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Theorem 1cubrlem 23372
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 670 . . . . 5  |-  ( 2  /  3 )  e.  RR
76recni 9597 . . . 4  |-  ( 2  /  3 )  e.  CC
8 cxpef 23217 . . . 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 1322 . . 3  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )
10 logm1 23145 . . . . . 6  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
1110oveq2i 6281 . . . . 5  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( ( 2  /  3
)  x.  ( _i  x.  pi ) )
12 ax-icn 9540 . . . . . 6  |-  _i  e.  CC
13 pire 23020 . . . . . . 7  |-  pi  e.  RR
1413recni 9597 . . . . . 6  |-  pi  e.  CC
157, 12, 14mul12i 9764 . . . . 5  |-  ( ( 2  /  3 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1611, 15eqtri 2483 . . . 4  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1716fveq2i 5851 . . 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 670 . . . . . . . 8  |-  ( pi 
/  6 )  e.  RR
2120recni 9597 . . . . . . 7  |-  ( pi 
/  6 )  e.  CC
22 coshalfpip 23056 . . . . . . 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 1322 . . . . . . . . 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 1322 . . . . . . . . 9  |-  ( pi 
/  6 )  =  ( ( 1  / 
6 )  x.  pi )
3227, 31oveq12i 6282 . . . . . . . 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 9596 . . . . . . . 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 460 . . . . . . . . 9  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
3837oveq1i 6280 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( 2  / 
3 )  x.  pi )
3932, 35, 383eqtr2i 2489 . . . . . . 7  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( 2  / 
3 )  x.  pi )
4039fveq2i 5851 . . . . . 6  |-  ( cos `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( cos `  ( ( 2  /  3 )  x.  pi ) )
41 sincos6thpi 23077 . . . . . . . . 9  |-  ( ( sin `  ( pi 
/  6 ) )  =  ( 1  / 
2 )  /\  ( cos `  ( pi  / 
6 ) )  =  ( ( sqr `  3
)  /  2 ) )
4241simpli 456 . . . . . . . 8  |-  ( sin `  ( pi  /  6
) )  =  ( 1  /  2 )
4342negeqi 9804 . . . . . . 7  |-  -u ( sin `  ( pi  / 
6 ) )  = 
-u ( 1  / 
2 )
44 ax-1cn 9539 . . . . . . . 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 1322 . . . . . . 7  |-  -u (
1  /  2 )  =  ( -u 1  /  2 )
4743, 46eqtri 2483 . . . . . 6  |-  -u ( sin `  ( pi  / 
6 ) )  =  ( -u 1  / 
2 )
4823, 40, 473eqtr3i 2491 . . . . 5  |-  ( cos `  ( ( 2  / 
3 )  x.  pi ) )  =  (
-u 1  /  2
)
49 sinhalfpip 23054 . . . . . . . . 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 5851 . . . . . . . 8  |-  ( sin `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( sin `  ( ( 2  /  3 )  x.  pi ) )
5241simpri 460 . . . . . . . 8  |-  ( cos `  ( pi  /  6
) )  =  ( ( sqr `  3
)  /  2 )
5350, 51, 523eqtr3i 2491 . . . . . . 7  |-  ( sin `  ( ( 2  / 
3 )  x.  pi ) )  =  ( ( sqr `  3
)  /  2 )
5453oveq2i 6281 . . . . . 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 13172 . . . . . . . . 9  |-  ( ( 3  e.  RR  /\  0  <_  3 )  -> 
( sqr `  3
)  e.  RR )
5955, 57, 58mp2an 670 . . . . . . . 8  |-  ( sqr `  3 )  e.  RR
6059recni 9597 . . . . . . 7  |-  ( sqr `  3 )  e.  CC
6112, 60, 24, 25divassi 10296 . . . . . 6  |-  ( ( _i  x.  ( sqr `  3 ) )  /  2 )  =  ( _i  x.  (
( sqr `  3
)  /  2 ) )
6254, 61eqtr4i 2486 . . . . 5  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( _i  x.  ( sqr `  3 ) )  /  2 )
6348, 62oveq12i 6282 . . . 4  |-  ( ( cos `  ( ( 2  /  3 )  x.  pi ) )  +  ( _i  x.  ( sin `  ( ( 2  /  3 )  x.  pi ) ) ) )  =  ( ( -u 1  / 
2 )  +  ( ( _i  x.  ( sqr `  3 ) )  /  2 ) )
647, 14mulcli 9590 . . . . 5  |-  ( ( 2  /  3 )  x.  pi )  e.  CC
65 efival 13972 . . . . 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 9590 . . . . 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 2493 . . 3  |-  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
709, 17, 693eqtri 2487 . 2  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
