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Theorem 1cubrlem 22900
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 10635 . . . 4  |-  -u 1  e.  CC
2 neg1ne0 10637 . . . 4  |-  -u 1  =/=  0
3 2re 10601 . . . . . 6  |-  2  e.  RR
4 3nn 10690 . . . . . 6  |-  3  e.  NN
5 nndivre 10567 . . . . . 6  |-  ( ( 2  e.  RR  /\  3  e.  NN )  ->  ( 2  /  3
)  e.  RR )
63, 4, 5mp2an 672 . . . . 5  |-  ( 2  /  3 )  e.  RR
76recni 9604 . . . 4  |-  ( 2  /  3 )  e.  CC
8 cxpef 22774 . . . 4  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^c  ( 2  /  3 ) )  =  ( exp `  (
( 2  /  3
)  x.  ( log `  -u 1 ) ) ) )
91, 2, 7, 8mp3an 1324 . . 3  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )
10 logm1 22701 . . . . . 6  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
1110oveq2i 6293 . . . . 5  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( ( 2  /  3
)  x.  ( _i  x.  pi ) )
12 ax-icn 9547 . . . . . 6  |-  _i  e.  CC
13 pire 22585 . . . . . . 7  |-  pi  e.  RR
1413recni 9604 . . . . . 6  |-  pi  e.  CC
157, 12, 14mul12i 9770 . . . . 5  |-  ( ( 2  /  3 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1611, 15eqtri 2496 . . . 4  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1716fveq2i 5867 . . 3  |-  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )  =  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )
18 6nn 10693 . . . . . . . . 9  |-  6  e.  NN
19 nndivre 10567 . . . . . . . . 9  |-  ( ( pi  e.  RR  /\  6  e.  NN )  ->  ( pi  /  6
)  e.  RR )
2013, 18, 19mp2an 672 . . . . . . . 8  |-  ( pi 
/  6 )  e.  RR
2120recni 9604 . . . . . . 7  |-  ( pi 
/  6 )  e.  CC
22 coshalfpip 22620 . . . . . . 7  |-  ( ( pi  /  6 )  e.  CC  ->  ( cos `  ( ( pi 
/  2 )  +  ( pi  /  6
) ) )  = 
-u ( sin `  (
pi  /  6 ) ) )
2321, 22ax-mp 5 . . . . . 6  |-  ( cos `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  -u ( sin `  ( pi 
/  6 ) )
24 2cn 10602 . . . . . . . . . 10  |-  2  e.  CC
25 2ne0 10624 . . . . . . . . . 10  |-  2  =/=  0
26 divrec2 10220 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
pi  /  2 )  =  ( ( 1  /  2 )  x.  pi ) )
2714, 24, 25, 26mp3an 1324 . . . . . . . . 9  |-  ( pi 
/  2 )  =  ( ( 1  / 
2 )  x.  pi )
28 6cn 10613 . . . . . . . . . 10  |-  6  e.  CC
2918nnne0i 10566 . . . . . . . . . 10  |-  6  =/=  0
30 divrec2 10220 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
pi  /  6 )  =  ( ( 1  /  6 )  x.  pi ) )
3114, 28, 29, 30mp3an 1324 . . . . . . . . 9  |-  ( pi 
/  6 )  =  ( ( 1  / 
6 )  x.  pi )
3227, 31oveq12i 6294 . . . . . . . 8  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
3324, 25reccli 10270 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
3428, 29reccli 10270 . . . . . . . . 9  |-  ( 1  /  6 )  e.  CC
3533, 34, 14adddiri 9603 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
36 halfpm6th 10756 . . . . . . . . . 10  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
3736simpri 462 . . . . . . . . 9  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
3837oveq1i 6292 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( 2  / 
3 )  x.  pi )
3932, 35, 383eqtr2i 2502 . . . . . . 7  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( 2  / 
3 )  x.  pi )
4039fveq2i 5867 . . . . . 6  |-  ( cos `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( cos `  ( ( 2  /  3 )  x.  pi ) )
41 sincos6thpi 22641 . . . . . . . . 9  |-  ( ( sin `  ( pi 
/  6 ) )  =  ( 1  / 
2 )  /\  ( cos `  ( pi  / 
6 ) )  =  ( ( sqr `  3
)  /  2 ) )
4241simpli 458 . . . . . . . 8  |-  ( sin `  ( pi  /  6
