MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1cubr Unicode version

Theorem 1cubr 20138
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 8795 . . . . . . 7  |-  1  e.  CC
3 neg1cn 9813 . . . . . . . . 9  |-  -u 1  e.  CC
4 ax-icn 8796 . . . . . . . . . 10  |-  _i  e.  CC
5 3cn 9818 . . . . . . . . . . 11  |-  3  e.  CC
6 sqrcl 11845 . . . . . . . . . . 11  |-  ( 3  e.  CC  ->  ( sqr `  3 )  e.  CC )
75, 6ax-mp 8 . . . . . . . . . 10  |-  ( sqr `  3 )  e.  CC
84, 7mulcli 8842 . . . . . . . . 9  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
93, 8addcli 8841 . . . . . . . 8  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
10 halfcl 9937 . . . . . . . 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 9122 . . . . . . . 8  |-  ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  e.  CC
13 halfcl 9937 . . . . . . . 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 2797 . . . . . . 7  |-  1  e.  _V
17 ovex 5883 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  _V
18 ovex 5883 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e. 
_V
1916, 17, 18tpss 3779 . . . . . 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 3208 . . . 4  |-  R  C_  CC
2221sseli 3176 . . 3  |-  ( A  e.  R  ->  A  e.  CC )
2322pm4.71ri 614 . 2  |-  ( A  e.  R  <->  ( A  e.  CC  /\  A  e.  R ) )
24 3nn 9878 . . . . 5  |-  3  e.  NN
25 cxpeq 20097 . . . . 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 3676 . . . . 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 2347 . . . . 5  |-  ( A  e.  R  <->  A  e.  { 1 ,  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) } )
29 2cn 9816 . . . . . . . . . . 11  |-  2  e.  CC
30 2p1e3 9847 . . . . . . . . . . . 12  |-  ( 2  +  1 )  =  3
3129, 2, 30addcomli 9004 . . . . . . . . . . 11  |-  ( 1  +  2 )  =  3
325, 2, 29, 31subaddrii 9135 . . . . . . . . . 10  |-  ( 3  -  1 )  =  2
3329addid2i 9000 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
3432, 33eqtr4i 2306 . . . . . . . . 9  |-  ( 3  -  1 )  =  ( 0  +  2 )
3534oveq2i 5869 . . . . . . . 8  |-  ( 0 ... ( 3  -  1 ) )  =  ( 0 ... (
0  +  2 ) )
36 0z 10035 . . . . . . . . 9  |-  0  e.  ZZ
37 fztp 10841 . . . . . . . . 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 2303 . . . . . . 7  |-  ( 0 ... ( 3  -  1 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
4039rexeqi 2741 . . . . . 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 9831 . . . . . . . . . . 11  |-  3  =/=  0
425, 41reccli 9490 . . . . . . . . . 10  |-  ( 1  /  3 )  e.  CC
43 1cxp 20019 . . . . . . . . . 10  |-  ( ( 1  /  3 )  e.  CC  ->  (
1  ^ c  ( 1  /  3 ) )  =  1 )
4442, 43ax-mp 8 . . . . . . . . 9  |-  ( 1  ^ c  ( 1  /  3 ) )  =  1
4544oveq1i 5868 . . . . . . . 8  |-  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )
4645eqeq2i 2293 . . . . . . 7  |-  ( A  =  ( ( 1  ^ c  ( 1  /  3 ) )  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) ) )
4746rexbii 2568 . . . . . 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 2797 . . . . . . 7  |-  0  e.  _V
49 ovex 5883 . . . . . . 7  |-  ( 0  +  1 )  e. 
_V
50 ovex 5883 . . . . . . 7  |-  ( 0  +  2 )  e. 
_V
51 oveq2 5866 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 0 ) )
5229, 5, 41divcli 9502 . . . . . . . . . . . . 13  |-  ( 2  /  3 )  e.  CC
53 cxpcl 20021 . . . . . . . . . . . . 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 11108 . . . . . . . . . . . 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 2331 . . . . . . . . . 10  |-  ( n  =  0  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  1 )
5857oveq2d 5874 . . . . . . . . 9  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  1 ) )
59 1t1e1 9870 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
6058, 59syl6eq 2331 . . . . . . . 8  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  1 )
6160eqeq2d 2294 . . . . . . 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 9000 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
6462, 63syl6eq 2331 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  1 )  ->  n  =  1 )
6564oveq2d 5874 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 ) )
66 exp1 11109 . . . . . . . . . . . 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 2331 . . . . . . . . . 10  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( -u
1  ^ c  ( 2  /  3 ) ) )
6968oveq2d 5874 . . . . . . . . 9  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( -u
1  ^ c  ( 2  /  3 ) ) ) )
7054mulid2i 8840 . . . . . . . . . 10  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( -u
1  ^ c  ( 2  /  3 ) )
71 1cubrlem 20137 . . . . . . . . . . 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 2303 . . . . . . . . 9  |-  ( 1  x.  ( -u 1  ^ c  ( 2  /  3 ) ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7469, 73syl6eq 2331 . . . . . . . 8  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
7574eqeq2d 2294 . . . . . . 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 2331 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  2 )  ->  n  =  2 )
7877oveq2d 5874 . . . . . . . . . 10  |-  ( n  =  ( 0  +  2 )  ->  (
( -u 1  ^ c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) )
7978oveq2d 5874 . . . . . . . . 9  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 ) ) )
8054sqcli 11184 . . . . . . . . . . 11  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )  e.  CC
8180mulid2i 8840 . . . . . . . . . 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 2303 . . . . . . . . 9  |-  ( 1  x.  ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8479, 83syl6eq 2331 . . . . . . . 8  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
8584eqeq2d 2294 . . . . . . 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 3689 . . . . . 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 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   {ctp 3642   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   3c3 9796   ZZcz 10024   ...cfz 10782   ^cexp 11104   sqrcsqr 11718    ^ c ccxp 19913
This theorem is referenced by:  cubic  20145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
  Copyright terms: Public domain W3C validator