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Theorem 1cubrlem 22364
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 10531 . . . 4  |-  -u 1  e.  CC
2 neg1ne0 10533 . . . 4  |-  -u 1  =/=  0
3 2re 10497 . . . . . 6  |-  2  e.  RR
4 3nn 10586 . . . . . 6  |-  3  e.  NN
5 nndivre 10463 . . . . . 6  |-  ( ( 2  e.  RR  /\  3  e.  NN )  ->  ( 2  /  3
)  e.  RR )
63, 4, 5mp2an 672 . . . . 5  |-  ( 2  /  3 )  e.  RR
76recni 9504 . . . 4  |-  ( 2  /  3 )  e.  CC
8 cxpef 22238 . . . 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 1315 . . 3  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )
10 logm1 22165 . . . . . 6  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
1110oveq2i 6206 . . . . 5  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( ( 2  /  3
)  x.  ( _i  x.  pi ) )
12 ax-icn 9447 . . . . . 6  |-  _i  e.  CC
13 pire 22049 . . . . . . 7  |-  pi  e.  RR
1413recni 9504 . . . . . 6  |-  pi  e.  CC
157, 12, 14mul12i 9670 . . . . 5  |-  ( ( 2  /  3 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1611, 15eqtri 2481 . . . 4  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1716fveq2i 5797 . . 3  |-  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )  =  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )
18 6nn 10589 . . . . . . . . 9  |-  6  e.  NN
19 nndivre 10463 . . . . . . . . 9  |-  ( ( pi  e.  RR  /\  6  e.  NN )  ->  ( pi  /  6
)  e.  RR )
2013, 18, 19mp2an 672 . . . . . . . 8  |-  ( pi 
/  6 )  e.  RR
2120recni 9504 . . . . . . 7  |-  ( pi 
/  6 )  e.  CC
22 coshalfpip 22084 . . . . . . 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 10498 . . . . . . . . . 10  |-  2  e.  CC
25 2ne0 10520 . . . . . . . . . 10  |-  2  =/=  0
26 divrec2 10117 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
pi  /  2 )  =  ( ( 1  /  2 )  x.  pi ) )
2714, 24, 25, 26mp3an 1315 . . . . . . . . 9  |-  ( pi 
/  2 )  =  ( ( 1  / 
2 )  x.  pi )
28 6cn 10509 . . . . . . . . . 10  |-  6  e.  CC
2918nnne0i 10462 . . . . . . . . . 10  |-  6  =/=  0
30 divrec2 10117 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
pi  /  6 )  =  ( ( 1  /  6 )  x.  pi ) )
3114, 28, 29, 30mp3an 1315 . . . . . . . . 9  |-  ( pi 
/  6 )  =  ( ( 1  / 
6 )  x.  pi )
3227, 31oveq12i 6207 . . . . . . . 8  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
3324, 25reccli 10167 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
3428, 29reccli 10167 . . . . . . . . 9  |-  ( 1  /  6 )  e.  CC
3533, 34, 14adddiri 9503 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
36 halfpm6th 10652 . . . . . . . . . 10  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
3736simpri 462 . . . . . . . . 9  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
3837oveq1i 6205 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( 2  / 
3 )  x.  pi )
3932, 35, 383eqtr2i 2487 . . . . . . 7  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( 2  / 
3 )  x.  pi )
4039fveq2i 5797 . . . . . 6  |-  ( cos `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( cos `  ( ( 2  /  3 )  x.  pi ) )
41 sincos6thpi 22105 . . . . . . . . 9  |-  ( ( sin `  ( pi 
/  6 ) )  =  ( 1  / 
2 )  /\  ( cos `  ( pi  / 
6 ) )  =  ( ( sqr `  3
)  /  2 ) )
4241simpli 458 . . . . . . . 8  |-  ( sin `  ( pi  /  6
) )  =  ( 1  /  2 )
4342negeqi 9709 . . . . . . 7  |-  -u ( sin `  ( pi  / 
6 ) )  = 
-u ( 1  / 
2 )
44 ax-1cn 9446 . . . . . . . 8  |-  1  e.  CC
