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Theorem 1cubr 20713
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 9079 . . . . . . 7  |-  1  e.  CC
3 neg1cn 10098 . . . . . . . . 9  |-  -u 1  e.  CC
4 ax-icn 9080 . . . . . . . . . 10  |-  _i  e.  CC
5 3cn 10103 . . . . . . . . . . 11  |-  3  e.  CC
6 sqrcl 12196 . . . . . . . . . . 11  |-  ( 3  e.  CC  ->  ( sqr `  3 )  e.  CC )
75, 6ax-mp 5 . . . . . . . . . 10  |-  ( sqr `  3 )  e.  CC
84, 7mulcli 9126 . . . . . . . . 9  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
93, 8addcli 9125 . . . . . . . 8  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
10 halfcl 10224 . . . . . . . 8  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  e.  CC  ->  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
119, 10ax-mp 5 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC
123, 8subcli 9407 . . . . . . . 8  |-  ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  e.  CC
13 halfcl 10224 . . . . . . . 8  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  e.  CC  ->  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
1412, 13ax-mp 5 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC
152, 11, 143pm3.2i 1133 . . . . . 6  |-  ( 1  e.  CC  /\  (
( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC  /\  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
162elexi 2971 . . . . . . 7  |-  1  e.  _V
17 ovex 6135 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  _V
18 ovex 6135 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e. 
_V
1916, 17, 18tpss 3988 . . . . . 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 201 . . . . 5  |-  { 1 ,  ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) ,  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) }  C_  CC
211, 20eqsstri 3364 . . . 4  |-  R  C_  CC
2221sseli 3330 . . 3  |-  ( A  e.  R  ->  A  e.  CC )
2322pm4.71ri 616 . 2  |-  ( A  e.  R  <->  ( A  e.  CC  /\  A  e.  R ) )
24 3nn 10165 . . . . 5  |-  3  e.  NN
25 cxpeq 20672 . . . . 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 1272 . . . 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 3875 . . . . 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 2506 . . . . 5  |-  ( A  e.  R  <->  A  e.  { 1 ,  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) } )
29 3m1e2 10127 . . . . . . . . . 10  |-  ( 3  -  1 )  =  2
30 2cn 10101 . . . . . . . . . . 11  |-  2  e.  CC
3130addid2i 9285 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
3229, 31eqtr4i 2465 . . . . . . . . 9  |-  ( 3  -  1 )  =  ( 0  +  2 )
3332oveq2i 6121 . . . . . . . 8  |-  ( 0 ... ( 3  -  1 ) )  =  ( 0 ... (
0  +  2 ) )
34 0z 10324 . . . . . . . . 9  |-  0  e.  ZZ
35 fztp 11133 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  (
0 ... ( 0  +  2 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } )
3634, 35ax-mp 5 . . . . . . . 8  |-  ( 0 ... ( 0  +  2 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
3733, 36eqtri 2462 . . . . . . 7  |-  ( 0 ... ( 3  -  1 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
3837rexeqi 2915 . . . . . 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 10116 . . . . . . . . . . 11  |-  3  =/=  0
405, 39reccli 9775 . . . . . . . . . 10  |-  ( 1  /  3 )  e.  CC
41 1cxp 20594 . . . . . . . . . 10  |-  ( ( 1  /  3 )  e.  CC  ->  (
1  ^ c  ( 1  /  3 ) )  =  1 )
4240, 41ax-mp 5 . . . . . . . . 9  |-  ( 1  ^ c  ( 1  /  3 ) )  =  1
4342oveq1i 6120 . . . . . . . 8  |-  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )
4443eqeq2i 2452 . . . . . . 7  |-  ( A  =  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) ) )
4544rexbii 2736 . . . . . 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 2971 . . . . . . 7  |-  0  e.  _V
47 ovex 6135 . . . . . . 7  |-  ( 0  +  1 )  e. 
_V
48 ovex 6135 . . . . . . 7  |-  ( 0  +  2 )  e. 
