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Theorem 1cubr 20643
Description: The cube roots of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
1cubr.r  |-  R  =  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) }
Assertion
Ref Expression
1cubr  |-  ( A  e.  R  <->  ( A  e.  CC  /\  ( A ^ 3 )  =  1 ) )

Proof of Theorem 1cubr
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 1cubr.r . . . . 5  |-  R  =  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) }
2 ax-1cn 9012 . . . . . . 7  |-  1  e.  CC
3 neg1cn 10031 . . . . . . . . 9  |-  -u 1  e.  CC
4 ax-icn 9013 . . . . . . . . . 10  |-  _i  e.  CC
5 3cn 10036 . . . . . . . . . . 11  |-  3  e.  CC
6 sqrcl 12128 . . . . . . . . . . 11  |-  ( 3  e.  CC  ->  ( sqr `  3 )  e.  CC )
75, 6ax-mp 8 . . . . . . . . . 10  |-  ( sqr `  3 )  e.  CC
84, 7mulcli 9059 . . . . . . . . 9  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
93, 8addcli 9058 . . . . . . . 8  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
10 halfcl 10157 . . . . . . . 8  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  e.  CC  ->  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
119, 10ax-mp 8 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC
123, 8subcli 9340 . . . . . . . 8  |-  ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  e.  CC
13 halfcl 10157 . . . . . . . 8  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  e.  CC  ->  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
1412, 13ax-mp 8 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC
152, 11, 143pm3.2i 1132 . . . . . 6  |-  ( 1  e.  CC  /\  (
( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC  /\  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
162elexi 2933 . . . . . . 7  |-  1  e.  _V
17 ovex 6073 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  _V
18 ovex 6073 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e. 
_V
1916, 17, 18tpss 3932 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2
)  e.  CC  /\  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
)  e.  CC )  <->  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) }  C_  CC )
2015, 19mpbi 200 . . . . 5  |-  { 1 ,  ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) ,  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) }  C_  CC
211, 20eqsstri 3346 . . . 4  |-  R  C_  CC
2221sseli 3312 . . 3  |-  ( A  e.  R  ->  A  e.  CC )
2322pm4.71ri 615 . 2  |-  ( A  e.  R  <->  ( A  e.  CC  /\  A  e.  R ) )
24 3nn 10098 . . . . 5  |-  3  e.  NN
25 cxpeq 20602 . . . . 5  |-  ( ( A  e.  CC  /\  3  e.  NN  /\  1  e.  CC )  ->  (
( A ^ 3 )  =  1  <->  E. n  e.  ( 0 ... ( 3  -  1 ) ) A  =  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) ) ) )
2624, 2, 25mp3an23 1271 . . . 4  |-  ( A  e.  CC  ->  (
( A ^ 3 )  =  1  <->  E. n  e.  ( 0 ... ( 3  -  1 ) ) A  =  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) ) ) )
27 eltpg 3819 . . . . 5  |-  ( A  e.  CC  ->  ( A  e.  { 1 ,  ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) ,  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) }  <->  ( A  =  1  \/  A  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2
)  \/  A  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
) ) ) )
281eleq2i 2476 . . . . 5  |-  ( A  e.  R  <->  A  e.  { 1 ,  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) } )
29 3m1e2 10060 . . . . . . . . . 10  |-  ( 3  -  1 )  =  2
30 2cn 10034 . . . . . . . . . . 11  |-  2  e.  CC
3130addid2i 9218 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
3229, 31eqtr4i 2435 . . . . . . . . 9  |-  ( 3  -  1 )  =  ( 0  +  2 )
3332oveq2i 6059 . . . . . . . 8  |-  ( 0 ... ( 3  -  1 ) )  =  ( 0 ... (
0  +  2 ) )
34 0z 10257 . . . . . . . . 9  |-  0  e.  ZZ
35 fztp 11066 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  (
0 ... ( 0  +  2 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } )
3634, 35ax-mp 8 . . . . . . . 8  |-  ( 0 ... ( 0  +  2 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
3733, 36eqtri 2432 . . . . . . 7  |-  ( 0 ... ( 3  -  1 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
3837rexeqi 2877 . . . . . 6  |-  ( E. n  e.  ( 0 ... ( 3  -  1 ) ) A  =  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  E. n  e.  {
0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } A  =  ( ( 1  ^ c  ( 1  / 
3 ) )  x.  ( ( -u 1  ^ c  ( 2  /  3 ) ) ^ n ) ) )
39 3ne0 10049 . . . . . . . . . . 11  |-  3  =/=  0
405, 39reccli 9708 . . . . . . . . . 10  |-  ( 1  /  3 )  e.  CC
41 1cxp 20524 . . . . . . . . . 10  |-  ( ( 1  /  3 )  e.  CC  ->  (
1  ^ c  ( 1  /  3 ) )  =  1 )
4240, 41ax-mp 8 . . . . . . . . 9  |-  ( 1  ^ c  ( 1  /  3 ) )  =  1
4342oveq1i 6058 . . . . . . . 8  |-  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )
4443eqeq2i 2422 . . . . . . 7  |-  ( A  =  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) ) )
4544rexbii 2699 . . . . . 6  |-  ( E. n  e.  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } A  =  ( ( 1  ^ c 
( 1  /  3
) )  x.  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
) )  <->  E. n  e.  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } A  =  ( 1  x.  ( ( -u 1  ^ c  ( 2  /  3 ) ) ^ n ) ) )
4634elexi 2933 . . . . . . 7  |-  0  e.  _V
47 ovex 6073 . . . . . . 7  |-  ( 0  +  1 )  e. 
