  df-lp 18752  df-perf 18753  df-cn 18843  df-cnp 18844  df-haus 18931  df-tx 19147  df-hmeo 19340  df-fil 19431  df-fm 19523  df-flim 19524  df-flf 19525  df-xms 19907  df-ms 19908  df-tms 19909  df-cncf 20466  df-limc 21353  df-dv 21354
This theorem is referenced by:  efif1olem4  22013  eflogeq  22062  root1eq1  22205  ang180lem1  22217  proot1ex  29581
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