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Theorem 1cubr 20542
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 8974 . . . . . . 7  |-  1  e.  CC
3 neg1cn 9992 . . . . . . . . 9  |-  -u 1  e.  CC
4 ax-icn 8975 . . . . . . . . . 10  |-  _i  e.  CC
5 3cn 9997 . . . . . . . . . . 11  |-  3  e.  CC
6 sqrcl 12085 . . . . . . . . . . 11  |-  ( 3  e.  CC  ->  ( sqr `  3 )  e.  CC )
75, 6ax-mp 8 . . . . . . . . . 10  |-  ( sqr `  3 )  e.  CC
84, 7mulcli 9021 . . . . . . . . 9  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
93, 8addcli 9020 . . . . . . . 8  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
10 halfcl 10118 . . . . . . . 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 9301 . . . . . . . 8  |-  ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  e.  CC
13 halfcl 10118 . . . . . . . 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 2901 . . . . . . 7  |-  1  e.  _V
17 ovex 6038 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  _V
18 ovex 6038 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e. 
_V
1916, 17, 18tpss 3900 . . . . . 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 3314 . . . 4  |-  R  C_  CC
2221sseli 3280 . . 3  |-  ( A  e.  R  ->  A  e.  CC )
2322pm4.71ri 615 . 2  |-  ( A  e.  R  <->  ( A  e.  CC  /\  A  e.  R ) )
24 3nn 10059 . . . . 5  |-  3  e.  NN
25 cxpeq 20501 . . . . 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 3787 . . . . 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 2444 . . . . 5  |-  ( A  e.  R  <->  A  e.  { 1 ,  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) } )
29 3m1e2 10021 . . . . . . . . . 10  |-  ( 3  -  1 )  =  2
30 2cn 9995 . . . . . . . . . . 11  |-  2  e.  CC
3130addid2i 9179 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
3229, 31eqtr4i 2403 . . . . . . . . 9  |-  ( 3  -  1 )  =  ( 0  +  2 )
3332oveq2i 6024 . . . . . . . 8  |-  ( 0 ... ( 3  -  1 ) )  =  ( 0 ... (
0  +  2 ) )
34 0z 10218 . . . . . . . . 9  |-  0  e.  ZZ
35 fztp 11027 . . . . . . . . 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 2400 . . . . . . 7  |-  ( 0 ... ( 3  -  1 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
3837rexeqi 2845 . . . . . 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 10010 . . . . . . . . . . 11  |-  3  =/=  0
405, 39reccli 9669 . . . . . . . . . 10  |-  ( 1  /  3 )  e.  CC
41 1cxp 20423 . . . . . . . . . 10  |-  ( ( 1  /  3 )  e.  CC  ->  (
1  ^ c  ( 1  /  3 ) )  =  1 )
4240, 41ax-mp 8 . . . . . . . . 9  |-  ( 1  ^ c  ( 1  /  3 ) )  =  1
4342oveq1i 6023 . . . . . . . 8  |-  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )
4443eqeq2i 2390 . . . . . . 7  |-  ( A  =  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) ) )
4544rexbii 2667 . . . . . 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 2901 . . . . . . 7  |-  0  e.  _V
47 ovex 6038 . . . . . . 7  |-  ( 0  +  1 )  e. 
_V
48 ovex 6038 . . . . . . 7  |-  ( 0  +  2 )  e. 
_V
49 oveq2 6021 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 0 ) )
5030, 5, 39divcli 9681 . . . . . . . . . . . . 13  |-  ( 2  /  3 )  e.  CC
51 cxpcl 20425 . . . . . . . . . . . . 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 11306 . . . . . . . . . . . 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 2428 . . . . . . . . . 10  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  1 )
5655oveq2d 6029 . . . . . . . . 9  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  1 ) )
57 1t1e1 10051 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
5856, 57syl6eq 2428 . . . . . . . 8  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  1 )
5958eqeq2d 2391 . . . . . . 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 9179 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
6260, 61syl6eq 2428 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  1 )  ->  n  =  1 )
6362oveq2d 6029 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 ) )
64 exp1 11307 . . . . . . . . . . . 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 2428 . . . . . . . . . 10  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( -u
1  ^ c  ( 2  /  3 ) ) )
6766oveq2d 6029 . . . . . . . . 9  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( -u
1  ^ c  ( 2  /  3 ) ) ) )
6852mulid2i 9019 . . . . . . . . . 10  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( -u
1  ^ c  ( 2  /  3 ) )
69 1cubrlem 20541 . . . . . . . . . . 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 2400 . . . . . . . . 9  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7267, 71syl6eq 2428 . . . . . . . 8  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
7372eqeq2d 2391 . . . . . . 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 2428 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  2 )  ->  n  =  2 )
7675oveq2d 6029 . . . . . . . . . 10  |-  ( n  =  ( 0  +  2 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) )
7776oveq2d 6029 . . . . . . . . 9  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) ) )
7852sqcli 11382 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )  e.  CC
7978mulid2i 9019 . . . . . . . . . 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 2400 . . . . . . . . 9  |-  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8277, 81syl6eq 2428 . . . . . . . 8  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
8382eqeq2d 2391 . . . . . . 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 3800 . . . . . 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 1717   E.wrex 2643    C_ wss 3256   {ctp 3752   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917   _ici 8918    + caddc 8919    x. cmul 8921    - cmin 9216   -ucneg 9217    / cdiv 9602   NNcn 9925   2c2 9974   3c3 9975   ZZcz 10207   ...cfz 10968   ^cexp 11302   sqrcsqr 11958    ^ c ccxp 20313
This theorem is referenced by:  cubic  20549
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-fi 7344  df-sup 7374  df-oi 7405  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ioo 10845  df-ioc 10846  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-shft 11802  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-limsup 12185  df-clim 12202  df-rlim 12203  df-sum 12400  df-ef 12590  df-sin 12592  df-cos 12593  df-pi 12595  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-hom 13473  df-cco 13474  df-rest 13570  df-topn 13571  df-topgen 13587  df-pt 13588  df-prds 13591  df-xrs 13646  df-0g 13647  df-gsum 13648  df-qtop 13653  df-imas 13654  df-xps 13656  df-mre 13731  df-mrc 13732  df-acs 13734  df-mnd 14610  df-submnd 14659  df-mulg 14735  df-cntz 15036  df-cmn 15334  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-fbas 16616  df-fg 16617  df-cnfld 16620  df-top 16879  df-bases 16881  df-topon 16882  df-topsp 16883  df-cld 16999  df-ntr 17000  df-cls 17001  df-nei 17078  df-lp 17116  df-perf 17117  df-cn 17206  df-cnp 17207  df-haus 17294  df-tx 17508  df-hmeo 17701  df-fil 17792  df-fm 17884  df-flim 17885  df-flf 17886  df-xms 18252  df-ms 18253  df-tms 18254  df-cncf 18772  df-limc 19613  df-dv 19614  df-log 20314  df-cxp 20315
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