) )  =  ( 1  /  2 )
4342negeqi 9809 . . . . . . 7  |-  -u ( sin `  ( pi  / 
6 ) )  = 
-u ( 1  / 
2 )
44 ax-1cn 9546 . . . . . . . 8  |-  1  e.  CC
45 divneg 10235 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
4644, 24, 25, 45mp3an 1324 . . . . . . 7  |-  -u (
1  /  2 )  =  ( -u 1  /  2 )
4743, 46eqtri 2496 . . . . . 6  |-  -u ( sin `  ( pi  / 
6 ) )  =  ( -u 1  / 
2 )
4823, 40, 473eqtr3i 2504 . . . . 5  |-  ( cos `  ( ( 2  / 
3 )  x.  pi ) )  =  (
-u 1  /  2
)
49 sinhalfpip 22618 . . . . . . . . 9  |-  ( ( pi  /  6 )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  +  ( pi  /  6
) ) )  =  ( cos `  (
pi  /  6 ) ) )
5021, 49ax-mp 5 . . . . . . . 8  |-  ( sin `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( cos `  ( pi 
/  6 ) )
5139fveq2i 5867 . . . . . . . 8  |-  ( sin `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( sin `  ( ( 2  /  3 )  x.  pi ) )
5241simpri 462 . . . . . . . 8  |-  ( cos `  ( pi  /  6
) )  =  ( ( sqr `  3
)  /  2 )
5350, 51, 523eqtr3i 2504 . . . . . . 7  |-  ( sin `  ( ( 2  / 
3 )  x.  pi ) )  =  ( ( sqr `  3
)  /  2 )
5453oveq2i 6293 . . . . . 6  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( _i  x.  ( ( sqr `  3 )  /  2 ) )
55 3re 10605 . . . . . . . . 9  |-  3  e.  RR
56 3nn0 10809 . . . . . . . . . 10  |-  3  e.  NN0
5756nn0ge0i 10819 . . . . . . . . 9  |-  0  <_  3
58 resqrtcl 13046 . . . . . . . . 9  |-  ( ( 3  e.  RR  /\  0  <_  3 )  -> 
( sqr `  3
)  e.  RR )
5955, 57, 58mp2an 672 . . . . . . . 8  |-  ( sqr `  3 )  e.  RR
6059recni 9604 . . . . . . 7  |-  ( sqr `  3 )  e.  CC
6112, 60, 24, 25divassi 10296 . . . . . 6  |-  ( ( _i  x.  ( sqr `  3 ) )  /  2 )  =  ( _i  x.  (
( sqr `  3
)  /  2 ) )
6254, 61eqtr4i 2499 . . . . 5  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( _i  x.  ( sqr `  3 ) )  /  2 )
6348, 62oveq12i 6294 . . . 4  |-  ( ( cos `  ( ( 2  /  3 )  x.  pi ) )  +  ( _i  x.  ( sin `  ( ( 2  /  3 )  x.  pi ) ) ) )  =  ( ( -u 1  / 
2 )  +  ( ( _i  x.  ( sqr `  3 ) )  /  2 ) )
647, 14mulcli 9597 . . . . 5  |-  ( ( 2  /  3 )  x.  pi )  e.  CC
65 efival 13744 . . . . 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 ) ) ) ) )
6664, 65ax-mp 5 . . . 4  |-  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( cos `  (
( 2  /  3
)  x.  pi ) )  +  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) ) )
6712, 60mulcli 9597 . . . . 5  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
681, 67, 24, 25divdiri 10297 . . . 4  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  =  ( ( -u
1  /  2 )  +  ( ( _i  x.  ( sqr `  3
) )  /  2
) )
6963, 66, 683eqtr4i 2506 . . 3  |-  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
709, 17, 693eqtri 2500 . 2  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
71 1z 10890 . . . 4  |-  1  e.  ZZ
72 root1cj 22858 . . . 4  |-  ( ( 3  e.  NN  /\  1  e.  ZZ )  ->  ( * `  (
( -u 1  ^c 
( 2  /  3
) ) ^ 1 ) )  =  ( ( -u 1  ^c  ( 2  / 
3 ) ) ^
( 3  -  1 ) ) )
734, 71, 72mp2an 672 . . 3  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ (
3  -  1 ) )
74 cxpcl 22783 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^c  ( 2  /  3 ) )  e.  CC )
751, 7, 74mp2an 672 . . . . . . 7  |-  ( -u
1  ^c  ( 2  /  3 ) )  e.  CC
76 exp1 12136 . . . . . . 7  |-  ( (
-u 1  ^c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^c  ( 2  /  3 ) ) ^ 1 )  =  ( -u 1  ^c  ( 2  / 
3 ) ) )
7775, 76ax-mp 5 . . . . . 6  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 1 )  =  ( -u
1  ^c  ( 2  /  3 ) )
