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Theorem efeq1 19998
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 10029 . . . 4  |-  ( A  e.  CC  ->  ( A  /  2 )  e.  CC )
2 ax-icn 8886 . . . . 5  |-  _i  e.  CC
3 ine0 9305 . . . . 5  |-  _i  =/=  0
4 divcl 9520 . . . . 5  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( A  /  2
)  /  _i )  e.  CC )
52, 3, 4mp3an23 1269 . . . 4  |-  ( ( A  /  2 )  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
61, 5syl 15 . . 3  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  e.  CC )
7 sineq0 19996 . . 3  |-  ( ( ( A  /  2
)  /  _i )  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  (
( ( A  / 
2 )  /  _i )  /  pi )  e.  ZZ ) )
86, 7syl 15 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  (
( ( A  / 
2 )  /  _i )  /  pi )  e.  ZZ ) )
9 sinval 12499 . . . . . 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 15 . . . . 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 9522 . . . . . . . . . 10  |-  ( ( ( A  /  2
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  ( A  /  2 ) )
122, 3, 11mp3an23 1269 . . . . . . . . 9  |-  ( ( A  /  2 )  e.  CC  ->  (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  ( A  /  2 ) )
131, 12syl 15 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  ( A  /  2 ) )
1413fveq2d 5612 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  ( A  /  2 ) ) )
15 mulneg1 9306 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  ( ( A  / 
2 )  /  _i )  e.  CC )  ->  ( -u _i  x.  ( ( A  / 
2 )  /  _i ) )  =  -u ( _i  x.  (
( A  /  2
)  /  _i ) ) )
162, 6, 15sylancr 644 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( _i  x.  ( ( A  / 
2 )  /  _i ) ) )
1713negeqd 9136 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
_i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  /  2
) )
1816, 17eqtrd 2390 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A  /  2 )  /  _i ) )  =  -u ( A  / 
2 ) )
1918fveq2d 5612 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( ( A  / 
2 )  /  _i ) ) )  =  ( exp `  -u ( A  /  2 ) ) )
2014, 19oveq12d 5963 . . . . . 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 5960 . . . . 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 2390 . . . 4  |-  ( A  e.  CC  ->  ( sin `  ( ( A  /  2 )  /  _i ) )  =  ( ( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) ) )
2322eqeq1d 2366 . . 3  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( 2  x.  _i ) )  =  0 ) )
24 efcl 12461 . . . . . 6  |-  ( ( A  /  2 )  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
251, 24syl 15 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  / 
2 ) )  e.  CC )
261negcld 9234 . . . . . 6  |-  ( A  e.  CC  ->  -u ( A  /  2 )  e.  CC )
27 efcl 12461 . . . . . 6  |-  ( -u ( A  /  2
)  e.  CC  ->  ( exp `  -u ( A  /  2 ) )  e.  CC )
2826, 27syl 15 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  -u ( A  / 
2 ) )  e.  CC )
2925, 28subcld 9247 . . . 4  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  e.  CC )
30 2cn 9906 . . . . . 6  |-  2  e.  CC
3130, 2mulcli 8932 . . . . 5  |-  ( 2  x.  _i )  e.  CC
32 2ne0 9919 . . . . . 6  |-  2  =/=  0
3330, 2, 32, 3mulne0i 9501 . . . . 5  |-  ( 2  x.  _i )  =/=  0
34 diveq0 9524 . . . . 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 1269 . . . 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 15 . . 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 12474 . . . . . . . 8  |-  ( -u ( A  /  2
)  e.  CC  ->  ( exp `  -u ( A  /  2 ) )  =/=  0 )
3826, 37syl 15 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  -u ( A  / 
2 ) )  =/=  0 )
3925, 28, 28, 38divsubdird 9665 . . . . . 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 12477 . . . . . . . . 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 642 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( ( A  /  2 )  -  -u ( A  /  2
) ) )  =  ( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )
421, 1subnegd 9254 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  ( ( A  /  2 )  +  ( A  /  2
) ) )
43 2halves 10032 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A  /  2
)  +  ( A  /  2 ) )  =  A )
4442, 43eqtrd 2390 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  /  2
)  -  -u ( A  /  2 ) )  =  A )
4544fveq2d 5612 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( ( A  /  2 )  -  -u ( A  /  2
) ) )  =  ( exp `  A
) )
4641, 45eqtr3d 2392 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  ( exp `  A ) )
4728, 38dividd 9624 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  1 )
4846, 47oveq12d 5963 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) )  -  ( ( exp `  -u ( A  /  2 ) )  /  ( exp `  -u ( A  /  2 ) ) ) )  =  ( ( exp `  A
)  -  1 ) )
4939, 48eqtrd 2390 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  /  ( exp `  -u ( A  / 
