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Theorem 1cubrlem 20153
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 9829 . . . 4  |-  -u 1  e.  CC
2 ax-1cn 8811 . . . . 5  |-  1  e.  CC
3 ax-1ne0 8822 . . . . 5  |-  1  =/=  0
42, 3negne0i 9137 . . . 4  |-  -u 1  =/=  0
5 2re 9831 . . . . . 6  |-  2  e.  RR
6 3nn 9894 . . . . . 6  |-  3  e.  NN
7 nndivre 9797 . . . . . 6  |-  ( ( 2  e.  RR  /\  3  e.  NN )  ->  ( 2  /  3
)  e.  RR )
85, 6, 7mp2an 653 . . . . 5  |-  ( 2  /  3 )  e.  RR
98recni 8865 . . . 4  |-  ( 2  /  3 )  e.  CC
10 cxpef 20028 . . . 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 19958 . . . . . 6  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
1312oveq2i 5885 . . . . 5  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( ( 2  /  3
)  x.  ( _i  x.  pi ) )
14 ax-icn 8812 . . . . . 6  |-  _i  e.  CC
15 pire 19848 . . . . . . 7  |-  pi  e.  RR
1615recni 8865 . . . . . 6  |-  pi  e.  CC
179, 14, 16mul12i 9023 . . . . 5  |-  ( ( 2  /  3 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1813, 17eqtri 2316 . . . 4  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1918fveq2i 5544 . . 3  |-  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )  =  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )
20 6nn 9897 . . . . . . . . 9  |-  6  e.  NN
21 nndivre 9797 . . . . . . . . 9  |-  ( ( pi  e.  RR  /\  6  e.  NN )  ->  ( pi  /  6
)  e.  RR )
2215, 20, 21mp2an 653 . . . . . . . 8  |-  ( pi 
/  6 )  e.  RR
2322recni 8865 . . . . . . 7  |-  ( pi 
/  6 )  e.  CC
24 coshalfpip 19878 . . . . . . 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 9832 . . . . . . . . . 10  |-  2  e.  CC
27 2ne0 9845 . . . . . . . . . 10  |-  2  =/=  0
28 divrec2 9457 . . . . . . . . . 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 9772 . . . . . . . . . 10  |-  6  e.  CC
3120nnne0i 9796 . . . . . . . . . 10  |-  6  =/=  0
32 divrec2 9457 . . . . . . . . . 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 5886 . . . . . . . 8  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
3526, 27reccli 9506 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
3630, 31reccli 9506 . . . . . . . . 9  |-  ( 1  /  6 )  e.  CC
3735, 36, 16adddiri 8864 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
38 halfpm6th 9952 . . . . . . . . . 10  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
3938simpri 448 . . . . . . . . 9  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
4039oveq1i 5884 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( 2  / 
3 )  x.  pi )
4134, 37, 403eqtr2i 2322 . . . . . . 7  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( 2  / 
3 )  x.  pi )
4241fveq2i 5544 . . . . . 6  |-  ( cos `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( cos `  ( ( 2  /  3 )  x.  pi ) )
43 sincos6thpi 19899 . . . . . . . . 9  |-  ( ( sin `  ( pi 
/  6 ) )  =  ( 1  / 
2 )  /\  ( cos `  ( pi  / 
6 ) )  =  ( ( sqr `  3
)  /  2 ) )
4443simpli 444 . . . . . . . 8  |-  ( sin `  ( pi  /  6
) )  =  ( 1  /  2 )
4544negeqi 9061 . . . . . . 7  |-  -u ( sin `  ( pi  / 
6 ) )  = 
-u ( 1  / 
2 )
46 divneg 9471 . . . . . . . 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 2316 . . . . . 6  |-  -u ( sin `  ( pi  / 
6 ) )  =  ( -u 1  / 
2 )
4925, 42, 483eqtr3i 2324 . . . . 5  |-  ( cos `  ( ( 2  / 
3 )  x.  pi ) )  =  (
-u 1  /  2
)
50 sinhalfpip 19876 . . . . . . . . 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 5544 . . . . . . . 8  |-  ( sin `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( sin `  ( ( 2  /  3 )  x.  pi ) )
5343simpri 448 . . . . . . . 8  |-  ( cos `  ( pi  /  6
) )  =  ( ( sqr `  3
)  /  2 )
5451, 52, 533eqtr3i 2324 . . . . . . 7  |-  ( sin `  ( ( 2  / 
3 )  x.  pi ) )  =  ( ( sqr `  3
)  /  2 )
5554oveq2i 5885 . . . . . 6  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( _i  x.  ( ( sqr `  3 )  /  2 ) )
56 3re 9833 . . . . . . . . 9  |-  3  e.  RR
57 3nn0 9999 . . . . . . . . . 10  |-  3  e.  NN0
5857nn0ge0i 10009 . . . . . . . . 9  |-  0  <_  3
59 resqrcl 11755 . . . . . . . . 9  |-  ( ( 3  e.  RR  /\  0  <_  3 )  -> 
( sqr `  3
)  e.  RR )
6056, 58, 59mp2an 653 . . . . . . . 8  |-  ( sqr `  3 )  e.  RR
6160recni 8865 . . . . . . 7  |-  ( sqr `  3 )  e.  CC
6214, 61, 26, 27divassi 9532 . . . . . 6  |-  ( ( _i  x.  ( sqr `  3 ) )  /  2 )  =  ( _i  x.  (
