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Theorem 1cubr 20249
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 8885 . . . . . . 7  |-  1  e.  CC
3 neg1cn 9903 . . . . . . . . 9  |-  -u 1  e.  CC
4 ax-icn 8886 . . . . . . . . . 10  |-  _i  e.  CC
5 3cn 9908 . . . . . . . . . . 11  |-  3  e.  CC
6 sqrcl 11941 . . . . . . . . . . 11  |-  ( 3  e.  CC  ->  ( sqr `  3 )  e.  CC )
75, 6ax-mp 8 . . . . . . . . . 10  |-  ( sqr `  3 )  e.  CC
84, 7mulcli 8932 . . . . . . . . 9  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
93, 8addcli 8931 . . . . . . . 8  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
10 halfcl 10029 . . . . . . . 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 9212 . . . . . . . 8  |-  ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  e.  CC
13 halfcl 10029 . . . . . . . 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 1130 . . . . . 6  |-  ( 1  e.  CC  /\  (
( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC  /\  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
162elexi 2873 . . . . . . 7  |-  1  e.  _V
17 ovex 5970 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  _V
18 ovex 5970 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e. 
_V
1916, 17, 18tpss 3860 . . . . . 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 199 . . . . 5  |-  { 1 ,  ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) ,  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) }  C_  CC
211, 20eqsstri 3284 . . . 4  |-  R  C_  CC
2221sseli 3252 . . 3  |-  ( A  e.  R  ->  A  e.  CC )
2322pm4.71ri 614 . 2  |-  ( A  e.  R  <->  ( A  e.  CC  /\  A  e.  R ) )
24 3nn 9970 . . . . 5  |-  3  e.  NN
25 cxpeq 20208 . . . . 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 1269 . . . 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 3752 . . . . 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 2422 . . . . 5  |-  ( A  e.  R  <->  A  e.  { 1 ,  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) } )
29 2cn 9906 . . . . . . . . . . 11  |-  2  e.  CC
30 2p1e3 9939 . . . . . . . . . . . 12  |-  ( 2  +  1 )  =  3
3129, 2, 30addcomli 9094 . . . . . . . . . . 11  |-  ( 1  +  2 )  =  3
325, 2, 29, 31subaddrii 9225 . . . . . . . . . 10  |-  ( 3  -  1 )  =  2
3329addid2i 9090 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
3432, 33eqtr4i 2381 . . . . . . . . 9  |-  ( 3  -  1 )  =  ( 0  +  2 )
3534oveq2i 5956 . . . . . . . 8  |-  ( 0 ... ( 3  -  1 ) )  =  ( 0 ... (
0  +  2 ) )
36 0z 10127 . . . . . . . . 9  |-  0  e.  ZZ
37 fztp 10933 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  (
0 ... ( 0  +  2 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } )
3836, 37ax-mp 8 . . . . . . . 8  |-  ( 0 ... ( 0  +  2 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
3935, 38eqtri 2378 . . . . . . 7  |-  ( 0 ... ( 3  -  1 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
4039rexeqi 2817 . . . . . 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 ) ) )
41 3ne0 9921 . . . . . . . . . . 11  |-  3  =/=  0
425, 41reccli 9580 . . . . . . . . . 10  |-  ( 1  /  3 )  e.  CC
43 1cxp 20130 . . . . . . . . . 10  |-  ( ( 1  /  3 )  e.  CC  ->  (
1  ^ c  ( 1  /  3 ) )  =  1 )
4442, 43ax-mp 8 . . . . . . . . 9  |-  ( 1  ^ c  ( 1  /  3 ) )  =  1
4544oveq1i 5955 . . . . . . . 8  |-  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )
4645eqeq2i 2368 . . . . . . 7  |-  ( A  =  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) ) )
4746rexbii 2644 . . . . . 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 ) ) )
4836elexi 2873 . . . . . . 7  |-  0  e.  _V
49 ovex 5970 . . . . . . 7  |-  ( 0  +  1 )  e. 
_V
50 ovex 5970 . . . . . . 7  |-  ( 0  +  2 )  e. 
