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Theorem 1cubr 23847
Description: The cube roots of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
1cubr.r  |-  R  =  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) }
Assertion
Ref Expression
1cubr  |-  ( A  e.  R  <->  ( A  e.  CC  /\  ( A ^ 3 )  =  1 ) )

Proof of Theorem 1cubr
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 1cubr.r . . . . 5  |-  R  =  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) }
2 ax-1cn 9615 . . . . . . 7  |-  1  e.  CC
3 neg1cn 10735 . . . . . . . . 9  |-  -u 1  e.  CC
4 ax-icn 9616 . . . . . . . . . 10  |-  _i  e.  CC
5 3cn 10706 . . . . . . . . . . 11  |-  3  e.  CC
6 sqrtcl 13501 . . . . . . . . . . 11  |-  ( 3  e.  CC  ->  ( sqr `  3 )  e.  CC )
75, 6ax-mp 5 . . . . . . . . . 10  |-  ( sqr `  3 )  e.  CC
84, 7mulcli 9666 . . . . . . . . 9  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
93, 8addcli 9665 . . . . . . . 8  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
10 halfcl 10861 . . . . . . . 8  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  e.  CC  ->  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
119, 10ax-mp 5 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC
123, 8subcli 9970 . . . . . . . 8  |-  ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  e.  CC
13 halfcl 10861 . . . . . . . 8  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  e.  CC  ->  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
1412, 13ax-mp 5 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC
152, 11, 143pm3.2i 1208 . . . . . 6  |-  ( 1  e.  CC  /\  (
( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC  /\  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
162elexi 3041 . . . . . . 7  |-  1  e.  _V
17 ovex 6336 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  _V
18 ovex 6336 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e. 
_V
1916, 17, 18tpss 4129 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2
)  e.  CC  /\  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
)  e.  CC )  <->  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) }  C_  CC )
2015, 19mpbi 213 . . . . 5  |-  { 1 ,  ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) ,  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) }  C_  CC
211, 20eqsstri 3448 . . . 4  |-  R  C_  CC
2221sseli 3414 . . 3  |-  ( A  e.  R  ->  A  e.  CC )
2322pm4.71ri 645 . 2  |-  ( A  e.  R  <->  ( A  e.  CC  /\  A  e.  R ) )
24 3nn 10791 . . . . 5  |-  3  e.  NN
25 cxpeq 23776 . . . . 5  |-  ( ( A  e.  CC  /\  3  e.  NN  /\  1  e.  CC )  ->  (
( A ^ 3 )  =  1  <->  E. n  e.  ( 0 ... ( 3  -  1 ) ) A  =  ( ( 1  ^c  ( 1  /  3 ) )  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ n ) ) ) )
2624, 2, 25mp3an23 1382 . . . 4  |-  ( A  e.  CC  ->  (
( A ^ 3 )  =  1  <->  E. n  e.  ( 0 ... ( 3  -  1 ) ) A  =  ( ( 1  ^c  ( 1  /  3 ) )  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ n ) ) ) )
27 eltpg 4005 . . . . 5  |-  ( A  e.  CC  ->  ( A  e.  { 1 ,  ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) ,  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) }  <->  ( A  =  1  \/  A  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2
)  \/  A  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
) ) ) )
281eleq2i 2541 . . . . 5  |-  ( A  e.  R  <->  A  e.  { 1 ,  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) } )
29 3m1e2 10748 . . . . . . . . . 10  |-  ( 3  -  1 )  =  2
30 2cn 10702 . . . . . . . . . . 11  |-  2  e.  CC
3130addid2i 9839 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
3229, 31eqtr4i 2496 . . . . . . . . 9  |-  ( 3  -  1 )  =  ( 0  +  2 )
3332oveq2i 6319 . . . . . . . 8  |-  ( 0 ... ( 3  -  1 ) )  =  ( 0 ... (
0  +  2 ) )
34 0z 10972 . . . . . . . . 9  |-  0  e.  ZZ
35 fztp 11878 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  (
0 ... ( 0  +  2 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } )
3634, 35ax-mp 5 . . . . . . . 8  |-  ( 0 ... ( 0  +  2 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
3733, 36eqtri 2493 . . . . . . 7  |-  ( 0 ... ( 3  -  1 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
3837rexeqi 2978 . . . . . 6  |-  ( E. n  e.  ( 0 ... ( 3  -  1 ) ) A  =  ( ( 1  ^c  ( 1  /  3 ) )  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ n ) )  <->  E. n  e.  {
0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } A  =  ( ( 1  ^c  ( 1  / 
3 ) )  x.  ( ( -u 1  ^c  ( 2  /  3 ) ) ^ n ) ) )
39 3ne0 10726 . . . . . . . . . . 11  |-  3  =/=  0
405, 39reccli 10359 . . . . . . . . . 10  |-  ( 1  /  3 )  e.  CC
41 1cxp 23696 . . . . . . . . . 10  |-  ( ( 1  /  3 )  e.  CC  ->  (
1  ^c  ( 1  /  3 ) )  =  1 )
4240, 41ax-mp 5 . . . . . . . . 9  |-  ( 1  ^c  ( 1  /  3 ) )  =  1
4342oveq1i 6318 . . . . . . . 8  |-  ( ( 1  ^c  ( 1  /  3 ) )  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )
4443eqeq2i 2483 . . . . . . 7  |-  ( A  =  ( ( 1  ^c  ( 1  /  3 ) )  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) ) )
4544rexbii 2881 . . . . . 6  |-  ( E. n  e.  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } A  =  ( ( 1  ^c 
( 1  /  3
) )  x.  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
) )  <->  E. n  e.  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } A  =  ( 1  x.  ( ( -u 1  ^c  ( 2  /  3 ) ) ^ n ) ) )
4634elexi 3041 . . . . . . 7  |-  0  e.  _V
47 ovex 6336 . . . . . . 7  |-  ( 0  +  1 )  e. 