( sqr `  3
)  /  2 ) )
6355, 62eqtr4i 2319 . . . . 5  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( _i  x.  ( sqr `  3 ) )  /  2 )
6449, 63oveq12i 5886 . . . 4  |-  ( ( cos `  ( ( 2  /  3 )  x.  pi ) )  +  ( _i  x.  ( sin `  ( ( 2  /  3 )  x.  pi ) ) ) )  =  ( ( -u 1  / 
2 )  +  ( ( _i  x.  ( sqr `  3 ) )  /  2 ) )
659, 16mulcli 8858 . . . . 5  |-  ( ( 2  /  3 )  x.  pi )  e.  CC
66 efival 12448 . . . . 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 8858 . . . . 5  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
691, 68, 26, 27divdiri 9533 . . . 4  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  =  ( ( -u
1  /  2 )  +  ( ( _i  x.  ( sqr `  3
) )  /  2
) )
7064, 67, 693eqtr4i 2326 . . 3  |-  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7111, 19, 703eqtri 2320 . 2  |-  ( -u
1  ^ c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
72 1z 10069 . . . 4  |-  1  e.  ZZ
73 root1cj 20112 . . . 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 20037 . . . . . . . 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 11125 . . . . . . 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 2316 . . . . 5  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 1 )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
8079fveq2i 5544 . . . 4  |-  ( * `
 ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 1 ) )  =  ( * `
 ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )
811, 68addcli 8857 . . . . . 6  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
8281, 26cjdivi 11692 . . . . 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 11689 . . . . . 6  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( ( * `  -u 1 )  +  ( * `  ( _i  x.  ( sqr `  3
) ) ) )
85 1re 8853 . . . . . . . . 9  |-  1  e.  RR
8685renegcli 9124 . . . . . . . 8  |-  -u 1  e.  RR
87 cjre 11640 . . . . . . . 8  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
8886, 87ax-mp 8 . . . . . . 7  |-  ( * `
 -u 1 )  = 
-u 1
8914, 61cjmuli 11690 . . . . . . . 8  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  ( ( * `  _i )  x.  ( * `  ( sqr `  3
) ) )
90 cji 11660 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
91 cjre 11640 . . . . . . . . . 10  |-  ( ( sqr `  3 )  e.  RR  ->  (
* `  ( sqr `  3 ) )  =  ( sqr `  3
) )
9260, 91ax-mp 8 . . . . . . . . 9  |-  ( * `
 ( sqr `  3
) )  =  ( sqr `  3 )
9390, 92oveq12i 5886 . . . . . . . 8  |-  ( ( * `  _i )  x.  ( * `  ( sqr `  3 ) ) )  =  (
-u _i  x.  ( sqr `  3 ) )
9414, 61mulneg1i 9241 . . . . . . . 8  |-  ( -u _i  x.  ( sqr `  3
) )  =  -u ( _i  x.  ( sqr `  3 ) )
9589, 93, 943eqtri 2320 . . . . . . 7  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  -u ( _i  x.  ( sqr `  3 ) )
9688, 95oveq12i 5886 . . . . . 6  |-  ( ( * `  -u 1
)  +  ( * `
 ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  + 
-u ( _i  x.  ( sqr `  3 ) ) )
971, 68negsubi 9140 . . . . . 6  |-  ( -u
1  +  -u (
_i  x.  ( sqr `  3 ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
9884, 96, 973eqtri 2320 . . . . 5  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
99 cjre 11640 . . . . . 6  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
1005, 99ax-mp 8 . . . . 5  |-  ( * `
 2 )  =  2
10198, 100oveq12i 5886 . . . 4  |-  ( ( * `  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) ) )  /  ( * ` 
2 ) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
)
10280, 83, 1013eqtri 2320 . . 3  |-  ( * `
 ( ( -u
1  ^ c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
103 3cn 9834 . . . . 5  |-  3  e.  CC
104 2p1e3 9863 . . . . . 6  |-  ( 2  +  1 )  =  3
10526, 2, 104addcomli 9020 . . . . 5  |-  ( 1  +  2 )  =  3
106103, 2, 26, 105subaddrii 9151 . . . 4  |-  ( 3  -  1 )  =  2
107106oveq2i 5885 . . 3  |-  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ (
3  -  1 ) )  =  ( (
-u 1  ^ c 
( 2  /  3
) ) ^ 2 )
10874, 102, 1073eqtr3ri 2325 . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   3c3 9812   6c6 9815   ZZcz 10040   ^cexp 11120   *ccj 11597   sqrcsqr 11734   expce 12359   sincsin 12361   cosccos 12362   picpi 12364   logclog 19928    ^ c ccxp 19929
This theorem is referenced by:  1cubr  20154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931
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