_V
49 oveq2 6118 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 0 ) )
5030, 5, 39divcli 9787 . . . . . . . . . . . . 13  |-  ( 2  /  3 )  e.  CC
51 cxpcl 20596 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^ c  ( 2  /  3 ) )  e.  CC )
523, 50, 51mp2an 655 . . . . . . . . . . . 12  |-  ( -u
1  ^ c  ( 2  /  3 ) )  e.  CC
53 exp0 11417 . . . . . . . . . . . 12  |-  ( (
-u 1  ^ c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^ c  ( 2  /  3 ) ) ^ 0 )  =  1 )
5452, 53ax-mp 5 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 0 )  =  1
5549, 54syl6eq 2490 . . . . . . . . . 10  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  1 )
5655oveq2d 6126 . . . . . . . . 9  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  1 ) )
57 1t1e1 10157 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
5856, 57syl6eq 2490 . . . . . . . 8  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  1 )
5958eqeq2d 2453 . . . . . . 7  |-  ( n  =  0  ->  ( A  =  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  1
) )
60 id 21 . . . . . . . . . . . . 13  |-  ( n  =  ( 0  +  1 )  ->  n  =  ( 0  +  1 ) )
612addid2i 9285 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
6260, 61syl6eq 2490 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  1 )  ->  n  =  1 )
6362oveq2d 6126 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 ) )
64 exp1 11418 . . . . . . . . . . . 12  |-  ( (
-u 1  ^ c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^ c  ( 2  /  3 ) ) ^ 1 )  =  ( -u 1  ^ c  ( 2  / 
3 ) ) )
6552, 64ax-mp 5 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 )  =  ( -u
1  ^ c  ( 2  /  3 ) )
6663, 65syl6eq 2490 . . . . . . . . . 10  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( -u
1  ^ c  ( 2  /  3 ) ) )
6766oveq2d 6126 . . . . . . . . 9  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( -u
1  ^ c  ( 2  /  3 ) ) ) )
6852mulid2i 9124 . . . . . . . . . 10  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( -u
1  ^ c  ( 2  /  3 ) )
69 1cubrlem 20712 . . . . . . . . . . 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 446 . . . . . . . . . 10  |-  ( -u
1  ^ c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7168, 70eqtri 2462 . . . . . . . . 9  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7267, 71syl6eq 2490 . . . . . . . 8  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
7372eqeq2d 2453 . . . . . . 7  |-  ( n  =  ( 0  +  1 )  ->  ( A  =  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ) )
74 id 21 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  2 )  ->  n  =  ( 0  +  2 ) )
7574, 31syl6eq 2490 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  2 )  ->  n  =  2 )
7675oveq2d 6126 . . . . . . . . . 10  |-  ( n  =  ( 0  +  2 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) )
7776oveq2d 6126 . . . . . . . . 9  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) ) )
7852sqcli 11493 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )  e.  CC
7978mulid2i 9124 . . . . . . . . . 10  |-  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )
8069simpri 450 . . . . . . . . . 10  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8179, 80eqtri 2462 . . . . . . . . 9  |-  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8277, 81syl6eq 2490 . . . . . . . 8  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
8382eqeq2d 2453 . . . . . . 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 3888 . . . . . 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 264 . . . . 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 281 . . . 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 249 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 3 )  =  1  <->  A  e.  R ) )
8887pm5.32i 620 . 2  |-  ( ( A  e.  CC  /\  ( A ^ 3 )  =  1 )  <->  ( A  e.  CC  /\  A  e.  R ) )
8923, 88bitr4i 245 1  |-  ( A  e.  R  <->  ( A  e.  CC  /\  ( A ^ 3 )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    \/ w3o 936    /\ w3a 937    = wceq 1653    e. wcel 1727   E.wrex 2712    C_ wss 3306   {ctp 3840   ` cfv 5483  (class class class)co 6110   CCcc 9019   0cc0 9021   1c1 9022   _ici 9023    + caddc 9024    x. cmul 9026    - cmin 9322   -ucneg 9323    / cdiv 9708   NNcn 10031   2c2 10080   3c3 10081   ZZcz 10313   ...cfz 11074   ^cexp 11413   sqrcsqr 12069    ^ c ccxp 20484
This theorem is referenced by:  cubic  20720
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 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099  ax-addf 9100  ax-mulf 9101
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-pm 7050  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-fi 7445  df-sup 7475  df-oi 7508  df-card 7857  df-cda 8079  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-q 10606  df-rp 10644  df-xneg 10741  df-xadd 10742  df-xmul 10743  df-ioo 10951  df-ioc 10952  df-ico 10953  df-icc 10954  df-fz 11075  df-fzo 11167  df-fl 11233  df-mod 11282  df-seq 11355  df-exp 11414  df-fac 11598  df-bc 11625  df-hash 11650  df-shft 11913  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-limsup 12296  df-clim 12313  df-rlim 12314  df-sum 12511  df-ef 12701  df-sin 12703  df-cos 12704  df-pi 12706  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-starv 13575  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-unif 13583  df-hom 13584  df-cco 13585  df-rest 13681  df-topn 13682  df-topgen 13698  df-pt 13699  df-prds 13702  df-xrs 13757  df-0g 13758  df-gsum 13759  df-qtop 13764  df-imas 13765  df-xps 13767  df-mre 13842  df-mrc 13843  df-acs 13845  df-mnd 14721  df-submnd 14770  df-mulg 14846  df-cntz 15147  df-cmn 15445  df-psmet 16725  df-xmet 16726  df-met 16727  df-bl 16728  df-mopn 16729  df-fbas 16730  df-fg 16731  df-cnfld 16735  df-top 16994  df-bases 16996  df-topon 16997  df-topsp 16998  df-cld 17114  df-ntr 17115  df-cls 17116  df-nei 17193  df-lp 17231  df-perf 17232  df-cn 17322  df-cnp 17323  df-haus 17410  df-tx 17625  df-hmeo 17818  df-fil 17909  df-fm 18001  df-flim 18002  df-flf 18003  df-xms 18381  df-ms 18382  df-tms 18383  df-cncf 18939  df-limc 19784  df-dv 19785  df-log 20485  df-cxp 20486
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