_V
48 ovex 6073 . . . . . . 7  |-  ( 0  +  2 )  e. 
_V
49 oveq2 6056 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 0 ) )
5030, 5, 39divcli 9720 . . . . . . . . . . . . 13  |-  ( 2  /  3 )  e.  CC
51 cxpcl 20526 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^ c  ( 2  /  3 ) )  e.  CC )
523, 50, 51mp2an 654 . . . . . . . . . . . 12  |-  ( -u
1  ^ c  ( 2  /  3 ) )  e.  CC
53 exp0 11349 . . . . . . . . . . . 12  |-  ( (
-u 1  ^ c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^ c  ( 2  /  3 ) ) ^ 0 )  =  1 )
5452, 53ax-mp 8 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 0 )  =  1
5549, 54syl6eq 2460 . . . . . . . . . 10  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  1 )
5655oveq2d 6064 . . . . . . . . 9  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  1 ) )
57 1t1e1 10090 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
5856, 57syl6eq 2460 . . . . . . . 8  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  1 )
5958eqeq2d 2423 . . . . . . 7  |-  ( n  =  0  ->  ( A  =  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  1
) )
60 id 20 . . . . . . . . . . . . 13  |-  ( n  =  ( 0  +  1 )  ->  n  =  ( 0  +  1 ) )
612addid2i 9218 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
6260, 61syl6eq 2460 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  1 )  ->  n  =  1 )
6362oveq2d 6064 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 ) )
64 exp1 11350 . . . . . . . . . . . 12  |-  ( (
-u 1  ^ c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^ c  ( 2  /  3 ) ) ^ 1 )  =  ( -u 1  ^ c  ( 2  / 
3 ) ) )
6552, 64ax-mp 8 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 )  =  ( -u
1  ^ c  ( 2  /  3 ) )
6663, 65syl6eq 2460 . . . . . . . . . 10  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( -u
1  ^ c  ( 2  /  3 ) ) )
6766oveq2d 6064 . . . . . . . . 9  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( -u
1  ^ c  ( 2  /  3 ) ) ) )
6852mulid2i 9057 . . . . . . . . . 10  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( -u
1  ^ c  ( 2  /  3 ) )
69 1cubrlem 20642 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  /\  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 2 )  =  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )
7069simpli 445 . . . . . . . . . 10  |-  ( -u
1  ^ c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7168, 70eqtri 2432 . . . . . . . . 9  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7267, 71syl6eq 2460 . . . . . . . 8  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
7372eqeq2d 2423 . . . . . . 7  |-  ( n  =  ( 0  +  1 )  ->  ( A  =  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ) )
74 id 20 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  2 )  ->  n  =  ( 0  +  2 ) )
7574, 31syl6eq 2460 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  2 )  ->  n  =  2 )
7675oveq2d 6064 . . . . . . . . . 10  |-  ( n  =  ( 0  +  2 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) )
7776oveq2d 6064 . . . . . . . . 9  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) ) )
7852sqcli 11425 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )  e.  CC
7978mulid2i 9057 . . . . . . . . . 10  |-  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )
8069simpri 449 . . . . . . . . . 10  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8179, 80eqtri 2432 . . . . . . . . 9  |-  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8277, 81syl6eq 2460 . . . . . . . 8  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
8382eqeq2d 2423 . . . . . . 7  |-  ( n  =  ( 0  +  2 )  ->  ( A  =  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 ) ) )
8446, 47, 48, 59, 73, 83rextp 3832 . . . . . 6  |-  ( E. n  e.  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } A  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  <->  ( A  =  1  \/  A  =  ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 )  \/  A  =  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) ) )
8538, 45, 843bitri 263 . . . . 5  |-  ( E. n  e.  ( 0 ... ( 3  -  1 ) ) A  =  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  ( A  =  1  \/  A  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2
)  \/  A  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
) ) )
8627, 28, 853bitr4g 280 . . . 4  |-  ( A  e.  CC  ->  ( A  e.  R  <->  E. n  e.  ( 0 ... (
3  -  1 ) ) A  =  ( ( 1  ^ c 
( 1  /  3
) )  x.  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
) ) ) )
8726, 86bitr4d 248 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 3 )  =  1  <->  A  e.  R ) )
8887pm5.32i 619 . 2  |-  ( ( A  e.  CC  /\  ( A ^ 3 )  =  1 )  <->  ( A  e.  CC  /\  A  e.  R ) )
8923, 88bitr4i 244 1  |-  ( A  e.  R  <->  ( A  e.  CC  /\  ( A ^ 3 )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675    C_ wss 3288   {ctp 3784   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954   1c1 8955   _ici 8956    + caddc 8957    x. cmul 8959    - cmin 9255   -ucneg 9256    / cdiv 9641   NNcn 9964   2c2 10013   3c3 10014   ZZcz 10246   ...cfz 11007   ^cexp 11345   sqrcsqr 12001    ^ c ccxp 20414
This theorem is referenced by:  cubic  20650
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ioc 10885  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633  df-sin 12635  df-cos 12636  df-pi 12638  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-limc 19714  df-dv 19715  df-log 20415  df-cxp 20416
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