45 divneg 10132 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
4644, 24, 25, 45mp3an 1315 . . . . . . 7  |-  -u (
1  /  2 )  =  ( -u 1  /  2 )
4743, 46eqtri 2481 . . . . . 6  |-  -u ( sin `  ( pi  / 
6 ) )  =  ( -u 1  / 
2 )
4823, 40, 473eqtr3i 2489 . . . . 5  |-  ( cos `  ( ( 2  / 
3 )  x.  pi ) )  =  (
-u 1  /  2
)
49 sinhalfpip 22082 . . . . . . . . 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 5797 . . . . . . . 8  |-  ( sin `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( sin `  ( ( 2  /  3 )  x.  pi ) )
5241simpri 462 . . . . . . . 8  |-  ( cos `  ( pi  /  6
) )  =  ( ( sqr `  3
)  /  2 )
5350, 51, 523eqtr3i 2489 . . . . . . 7  |-  ( sin `  ( ( 2  / 
3 )  x.  pi ) )  =  ( ( sqr `  3
)  /  2 )
5453oveq2i 6206 . . . . . 6  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( _i  x.  ( ( sqr `  3 )  /  2 ) )
55 3re 10501 . . . . . . . . 9  |-  3  e.  RR
56 3nn0 10703 . . . . . . . . . 10  |-  3  e.  NN0
5756nn0ge0i 10713 . . . . . . . . 9  |-  0  <_  3
58 resqrcl 12856 . . . . . . . . 9  |-  ( ( 3  e.  RR  /\  0  <_  3 )  -> 
( sqr `  3
)  e.  RR )
5955, 57, 58mp2an 672 . . . . . . . 8  |-  ( sqr `  3 )  e.  RR
6059recni 9504 . . . . . . 7  |-  ( sqr `  3 )  e.  CC
6112, 60, 24, 25divassi 10193 . . . . . 6  |-  ( ( _i  x.  ( sqr `  3 ) )  /  2 )  =  ( _i  x.  (
( sqr `  3
)  /  2 ) )
6254, 61eqtr4i 2484 . . . . 5  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( _i  x.  ( sqr `  3 ) )  /  2 )
6348, 62oveq12i 6207 . . . 4  |-  ( ( cos `  ( ( 2  /  3 )  x.  pi ) )  +  ( _i  x.  ( sin `  ( ( 2  /  3 )  x.  pi ) ) ) )  =  ( ( -u 1  / 
2 )  +  ( ( _i  x.  ( sqr `  3 ) )  /  2 ) )
647, 14mulcli 9497 . . . . 5  |-  ( ( 2  /  3 )  x.  pi )  e.  CC
65 efival 13549 . . . . 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 9497 . . . . 5  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
681, 67, 24, 25divdiri 10194 . . . 4  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  =  ( ( -u
1  /  2 )  +  ( ( _i  x.  ( sqr `  3
) )  /  2
) )
6963, 66, 683eqtr4i 2491 . . 3  |-  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
709, 17, 693eqtri 2485 . 2  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
71 1z 10782 . . . 4  |-  1  e.  ZZ
72 root1cj 22322 . . . 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 22247 . . . . . . . 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 11983 . . . . . . 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 2481 . . . . 5  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 1 )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7978fveq2i 5797 . . . 4  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( * `
 ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )
801, 67addcli 9496 . . . . . 6  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
8180, 24cjdivi 12793 . . . . 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 12790 . . . . . 6  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( ( * `  -u 1 )  +  ( * `  ( _i  x.  ( sqr `  3
) ) ) )
84 neg1rr 10532 . . . . . . . 8  |-  -u 1  e.  RR
85 cjre 12741 . . . . . . . 8  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
8684, 85ax-mp 5 . . . . . . 7  |-  ( * `
 -u 1 )  = 
-u 1
8712, 60cjmuli 12791 . . . . . . . 8  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  ( ( * `  _i )  x.  ( * `  ( sqr `  3
) ) )
88 cji 12761 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