71 1z 10890 . . . 4  |-  1  e.  ZZ
72 root1cj 23301 . . . 4  |-  ( ( 3  e.  NN  /\  1  e.  ZZ )  ->  ( * `  (
( -u 1  ^c 
( 2  /  3
) ) ^ 1 ) )  =  ( ( -u 1  ^c  ( 2  / 
3 ) ) ^
( 3  -  1 ) ) )
734, 71, 72mp2an 670 . . 3  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ (
3  -  1 ) )
74 cxpcl 23226 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^c  ( 2  /  3 ) )  e.  CC )
751, 7, 74mp2an 670 . . . . . . 7  |-  ( -u
1  ^c  ( 2  /  3 ) )  e.  CC
76 exp1 12157 . . . . . . 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 2483 . . . . 5  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 1 )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7978fveq2i 5851 . . . 4  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( * `
 ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )
801, 67addcli 9589 . . . . . 6  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
8180, 24cjdivi 13109 . . . . 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 13106 . . . . . 6  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( ( * `  -u 1 )  +  ( * `  ( _i  x.  ( sqr `  3
) ) ) )
84 neg1rr 10636 . . . . . . . 8  |-  -u 1  e.  RR
85 cjre 13057 . . . . . . . 8  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
8684, 85ax-mp 5 . . . . . . 7  |-  ( * `
 -u 1 )  = 
-u 1
8712, 60cjmuli 13107 . . . . . . . 8  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  ( ( * `  _i )  x.  ( * `  ( sqr `  3
) ) )
88 cji 13077 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
89 cjre 13057 . . . . . . . . . 10  |-  ( ( sqr `  3 )  e.  RR  ->  (
* `  ( sqr `  3 ) )  =  ( sqr `  3
) )
9059, 89ax-mp 5 . . . . . . . . 9  |-  ( * `
 ( sqr `  3
) )  =  ( sqr `  3 )
9188, 90oveq12i 6282 . . . . . . . 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 2487 . . . . . . 7  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  -u ( _i  x.  ( sqr `  3 ) )
9486, 93oveq12i 6282 . . . . . 6  |-  ( ( * `  -u 1
)  +  ( * `
 ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  + 
-u ( _i  x.  ( sqr `  3 ) ) )
951, 67negsubi 9888 . . . . . 6  |-  ( -u
1  +  -u (
_i  x.  ( sqr `  3 ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
9683, 94, 953eqtri 2487 . . . . 5  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
97 cjre 13057 . . . . . 6  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
983, 97ax-mp 5 . . . . 5  |-  ( * `
 2 )  =  2
9996, 98oveq12i 6282 . . . 4  |-  ( ( * `  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) ) )  /  ( * ` 
2 ) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
)
10079, 82, 993eqtri 2487 . . 3  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
101 3m1e2 10648 . . . 4  |-  ( 3  -  1 )  =  2
102101oveq2i 6281 . . 3  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ (
3  -  1 ) )  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )
10373, 100, 1023eqtr3ri 2492 . 2  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
10470, 103pm3.2i 453 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   _ici 9483    + caddc 9484    x. cmul 9486    <_ cle 9618    - cmin 9796   -ucneg 9797    / cdiv 10202   NNcn 10531   2c2 10581   3c3 10582   6c6 10585   ZZcz 10860   ^cexp 12151   *ccj 13014   sqrcsqrt 13151   expce 13882   sincsin 13884   cosccos 13885   picpi 13887   logclog 23111    ^c ccxp 23112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  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 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-shft 12985  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-limsup 13379  df-clim 13396  df-rlim 13397  df-sum 13594  df-ef 13888  df-sin 13890  df-cos 13891  df-pi 13893  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-haus 19986  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-cncf 21551  df-limc 22439  df-dv 22440  df-log 23113  df-cxp 23114
This theorem is referenced by:  1cubr  23373
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