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Theorem 1cubrlem 20137
Description: The cube roots of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
1cubrlem  |-  ( (
-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 ) )

Proof of Theorem 1cubrlem
StepHypRef Expression
1 neg1cn 9813 . . . 4  |-  -u 1  e.  CC
2 ax-1cn 8795 . . . . 5  |-  1  e.  CC
3 ax-1ne0 8806 . . . . 5  |-  1  =/=  0
42, 3negne0i 9121 . . . 4  |-  -u 1  =/=  0
5 2re 9815 . . . . . 6  |-  2  e.  RR
6 3nn 9878 . . . . . 6  |-  3  e.  NN
7 nndivre 9781 . . . . . 6  |-  ( ( 2  e.  RR  /\  3  e.  NN )  ->  ( 2  /  3
)  e.  RR )
85, 6, 7mp2an 653 . . . . 5  |-  ( 2  /  3 )  e.  RR
98recni 8849 . . . 4  |-  ( 2  /  3 )  e.  CC
10 cxpef 20012 . . . 4  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^ c  ( 2  /  3 ) )  =  ( exp `  (
( 2  /  3
)  x.  ( log `  -u 1 ) ) ) )
111, 4, 9, 10mp3an 1277 . . 3  |-  ( -u
1  ^ c  ( 2  /  3 ) )  =  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )
12 logm1 19942 . . . . . 6  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
1312oveq2i 5869 . . . . 5  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( ( 2  /  3
)  x.  ( _i  x.  pi ) )
14 ax-icn 8796 . . . . . 6  |-  _i  e.  CC
15 pire 19832 . . . . . . 7  |-  pi  e.  RR
1615recni 8849 . . . . . 6  |-  pi  e.  CC
179, 14, 16mul12i 9007 . . . . 5  |-  ( ( 2  /  3 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1813, 17eqtri 2303 . . . 4  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1918fveq2i 5528 . . 3  |-  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )  =  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )
20 6nn 9881 . . . . . . . . 9  |-  6  e.  NN
21 nndivre 9781 . . . . . . . . 9  |-  ( ( pi  e.  RR  /\  6  e.  NN )  ->  ( pi  /  6
)  e.  RR )
2215, 20, 21mp2an 653 . . . . . . . 8  |-  ( pi 
/  6 )  e.  RR
2322recni 8849 . . . . . . 7  |-  ( pi 
/  6 )  e.  CC
24 coshalfpip 19862 . . . . . . 7  |-  ( ( pi  /  6 )  e.  CC  ->  ( cos `  ( ( pi 
/  2 )  +  ( pi  /  6
) ) )  = 
-u ( sin `  (
pi  /  6 ) ) )
2523, 24ax-mp 8 . . . . . 6  |-  ( cos `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  -u ( sin `  ( pi 
/  6 ) )
26 2cn 9816 . . . . . . . . . 10  |-  2  e.  CC
27 2ne0 9829 . . . . . . . . . 10  |-  2  =/=  0
28 divrec2 9441 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
pi  /  2 )  =  ( ( 1  /  2 )  x.  pi ) )
2916, 26, 27, 28mp3an 1277 . . . . . . . . 9  |-  ( pi 
/  2 )  =  ( ( 1  / 
2 )  x.  pi )
3020nncni 9756 . . . . . . . . . 10  |-  6  e.  CC
3120nnne0i 9780 . . . . . . . . . 10  |-  6  =/=  0
32 divrec2 9441 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
pi  /  6 )  =  ( ( 1  /  6 )  x.  pi ) )
3316, 30, 31, 32mp3an 1277 . . . . . . . . 9  |-  ( pi 
/  6 )  =  ( ( 1  / 
6 )  x.  pi )
3429, 33oveq12i 5870 . . . . . . . 8  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
3526, 27reccli 9490 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
3630, 31reccli 9490 . . . . . . . . 9  |-  ( 1  /  6 )  e.  CC
3735, 36, 16adddiri 8848 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
38 halfpm6th 9936 . . . . . . . . . 10  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
3938simpri 448 . . . . . . . . 9  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
4039oveq1i 5868 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( 2  / 
3 )  x.  pi )
4134, 37, 403eqtr2i 2309 . . . . . . 7  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( 2  / 
3 )  x.  pi )
4241fveq2i 5528 . . . . . 6  |-  ( cos `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( cos `  ( ( 2  /  3 )  x.  pi ) )
43 sincos6thpi 19883 . . . . . . . . 9  |-  ( ( sin `  ( pi 