2 ) ) )  =  ( ( exp `  A )  -  1 ) )
5049eqeq1d 2366 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  (
( exp `  A
)  -  1 )  =  0 ) )
51 diveq0 9524 . . . . 5  |-  ( ( ( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  e.  CC  /\  ( exp `  -u ( A  /  2 ) )  e.  CC  /\  ( exp `  -u ( A  / 
2 ) )  =/=  0 )  ->  (
( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  (
( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  =  0 ) )
5229, 28, 38, 51syl3anc 1182 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( A  /  2
) )  -  ( exp `  -u ( A  / 
2 ) ) )  /  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  (
( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  =  0 ) )
53 efcl 12461 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
54 ax-1cn 8885 . . . . 5  |-  1  e.  CC
55 subeq0 9163 . . . . 5  |-  ( ( ( exp `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( exp `  A )  -  1 )  =  0  <->  ( exp `  A )  =  1 ) )
5653, 54, 55sylancl 643 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  1 )  =  0  <->  ( exp `  A )  =  1 ) )
5750, 52, 563bitr3d 274 . . 3  |-  ( A  e.  CC  ->  (
( ( exp `  ( A  /  2 ) )  -  ( exp `  -u ( A  /  2 ) ) )  =  0  <->  ( exp `  A )  =  1 ) )
5823, 36, 573bitrd 270 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
( A  /  2
)  /  _i ) )  =  0  <->  ( exp `  A )  =  1 ) )
5930, 32pm3.2i 441 . . . . . 6  |-  ( 2  e.  CC  /\  2  =/=  0 )
602, 3pm3.2i 441 . . . . . 6  |-  ( _i  e.  CC  /\  _i  =/=  0 )
61 divdiv32 9558 . . . . . 6  |-  ( ( A  e.  CC  /\  ( 2  e.  CC  /\  2  =/=  0 )  /\  ( _i  e.  CC  /\  _i  =/=  0
) )  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6259, 60, 61mp3an23 1269 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  2
)  /  _i )  =  ( ( A  /  _i )  / 
2 ) )
6362oveq1d 5960 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( ( ( A  /  _i )  / 
2 )  /  pi ) )
64 divcl 9520 . . . . . . 7  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  ( A  /  _i )  e.  CC )
652, 3, 64mp3an23 1269 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  _i )  e.  CC )
66 pire 19939 . . . . . . . . 9  |-  pi  e.  RR
6766recni 8939 . . . . . . . 8  |-  pi  e.  CC
68 pipos 19940 . . . . . . . . 9  |-  0  <  pi
6966, 68gt0ne0ii 9399 . . . . . . . 8  |-  pi  =/=  0
7067, 69pm3.2i 441 . . . . . . 7  |-  ( pi  e.  CC  /\  pi  =/=  0 )
71 divdiv1 9561 . . . . . . 7  |-  ( ( ( A  /  _i )  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 )  /\  ( pi  e.  CC  /\  pi  =/=  0 ) )  ->  ( (
( A  /  _i )  /  2 )  /  pi )  =  (
( A  /  _i )  /  ( 2  x.  pi ) ) )
7259, 70, 71mp3an23 1269 . . . . . 6  |-  ( ( A  /  _i )  e.  CC  ->  (
( ( A  /  _i )  /  2
)  /  pi )  =  ( ( A  /  _i )  / 
( 2  x.  pi ) ) )
7365, 72syl 15 . . . . 5  |-  ( A  e.  CC  ->  (
( ( A  /  _i )  /  2
)  /  pi )  =  ( ( A  /  _i )  / 
( 2  x.  pi ) ) )
7430, 67mulcli 8932 . . . . . . 7  |-  ( 2  x.  pi )  e.  CC
7530, 67, 32, 69mulne0i 9501 . . . . . . 7  |-  ( 2  x.  pi )  =/=  0
7674, 75pm3.2i 441 . . . . . 6  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
77 divdiv1 9561 . . . . . 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 ) ) ) )
7860, 76, 77mp3an23 1269 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  _i )  /  ( 2  x.  pi ) )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
7973, 78eqtrd 2390 . . . 4  |-  ( A  e.  CC  ->  (
( ( A  /  _i )  /  2
)  /  pi )  =  ( A  / 
( _i  x.  (
2  x.  pi ) ) ) )
8063, 79eqtrd 2390 . . 3  |-  ( A  e.  CC  ->  (
( ( A  / 
2 )  /  _i )  /  pi )  =  ( A  /  (
_i  x.  ( 2  x.  pi ) ) ) )
8180eleq1d 2424 . 2  |-  ( A  e.  CC  ->  (
( ( ( A  /  2 )  /  _i )  /  pi )  e.  ZZ  <->  ( A  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ ) )
828, 58, 813bitr3d 274 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 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   ` cfv 5337  (class class class)co 5945   CCcc 8825   0cc0 8827   1c1 8828   _ici 8829    + caddc 8830    x. cmul 8832    - cmin 9127   -ucneg 9128    / cdiv 9513   2c2 9885   ZZcz 10116   expce 12440   sincsin 12442   picpi 12445
This theorem is referenced by:  efif1olem4  20014  eflogeq  20063  root1eq1  20206  ang180lem1  20218  proot1ex  26843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ioc 10753  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-shft 11658  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-sum 12256  df-ef 12446  df-sin 12448  df-cos 12449  df-pi 12451  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-cn 17063  df-cnp 17064  df-haus 17149  df-tx 17363  df-hmeo 17552  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-tms 17989  df-cncf 18485  df-limc 19320  df-dv 19321
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