89 cjre 12741 . . . . . . . . . 10  |-  ( ( sqr `  3 )  e.  RR  ->  (
* `  ( sqr `  3 ) )  =  ( sqr `  3
) )
9059, 89ax-mp 5 . . . . . . . . 9  |-  ( * `
 ( sqr `  3
) )  =  ( sqr `  3 )
9188, 90oveq12i 6207 . . . . . . . 8  |-  ( ( * `  _i )  x.  ( * `  ( sqr `  3 ) ) )  =  (
-u _i  x.  ( sqr `  3 ) )
9212, 60mulneg1i 9896 . . . . . . . 8  |-  ( -u _i  x.  ( sqr `  3
) )  =  -u ( _i  x.  ( sqr `  3 ) )
9387, 91, 923eqtri 2485 . . . . . . 7  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  -u ( _i  x.  ( sqr `  3 ) )
9486, 93oveq12i 6207 . . . . . 6  |-  ( ( * `  -u 1
)  +  ( * `
 ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  + 
-u ( _i  x.  ( sqr `  3 ) ) )
951, 67negsubi 9792 . . . . . 6  |-  ( -u
1  +  -u (
_i  x.  ( sqr `  3 ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
9683, 94, 953eqtri 2485 . . . . 5  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
97 cjre 12741 . . . . . 6  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
983, 97ax-mp 5 . . . . 5  |-  ( * `
 2 )  =  2
9996, 98oveq12i 6207 . . . 4  |-  ( ( * `  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) ) )  /  ( * ` 
2 ) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
)
10079, 82, 993eqtri 2485 . . 3  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
101 3m1e2 10544 . . . 4  |-  ( 3  -  1 )  =  2
102101oveq2i 6206 . . 3  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ (
3  -  1 ) )  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )
10373, 100, 1023eqtr3ri 2490 . 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 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   CCcc 9386   RRcr 9387   0cc0 9388   1c1 9389   _ici 9390    + caddc 9391    x. cmul 9393    <_ cle 9525    - cmin 9701   -ucneg 9702    / cdiv 10099   NNcn 10428   2c2 10477   3c3 10478   6c6 10481   ZZcz 10752   ^cexp 11977   *ccj 12698   sqrcsqr 12835   expce 13460   sincsin 13462   cosccos 13463   picpi 13465   logclog 22134    ^c ccxp 22135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-fi 7767  df-sup 7797  df-oi 7830  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-ioo 11410  df-ioc 11411  df-ico 11412  df-icc 11413  df-fz 11550  df-fzo 11661  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-fac 12164  df-bc 12191  df-hash 12216  df-shft 12669  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-limsup 13062  df-clim 13079  df-rlim 13080  df-sum 13277  df-ef 13466  df-sin 13468  df-cos 13469  df-pi 13471  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-hom 14376  df-cco 14377  df-rest 14475  df-topn 14476  df-0g 14494  df-gsum 14495  df-topgen 14496  df-pt 14497  df-prds 14500  df-xrs 14554  df-qtop 14559  df-imas 14560  df-xps 14562  df-mre 14638  df-mrc 14639  df-acs 14641  df-mnd 15529  df-submnd 15579  df-mulg 15662  df-cntz 15949  df-cmn 16395  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-fbas 17934  df-fg 17935  df-cnfld 17939  df-top 18630  df-bases 18632  df-topon 18633  df-topsp 18634  df-cld 18750  df-ntr 18751  df-cls 18752  df-nei 18829  df-lp 18867  df-perf 18868  df-cn 18958  df-cnp 18959  df-haus 19046  df-tx 19262  df-hmeo 19455  df-fil 19546  df-fm 19638  df-flim 19639  df-flf 19640  df-xms 20022  df-ms 20023  df-tms 20024  df-cncf 20581  df-limc 21469  df-dv 21470  df-log 22136  df-cxp 22137
This theorem is referenced by:  1cubr  22365
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