/  6 ) )  =  ( 1  / 
2 )  /\  ( cos `  ( pi  / 
6 ) )  =  ( ( sqr `  3
)  /  2 ) )
4443simpli 444 . . . . . . . 8  |-  ( sin `  ( pi  /  6
) )  =  ( 1  /  2 )
4544negeqi 9045 . . . . . . 7  |-  -u ( sin `  ( pi  / 
6 ) )  = 
-u ( 1  / 
2 )
46 divneg 9455 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
472, 26, 27, 46mp3an 1277 . . . . . . 7  |-  -u (
1  /  2 )  =  ( -u 1  /  2 )
4845, 47eqtri 2303 . . . . . 6  |-  -u ( sin `  ( pi  / 
6 ) )  =  ( -u 1  / 
2 )
4925, 42, 483eqtr3i 2311 . . . . 5  |-  ( cos `  ( ( 2  / 
3 )  x.  pi ) )  =  (
-u 1  /  2
)
50 sinhalfpip 19860 . . . . . . . . 9  |-  ( ( pi  /  6 )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  +  ( pi  /  6
) ) )  =  ( cos `  (
pi  /  6 ) ) )
5123, 50ax-mp 8 . . . . . . . 8  |-  ( sin `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( cos `  ( pi 
/  6 ) )
5241fveq2i 5528 . . . . . . . 8  |-  ( sin `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( sin `  ( ( 2  /  3 )  x.  pi ) )
5343simpri 448 . . . . . . . 8  |-  ( cos `  ( pi  /  6
) )  =  ( ( sqr `  3
)  /  2 )
5451, 52, 533eqtr3i 2311 . . . . . . 7  |-  ( sin `  ( ( 2  / 
3 )  x.  pi ) )  =  ( ( sqr `  3
)  /  2 )
5554oveq2i 5869 . . . . . 6  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( _i  x.  ( ( sqr `  3 )  /  2 ) )
56 3re 9817 . . . . . . . . 9  |-  3  e.  RR
57 3nn0 9983 . . . . . . . . . 10  |-  3  e.  NN0
5857nn0ge0i 9993 . . . . . . . . 9  |-  0  <_  3
59 resqrcl 11739 . . . . . . . . 9  |-  ( ( 3  e.  RR  /\  0  <_  3 )  -> 
( sqr `  3
)  e.  RR )
6056, 58, 59mp2an 653 . . . . . . . 8  |-  ( sqr `  3 )  e.  RR
6160recni 8849 . . . . . . 7  |-  ( sqr `  3 )  e.  CC
6214, 61, 26, 27divassi 9516 . . . . . 6  |-  ( ( _i  x.  ( sqr `  3 ) )  /  2 )  =  ( _i  x.  (
( sqr `  3
)  /  2 ) )
6355, 62eqtr4i 2306 . . . . 5  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( _i  x.  ( sqr `  3 ) )  /  2 )
6449, 63oveq12i 5870 . . . 4  |-  ( ( cos `  ( ( 2  /  3 )  x.  pi ) )  +  ( _i  x.  ( sin `  ( ( 2  /  3 )  x.  pi ) ) ) )  =  ( ( -u 1  / 
2 )  +  ( ( _i  x.  ( sqr `  3 ) )  /  2 ) )
659, 16mulcli 8842 . . . . 5  |-  ( ( 2  /  3 )  x.  pi )  e.  CC
66 efival 12432 . . . . 5  |-  ( ( ( 2  /  3
)  x.  pi )  e.  CC  ->  ( exp `  ( _i  x.  ( ( 2  / 
3 )  x.  pi ) ) )  =  ( ( cos `  (
( 2  /  3
)  x.  pi ) )  +  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) ) ) )
6765, 66ax-mp 8 . . . 4  |-  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( cos `  (
( 2  /  3
)  x.  pi ) )  +  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) ) )
6814, 61mulcli 8842 . . . . 5  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
691, 68, 26, 27divdiri 9517 . . . 4  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  =  ( ( -u
1  /  2 )  +  ( ( _i  x.  ( sqr `  3
) )  /  2
) )
7064, 67, 693eqtr4i 2313 . . 3  |-  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7111, 19, 703eqtri 2307 . 2  |-  ( -u
1  ^ c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
72 1z 10053 . . . 4  |-  1  e.  ZZ
73 root1cj 20096 . . . 4  |-  ( ( 3  e.  NN  /\  1  e.  ZZ )  ->  ( * `  (
( -u 1  ^ c 
( 2  /  3
) ) ^ 1 ) )  =  ( ( -u 1  ^ c  ( 2  / 
3 ) ) ^
( 3  -  1 ) ) )
746, 72, 73mp2an 653 . . 3  |-  ( * `
 ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ (
3  -  1 ) )
75 cxpcl 20021 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^ c  ( 2  /  3 ) )  e.  CC )
761, 9, 75mp2an 653 . . . . . . 7  |-  ( -u
1  ^ c  ( 2  /  3 ) )  e.  CC
77 exp1 11109 . . . . . . 7  |-  ( (
-u 1  ^ c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^ c  ( 2  /  3 ) ) ^ 1 )  =  ( -u 1  ^ c  ( 2  / 
3 ) ) )
