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Theorem efeq1 22665
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 10763 . . . 4  |-  ( A  e.  CC  ->  ( A  /  2 )  e.  CC )
2 ax-icn 9550 . . . . 5  |-  _i  e.  CC
3 ine0 9991 . . . . 5  |-  _i  =/=  0
4 divcl 10212 . . . . 5  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( A  /  2
)  /  _i )  e.  CC )
52, 3, 4mp3an23 1316 . . . 4  |-  ( ( A  /  2 )  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
61, 5syl 16 . . 3  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
7 sineq0 22663 . . 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 13717 . . . . . 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 10214 . . . . . . . . . 10  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  ( A  /  2 ) )
122, 3, 11mp3an23 1316 . . . . . . . . 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 5869 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  ( A  /  2 ) ) )
15 mulneg1 9992 . . . . . . . . . 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 9813 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  /  2
) )
1816, 17eqtrd 2508 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  / 
2 ) )
1918fveq2d 5869 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  -u ( A  /  2 ) ) )
2014, 19oveq12d 6301 . . . . . 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 6298 . . . . 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 2508 . . . 4  |-  ( A  e.  CC  ->  ( sin `  ( ( A  /  2 )  /  _i ) )  =  ( ( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) ) )
2322eqeq1d 2469 . . 3  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) )  =  0 ) )
24 efcl 13679 . . . . . 6  |-  ( ( A  /  2 )  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
251, 24syl 16 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
261negcld 9916 . . . . . 6  |-  ( A  e.  CC  ->  -u ( A  /  2 )  e.  CC )
27 efcl 13679 . . . . . 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 9929 . . . 4  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  e.  CC )
30 2cn 10605 . . . . . 6  |-  2  e.  CC
3130, 2mulcli 9600 . . . . 5  |-  ( 2  x.  _i )  e.  CC
32 2ne0 10627 . . . . . 6  |-  2  =/=  0
3330, 2, 32, 3mulne0i 10191 . . . . 5  |-  ( 2  x.  _i )  =/=  0
34 diveq0 10216 . . . . 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 1316 . . . 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 13692 . . . . . . . 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 10358 . . . . . 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 13695 . . . . . . . . 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 9936 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  ( ( A  /  2 )  +  ( A  /  2
) ) )
43 2halves 10766 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  +  ( A  /  2 ) )  =  A )
4442, 43eqtrd 2508 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  A )
4544fveq2d 5869 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( ( A  /  2 )  -  -u ( A  /  2
) ) )  =  ( exp `  A
) )
4641, 45eqtr3d 2510 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  ( exp `  A ) )
4728, 38dividd 10317 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  1 )
4846, 47oveq12d 6301 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  -  ( ( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )  =  ( ( exp `  A
)  -  1 ) )
4939, 48eqtrd 2508 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( exp `  -u ( A  / 
2 ) ) )  =  ( ( exp `  A )  -  1 ) )
5049eqeq1d 2469 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  (
( exp `  A
)  -  1 )  =  0 ) )
5129, 28, 38diveq0ad 10329 . . . 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 13679 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
53 ax-1cn 9549 . . . . 5  |-  1  e.  CC
54 subeq0 9844 . . . . 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 10749 . . . . . 6  |-  ( 2  e.  CC  /\  2  =/=  0 )
592, 3pm3.2i 455 . . . . . 6  |-  ( _i  e.  CC  /\  _i  =/=  0 )
60 divdiv32 10251 . . . . . 6  |-  ( ( A  e.  CC  /\  ( 2  e.  CC  /\  2  =/=  0 )  /\  ( _i  e.  CC  /\  _i  =/=  0
) )  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6158, 59, 60mp3an23 1316 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6261oveq1d 6298 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( ( ( A  /  _i )  / 
2 )  /  pi ) )
63 divcl 10212 . . . . . . 7  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  ( A  /  _i )  e.  CC )
642, 3, 63mp3an23 1316 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  e.  CC )
65 picn 22602 . . . . . . . 8  |-  pi  e.  CC
66 pire 22601 . . . . . . . . 9  |-  pi  e.  RR
67 pipos 22603 . . . . . . . . 9  |-  0  <  pi
6866, 67gt0ne0ii 10088 . . . . . . . 8  |-  pi  =/=  0
6965, 68pm3.2i 455 . . . . . . 7  |-  ( pi  e.  CC  /\  pi  =/=  0 )
70 divdiv1 10254 . . . . . . 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 1316 . . . . . 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 9600 . . . . . . 7  |-  ( 2  x.  pi )  e.  CC
7430, 65, 32, 68mulne0i 10191 . . . . . . 7  |-  ( 2  x.  pi )  =/=  0
7573, 74pm3.2i 455 . . . . . 6  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
76 divdiv1 10254 . . . . . 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 1316 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  _i )  /  ( 2  x.  pi ) )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
7872, 77eqtrd 2508 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  /  _i )  /  2
)  /  pi )  =  ( A  / 
( _i  x.  (
2  x.  pi ) ) ) )
7962, 78eqtrd 2508 . . 3  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
8079eleq1d 2536 . 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 1379    e. wcel 1767    =/= wne 2662   ` cfv 5587  (class class class)co 6283   CCcc 9489   0cc0 9491   1c1 9492   _ici 9493    + caddc 9494    x. cmul 9496    - cmin 9804   -ucneg 9805    / cdiv 10205   2c2 10584   ZZcz 10863   expce 13658   sincsin 13660   picpi 13663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ioc 11533  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-fac 12321  df-bc 12348  df-hash 12373  df-shft 12862  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-limsup 13256  df-clim 13273  df-rlim 13274  df-sum 13471  df-ef 13664  df-sin 13666  df-cos 13667  df-pi 13669  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-fbas 18203  df-fg 18204  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-lp 19419  df-perf 19420  df-cn 19510  df-cnp 19511  df-haus 19598  df-tx 19814  df-hmeo 20007  df-fil 20098  df-fm 20190  df-flim 20191  df-flf 20192  df-xms 20574  df-ms 20575  df-tms 20576  df-cncf 21133  df-limc 22021  df-dv 22022
This theorem is referenced by:  efif1olem4  22681  eflogeq  22730  root1eq1  22873  ang180lem1  22885  proot1ex  30782
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