7877, 70eqtri 2496 . . . . 5  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 1 )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7978fveq2i 5867 . . . 4  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( * `
 ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )
801, 67addcli 9596 . . . . . 6  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
8180, 24cjdivi 12983 . . . . 5  |-  ( 2  =/=  0  ->  (
* `  ( ( -u 1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )  =  ( ( * `  ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  /  (
* `  2 )
) )
8225, 81ax-mp 5 . . . 4  |-  ( * `
 ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )  =  ( ( * `  ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  /  (
* `  2 )
)
831, 67cjaddi 12980 . . . . . 6  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( ( * `  -u 1 )  +  ( * `  ( _i  x.  ( sqr `  3
) ) ) )
84 neg1rr 10636 . . . . . . . 8  |-  -u 1  e.  RR
85 cjre 12931 . . . . . . . 8  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
8684, 85ax-mp 5 . . . . . . 7  |-  ( * `
 -u 1 )  = 
-u 1
8712, 60cjmuli 12981 . . . . . . . 8  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  ( ( * `  _i )  x.  ( * `  ( sqr `  3
) ) )
88 cji 12951 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
89 cjre 12931 . . . . . . . . . 10  |-  ( ( sqr `  3 )  e.  RR  ->  (
* `  ( sqr `  3 ) )  =  ( sqr `  3
) )
9059, 89ax-mp 5 . . . . . . . . 9  |-  ( * `
 ( sqr `  3
) )  =  ( sqr `  3 )
9188, 90oveq12i 6294 . . . . . . . 8  |-  ( ( * `  _i )  x.  ( * `  ( sqr `  3 ) ) )  =  (
-u _i  x.  ( sqr `  3 ) )
9212, 60mulneg1i 9998 . . . . . . . 8  |-  ( -u _i  x.  ( sqr `  3
) )  =  -u ( _i  x.  ( sqr `  3 ) )
9387, 91, 923eqtri 2500 . . . . . . 7  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  -u ( _i  x.  ( sqr `  3 ) )
9486, 93oveq12i 6294 . . . . . 6  |-  ( ( * `  -u 1
)  +  ( * `
 ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  + 
-u ( _i  x.  ( sqr `  3 ) ) )
951, 67negsubi 9893 . . . . . 6  |-  ( -u
1  +  -u (
_i  x.  ( sqr `  3 ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
9683, 94, 953eqtri 2500 . . . . 5  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
97 cjre 12931 . . . . . 6  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
983, 97ax-mp 5 . . . . 5  |-  ( * `
 2 )  =  2
9996, 98oveq12i 6294 . . . 4  |-  ( ( * `  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) ) )  /  ( * ` 
2 ) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
)
10079, 82, 993eqtri 2500 . . 3  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
101 3m1e2 10648 . . . 4  |-  ( 3  -  1 )  =  2
102101oveq2i 6293 . . 3  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ (
3  -  1 ) )  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )
10373, 100, 1023eqtr3ri 2505 . 2  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
10470, 103pm3.2i 455 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 setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   _ici 9490    + caddc 9491    x. cmul 9493    <_ cle 9625    - cmin 9801   -ucneg 9802    / cdiv 10202   NNcn 10532   2c2 10581   3c3 10582   6c6 10585   ZZcz 10860   ^cexp 12130   *ccj 12888   sqrcsqrt 13025   expce 13655   sincsin 13657   cosccos 13658   picpi 13660   logclog 22670    ^c ccxp 22671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-pi 13666  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006  df-log 22672  df-cxp 22673
This theorem is referenced by:  1cubr  22901
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