7876, 77ax-mp 8 . . . . . 6  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 )  =  ( -u
1  ^ c  ( 2  /  3 ) )
7978, 71eqtri 2303 . . . . 5  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
8079fveq2i 5528 . . . 4  |-  ( * `
 ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 1 ) )  =  ( * `
 ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )
811, 68addcli 8841 . . . . . 6  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
8281, 26cjdivi 11676 . . . . 5  |-  ( 2  =/=  0  ->  (
* `  ( ( -u 1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )  =  ( ( * `  ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  /  (
* `  2 )
) )
8327, 82ax-mp 8 . . . 4  |-  ( * `
 ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )  =  ( ( * `  ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  /  (
* `  2 )
)
841, 68cjaddi 11673 . . . . . 6  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( ( * `  -u 1 )  +  ( * `  ( _i  x.  ( sqr `  3
) ) ) )
85 1re 8837 . . . . . . . . 9  |-  1  e.  RR
8685renegcli 9108 . . . . . . . 8  |-  -u 1  e.  RR
87 cjre 11624 . . . . . . . 8  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
8886, 87ax-mp 8 . . . . . . 7  |-  ( * `
 -u 1 )  = 
-u 1
8914, 61cjmuli 11674 . . . . . . . 8  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  ( ( * `  _i )  x.  ( * `  ( sqr `  3
) ) )
90 cji 11644 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
91 cjre 11624 . . . . . . . . . 10  |-  ( ( sqr `  3 )  e.  RR  ->  (
* `  ( sqr `  3 ) )  =  ( sqr `  3
) )
9260, 91ax-mp 8 . . . . . . . . 9  |-  ( * `
 ( sqr `  3
) )  =  ( sqr `  3 )
9390, 92oveq12i 5870 . . . . . . . 8  |-  ( ( * `  _i )  x.  ( * `  ( sqr `  3 ) ) )  =  (
-u _i  x.  ( sqr `  3 ) )
9414, 61mulneg1i 9225 . . . . . . . 8  |-  ( -u _i  x.  ( sqr `  3
) )  =  -u ( _i  x.  ( sqr `  3 ) )
9589, 93, 943eqtri 2307 . . . . . . 7  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  -u ( _i  x.  ( sqr `  3 ) )
9688, 95oveq12i 5870 . . . . . 6  |-  ( ( * `  -u 1
)  +  ( * `
 ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  + 
-u ( _i  x.  ( sqr `  3 ) ) )
971, 68negsubi 9124 . . . . . 6  |-  ( -u
1  +  -u (
_i  x.  ( sqr `  3 ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
9884, 96, 973eqtri 2307 . . . . 5  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
99 cjre 11624 . . . . . 6  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
1005, 99ax-mp 8 . . . . 5  |-  ( * `
 2 )  =  2
10198, 100oveq12i 5870 . . . 4  |-  ( ( * `  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) ) )  /  ( * ` 
2 ) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
)
10280, 83, 1013eqtri 2307 . . 3  |-  ( * `
 ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
103 3cn 9818 . . . . 5  |-  3  e.  CC
104 2p1e3 9847 . . . . . 6  |-  ( 2  +  1 )  =  3
10526, 2, 104addcomli 9004 . . . . 5  |-  ( 1  +  2 )  =  3
106103, 2, 26, 105subaddrii 9135 . . . 4  |-  ( 3  -  1 )  =  2
107106oveq2i 5869 . . 3  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ (
3  -  1 ) )  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )
10874, 102, 1073eqtr3ri 2312 . 2  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
10971, 108pm3.2i 441 1  |-  ( (
-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 ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   3c3 9796   6c6 9799   ZZcz 10024   ^cexp 11104   *ccj 11581   sqrcsqr 11718   expce 12343   sincsin 12345   cosccos 12346   picpi 12348   logclog 19912    ^ c ccxp 19913
This theorem is referenced by:  1cubr  20138
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
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