_V
51 oveq2 5953 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 0 ) )
5229, 5, 41divcli 9592 . . . . . . . . . . . . 13  |-  ( 2  /  3 )  e.  CC
53 cxpcl 20132 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^ c  ( 2  /  3 ) )  e.  CC )
543, 52, 53mp2an 653 . . . . . . . . . . . 12  |-  ( -u
1  ^ c  ( 2  /  3 ) )  e.  CC
55 exp0 11201 . . . . . . . . . . . 12  |-  ( (
-u 1  ^ c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^ c  ( 2  /  3 ) ) ^ 0 )  =  1 )
5654, 55ax-mp 8 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 0 )  =  1
5751, 56syl6eq 2406 . . . . . . . . . 10  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  1 )
5857oveq2d 5961 . . . . . . . . 9  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  1 ) )
59 1t1e1 9962 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
6058, 59syl6eq 2406 . . . . . . . 8  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  1 )
6160eqeq2d 2369 . . . . . . 7  |-  ( n  =  0  ->  ( A  =  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  1
) )
62 id 19 . . . . . . . . . . . . 13  |-  ( n  =  ( 0  +  1 )  ->  n  =  ( 0  +  1 ) )
632addid2i 9090 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
6462, 63syl6eq 2406 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  1 )  ->  n  =  1 )
6564oveq2d 5961 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 ) )
66 exp1 11202 . . . . . . . . . . . 12  |-  ( (
-u 1  ^ c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^ c  ( 2  /  3 ) ) ^ 1 )  =  ( -u 1  ^ c  ( 2  / 
3 ) ) )
6754, 66ax-mp 8 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 )  =  ( -u
1  ^ c  ( 2  /  3 ) )
6865, 67syl6eq 2406 . . . . . . . . . 10  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( -u
1  ^ c  ( 2  /  3 ) ) )
6968oveq2d 5961 . . . . . . . . 9  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( -u
1  ^ c  ( 2  /  3 ) ) ) )
7054mulid2i 8930 . . . . . . . . . 10  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( -u
1  ^ c  ( 2  /  3 ) )
71 1cubrlem 20248 . . . . . . . . . . 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 ) )
7271simpli 444 . . . . . . . . . 10  |-  ( -u
1  ^ c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7370, 72eqtri 2378 . . . . . . . . 9  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7469, 73syl6eq 2406 . . . . . . . 8  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
7574eqeq2d 2369 . . . . . . 7  |-  ( n  =  ( 0  +  1 )  ->  ( A  =  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ) )
76 id 19 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  2 )  ->  n  =  ( 0  +  2 ) )
7776, 33syl6eq 2406 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  2 )  ->  n  =  2 )
7877oveq2d 5961 . . . . . . . . . 10  |-  ( n  =  ( 0  +  2 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) )
7978oveq2d 5961 . . . . . . . . 9  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) ) )
8054sqcli 11277 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )  e.  CC
8180mulid2i 8930 . . . . . . . . . 10  |-  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )
8271simpri 448 . . . . . . . . . 10  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8381, 82eqtri 2378 . . . . . . . . 9  |-  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8479, 83syl6eq 2406 . . . . . . . 8  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
8584eqeq2d 2369 . . . . . . 7  |-  ( n  =  ( 0  +  2 )  ->  ( A  =  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 ) ) )
8648, 49, 50, 61, 75, 85rextp 3765 . . . . . 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 ) ) )
8740, 47, 863bitri 262 . . . . 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
) ) )
8827, 28, 873bitr4g 279 . . . 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
) ) ) )
8926, 88bitr4d 247 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 3 )  =  1  <->  A  e.  R ) )
9089pm5.32i 618 . 2  |-  ( ( A  e.  CC  /\  ( A ^ 3 )  =  1 )  <->  ( A  e.  CC  /\  A  e.  R ) )
9123, 90bitr4i 243 1  |-  ( A  e.  R  <->  ( A  e.  CC  /\  ( A ^ 3 )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1642    e. wcel 1710   E.wrex 2620    C_ wss 3228   {ctp 3718   ` 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   NNcn 9836   2c2 9885   3c3 9886   ZZcz 10116   ...cfz 10874   ^cexp 11197   sqrcsqr 11814    ^ c ccxp 20020
This theorem is referenced by:  cubic  20256
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  df-log 20021  df-cxp 20022
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