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Theorem 1cubrlem 22195
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 10421 . . . 4  |-  -u 1  e.  CC
2 neg1ne0 10423 . . . 4  |-  -u 1  =/=  0
3 2re 10387 . . . . . 6  |-  2  e.  RR
4 3nn 10476 . . . . . 6  |-  3  e.  NN
5 nndivre 10353 . . . . . 6  |-  ( ( 2  e.  RR  /\  3  e.  NN )  ->  ( 2  /  3
)  e.  RR )
63, 4, 5mp2an 667 . . . . 5  |-  ( 2  /  3 )  e.  RR
76recni 9394 . . . 4  |-  ( 2  /  3 )  e.  CC
8 cxpef 22069 . . . 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 1309 . . 3  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )
10 logm1 21996 . . . . . 6  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
1110oveq2i 6101 . . . . 5  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( ( 2  /  3
)  x.  ( _i  x.  pi ) )
12 ax-icn 9337 . . . . . 6  |-  _i  e.  CC
13 pire 21880 . . . . . . 7  |-  pi  e.  RR
1413recni 9394 . . . . . 6  |-  pi  e.  CC
157, 12, 14mul12i 9560 . . . . 5  |-  ( ( 2  /  3 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1611, 15eqtri 2461 . . . 4  |-  ( ( 2  /  3 )  x.  ( log `  -u 1
) )  =  ( _i  x.  ( ( 2  /  3 )  x.  pi ) )
1716fveq2i 5691 . . 3  |-  ( exp `  ( ( 2  / 
3 )  x.  ( log `  -u 1 ) ) )  =  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )
18 6nn 10479 . . . . . . . . 9  |-  6  e.  NN
19 nndivre 10353 . . . . . . . . 9  |-  ( ( pi  e.  RR  /\  6  e.  NN )  ->  ( pi  /  6
)  e.  RR )
2013, 18, 19mp2an 667 . . . . . . . 8  |-  ( pi 
/  6 )  e.  RR
2120recni 9394 . . . . . . 7  |-  ( pi 
/  6 )  e.  CC
22 coshalfpip 21915 . . . . . . 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 10388 . . . . . . . . . 10  |-  2  e.  CC
25 2ne0 10410 . . . . . . . . . 10  |-  2  =/=  0
26 divrec2 10007 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
pi  /  2 )  =  ( ( 1  /  2 )  x.  pi ) )
2714, 24, 25, 26mp3an 1309 . . . . . . . . 9  |-  ( pi 
/  2 )  =  ( ( 1  / 
2 )  x.  pi )
28 6cn 10399 . . . . . . . . . 10  |-  6  e.  CC
2918nnne0i 10352 . . . . . . . . . 10  |-  6  =/=  0
30 divrec2 10007 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
pi  /  6 )  =  ( ( 1  /  6 )  x.  pi ) )
3114, 28, 29, 30mp3an 1309 . . . . . . . . 9  |-  ( pi 
/  6 )  =  ( ( 1  / 
6 )  x.  pi )
3227, 31oveq12i 6102 . . . . . . . 8  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
3324, 25reccli 10057 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
3428, 29reccli 10057 . . . . . . . . 9  |-  ( 1  /  6 )  e.  CC
3533, 34, 14adddiri 9393 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( ( 1  /  2 )  x.  pi )  +  ( ( 1  /  6
)  x.  pi ) )
36 halfpm6th 10542 . . . . . . . . . 10  |-  ( ( ( 1  /  2
)  -  ( 1  /  6 ) )  =  ( 1  / 
3 )  /\  (
( 1  /  2
)  +  ( 1  /  6 ) )  =  ( 2  / 
3 ) )
3736simpri 459 . . . . . . . . 9  |-  ( ( 1  /  2 )  +  ( 1  / 
6 ) )  =  ( 2  /  3
)
3837oveq1i 6100 . . . . . . . 8  |-  ( ( ( 1  /  2
)  +  ( 1  /  6 ) )  x.  pi )  =  ( ( 2  / 
3 )  x.  pi )
3932, 35, 383eqtr2i 2467 . . . . . . 7  |-  ( ( pi  /  2 )  +  ( pi  / 
6 ) )  =  ( ( 2  / 
3 )  x.  pi )
4039fveq2i 5691 . . . . . 6  |-  ( cos `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( cos `  ( ( 2  /  3 )  x.  pi ) )