_V
48 ovex 6336 . . . . . . 7  |-  ( 0  +  2 )  e. 
_V
49 oveq2 6316 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 0 ) )
5030, 5, 39divcli 10371 . . . . . . . . . . . . 13  |-  ( 2  /  3 )  e.  CC
51 cxpcl 23698 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^c  ( 2  /  3 ) )  e.  CC )
523, 50, 51mp2an 686 . . . . . . . . . . . 12  |-  ( -u
1  ^c  ( 2  /  3 ) )  e.  CC
53 exp0 12314 . . . . . . . . . . . 12  |-  ( (
-u 1  ^c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^c  ( 2  /  3 ) ) ^ 0 )  =  1 )
5452, 53ax-mp 5 . . . . . . . . . . 11  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 0 )  =  1
5549, 54syl6eq 2521 . . . . . . . . . 10  |-  ( n  =  0  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  1 )
5655oveq2d 6324 . . . . . . . . 9  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  1 ) )
57 1t1e1 10780 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
5856, 57syl6eq 2521 . . . . . . . 8  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  1 )
5958eqeq2d 2481 . . . . . . 7  |-  ( n  =  0  ->  ( A  =  ( 1  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  1
) )
60 id 22 . . . . . . . . . . . . 13  |-  ( n  =  ( 0  +  1 )  ->  n  =  ( 0  +  1 ) )
612addid2i 9839 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
6260, 61syl6eq 2521 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  1 )  ->  n  =  1 )
6362oveq2d 6324 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 1 ) )
64 exp1 12316 . . . . . . . . . . . 12  |-  ( (
-u 1  ^c 
( 2  /  3
) )  e.  CC  ->  ( ( -u 1  ^c  ( 2  /  3 ) ) ^ 1 )  =  ( -u 1  ^c  ( 2  / 
3 ) ) )
6552, 64ax-mp 5 . . . . . . . . . . 11  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 1 )  =  ( -u
1  ^c  ( 2  /  3 ) )
6663, 65syl6eq 2521 . . . . . . . . . 10  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  ( -u
1  ^c  ( 2  /  3 ) ) )
6766oveq2d 6324 . . . . . . . . 9  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( -u
1  ^c  ( 2  /  3 ) ) ) )
6852mulid2i 9664 . . . . . . . . . 10  |-  ( 1  x.  ( -u 1  ^c  ( 2  /  3 ) ) )  =  ( -u
1  ^c  ( 2  /  3 ) )
69 1cubrlem 23846 . . . . . . . . . . 11  |-  ( (
-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 ) )
7069simpli 465 . . . . . . . . . 10  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7168, 70eqtri 2493 . . . . . . . . 9  |-  ( 1  x.  ( -u 1  ^c  ( 2  /  3 ) ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7267, 71syl6eq 2521 . . . . . . . 8  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
7372eqeq2d 2481 . . . . . . 7  |-  ( n  =  ( 0  +  1 )  ->  ( A  =  ( 1  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ) )
74 id 22 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  2 )  ->  n  =  ( 0  +  2 ) )
7574, 31syl6eq 2521 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  2 )  ->  n  =  2 )
7675oveq2d 6324 . . . . . . . . . 10  |-  ( n  =  ( 0  +  2 )  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 ) )
7776oveq2d 6324 . . . . . . . . 9  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 ) ) )
7852sqcli 12393 . . . . . . . . . . 11  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )  e.  CC
7978mulid2i 9664 . . . . . . . . . 10  |-  ( 1  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )
8069simpri 469 . . . . . . . . . 10  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8179, 80eqtri 2493 . . . . . . . . 9  |-  ( 1  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8277, 81syl6eq 2521 . . . . . . . 8  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
8382eqeq2d 2481 . . . . . . 7  |-  ( n  =  ( 0  +  2 )  ->  ( A  =  ( 1  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 ) ) )
8446, 47, 48, 59, 73, 83rextp 4019 . . . . . 6  |-  ( E. n  e.  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) } A  =  ( 1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  <->  ( A  =  1  \/  A  =  ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 )  \/  A  =  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) ) )
8538, 45, 843bitri 279 . . . . 5  |-  ( E. n  e.  ( 0 ... ( 3  -  1 ) ) A  =  ( ( 1  ^c  ( 1  /  3 ) )  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ n ) )  <->  ( A  =  1  \/  A  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2
)  \/  A  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2
) ) )
8627, 28, 853bitr4g 296 . . . 4  |-  ( A  e.  CC  ->  ( A  e.  R  <->  E. n  e.  ( 0 ... (
3  -  1 ) ) A  =  ( ( 1  ^c 
( 1  /  3
) )  x.  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
) ) ) )
8726, 86bitr4d 264 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 3 )  =  1  <->  A  e.  R ) )
8887pm5.32i 649 . 2  |-  ( ( A  e.  CC  /\  ( A ^ 3 )  =  1 )  <->  ( A  e.  CC  /\  A  e.  R ) )
8923, 88bitr4i 260 1  |-  ( A  e.  R  <->  ( A  e.  CC  /\  ( A ^ 3 )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    \/ w3o 1006    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757    C_ wss 3390   {ctp 3963   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558   _ici 9559    + caddc 9560    x. cmul 9562    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   3c3 10682   ZZcz 10961   ...cfz 11810   ^cexp 12310   sqrcsqrt 13373    ^c ccxp 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586
This theorem is referenced by:  cubic  23854
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