41 sincos6thpi 21936 . . . . . . . . 9  |-  ( ( sin `  ( pi 
/  6 ) )  =  ( 1  / 
2 )  /\  ( cos `  ( pi  / 
6 ) )  =  ( ( sqr `  3
)  /  2 ) )
4241simpli 455 . . . . . . . 8  |-  ( sin `  ( pi  /  6
) )  =  ( 1  /  2 )
4342negeqi 9599 . . . . . . 7  |-  -u ( sin `  ( pi  / 
6 ) )  = 
-u ( 1  / 
2 )
44 ax-1cn 9336 . . . . . . . 8  |-  1  e.  CC
45 divneg 10022 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
4644, 24, 25, 45mp3an 1309 . . . . . . 7  |-  -u (
1  /  2 )  =  ( -u 1  /  2 )
4743, 46eqtri 2461 . . . . . 6  |-  -u ( sin `  ( pi  / 
6 ) )  =  ( -u 1  / 
2 )
4823, 40, 473eqtr3i 2469 . . . . 5  |-  ( cos `  ( ( 2  / 
3 )  x.  pi ) )  =  (
-u 1  /  2
)
49 sinhalfpip 21913 . . . . . . . . 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 5691 . . . . . . . 8  |-  ( sin `  ( ( pi  / 
2 )  +  ( pi  /  6 ) ) )  =  ( sin `  ( ( 2  /  3 )  x.  pi ) )
5241simpri 459 . . . . . . . 8  |-  ( cos `  ( pi  /  6
) )  =  ( ( sqr `  3
)  /  2 )
5350, 51, 523eqtr3i 2469 . . . . . . 7  |-  ( sin `  ( ( 2  / 
3 )  x.  pi ) )  =  ( ( sqr `  3
)  /  2 )
5453oveq2i 6101 . . . . . 6  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( _i  x.  ( ( sqr `  3 )  /  2 ) )
55 3re 10391 . . . . . . . . 9  |-  3  e.  RR
56 3nn0 10593 . . . . . . . . . 10  |-  3  e.  NN0
5756nn0ge0i 10603 . . . . . . . . 9  |-  0  <_  3
58 resqrcl 12739 . . . . . . . . 9  |-  ( ( 3  e.  RR  /\  0  <_  3 )  -> 
( sqr `  3
)  e.  RR )
5955, 57, 58mp2an 667 . . . . . . . 8  |-  ( sqr `  3 )  e.  RR
6059recni 9394 . . . . . . 7  |-  ( sqr `  3 )  e.  CC
6112, 60, 24, 25divassi 10083 . . . . . 6  |-  ( ( _i  x.  ( sqr `  3 ) )  /  2 )  =  ( _i  x.  (
( sqr `  3
)  /  2 ) )
6254, 61eqtr4i 2464 . . . . 5  |-  ( _i  x.  ( sin `  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( _i  x.  ( sqr `  3 ) )  /  2 )
6348, 62oveq12i 6102 . . . 4  |-  ( ( cos `  ( ( 2  /  3 )  x.  pi ) )  +  ( _i  x.  ( sin `  ( ( 2  /  3 )  x.  pi ) ) ) )  =  ( ( -u 1  / 
2 )  +  ( ( _i  x.  ( sqr `  3 ) )  /  2 ) )
647, 14mulcli 9387 . . . . 5  |-  ( ( 2  /  3 )  x.  pi )  e.  CC
65 efival 13432 . . . . 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 9387 . . . . 5  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
681, 67, 24, 25divdiri 10084 . . . 4  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  =  ( ( -u
1  /  2 )  +  ( ( _i  x.  ( sqr `  3
) )  /  2
) )
6963, 66, 683eqtr4i 2471 . . 3  |-  ( exp `  ( _i  x.  (
( 2  /  3
)  x.  pi ) ) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
709, 17, 693eqtri 2465 . 2  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
71 1z 10672 . . . 4  |-  1  e.  ZZ
72 root1cj 22153 . . . 4  |-  ( ( 3  e.  NN  /\  1  e.  ZZ )  ->  ( * `  (
( -u 1  ^c 
( 2  /  3
) ) ^ 1 ) )  =  ( ( -u 1  ^c  ( 2  / 
3 ) ) ^
( 3  -  1 ) ) )
734, 71, 72mp2an 667 . . 3  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ (
3  -  1 ) )
74 cxpcl 22078 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^c  ( 2  /  3 ) )  e.  CC )
751, 7, 74mp2an 667 . . . . . . 7  |-  ( -u
1  ^c  ( 2  /  3 ) )  e.  CC
76 exp1 11867 . . . . . . 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 2461 . . . . 5  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 1 )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7978fveq2i 5691 . . . 4  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( * `
 ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) )
801, 67addcli 9386 . . . . . 6  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
8180, 24cjdivi 12676 . . . . 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 12673 . . . . . 6  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( ( * `  -u 1 )  +  ( * `  ( _i  x.  ( sqr `  3
) ) ) )
84 neg1rr 10422 . . . . . . . 8  |-  -u 1  e.  RR
85 cjre 12624 . . . . . . . 8  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
8684, 85ax-mp 5 . . . . . . 7  |-  ( * `
 -u 1 )  = 
-u 1
8712, 60cjmuli 12674 . . . . . . . 8  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  ( ( * `  _i )  x.  ( * `  ( sqr `  3
) ) )
88 cji 12644 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
89 cjre 12624 . . . . . . . . . 10  |-  ( ( sqr `  3 )  e.  RR  ->  (
* `  ( sqr `  3 ) )  =  ( sqr `  3
) )
9059, 89ax-mp 5 . . . . . . . . 9  |-  ( * `
 ( sqr `  3
) )  =  ( sqr `  3 )
9188, 90oveq12i 6102 . . . . . . . 8  |-  ( ( * `  _i )  x.  ( * `  ( sqr `  3 ) ) )  =  (
-u _i  x.  ( sqr `  3 ) )
9212, 60mulneg1i 9786 . . . . . . . 8  |-  ( -u _i  x.  ( sqr `  3
) )  =  -u ( _i  x.  ( sqr `  3 ) )
9387, 91, 923eqtri 2465 . . . . . . 7  |-  ( * `
 ( _i  x.  ( sqr `  3 ) ) )  =  -u ( _i  x.  ( sqr `  3 ) )
9486, 93oveq12i 6102 . . . . . 6  |-  ( ( * `  -u 1
)  +  ( * `
 ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  + 
-u ( _i  x.  ( sqr `  3 ) ) )
951, 67negsubi 9682 . . . . . 6  |-  ( -u
1  +  -u (
_i  x.  ( sqr `  3 ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
9683, 94, 953eqtri 2465 . . . . 5  |-  ( * `
 ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) ) )  =  ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )
97 cjre 12624 . . . . . 6  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
983, 97ax-mp 5 . . . . 5  |-  ( * `
 2 )  =  2
9996, 98oveq12i 6102 . . . 4  |-  ( ( * `  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) ) )  /  ( * ` 
2 ) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
)
10079, 82, 993eqtri 2465 . . 3  |-  ( * `
 ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 1 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
101 3m1e2 10434 . . . 4  |-  ( 3  -  1 )  =  2
102101oveq2i 6101 . . 3  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ (
3  -  1 ) )  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )
10373, 100, 1023eqtr3ri 2470 . 2  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
10470, 103pm3.2i 452 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 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   _ici 9280    + caddc 9281    x. cmul 9283    <_ cle 9415    - cmin 9591   -ucneg 9592    / cdiv 9989   NNcn 10318   2c2 10367   3c3 10368   6c6 10371   ZZcz 10642   ^cexp 11861   *ccj 12581   sqrcsqr 12718   expce 13343   sincsin 13345   cosccos 13346   picpi 13348   logclog 21965    ^c ccxp 21966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-cxp 21968
This theorem is referenced by:  1cubr  22196
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