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Theorem 1cubr 22237
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 9340 . . . . . . 7  |-  1  e.  CC
3 neg1cn 10425 . . . . . . . . 9  |-  -u 1  e.  CC
4 ax-icn 9341 . . . . . . . . . 10  |-  _i  e.  CC
5 3cn 10396 . . . . . . . . . . 11  |-  3  e.  CC
6 sqrcl 12849 . . . . . . . . . . 11  |-  ( 3  e.  CC  ->  ( sqr `  3 )  e.  CC )
75, 6ax-mp 5 . . . . . . . . . 10  |-  ( sqr `  3 )  e.  CC
84, 7mulcli 9391 . . . . . . . . 9  |-  ( _i  x.  ( sqr `  3
) )  e.  CC
93, 8addcli 9390 . . . . . . . 8  |-  ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  e.  CC
10 halfcl 10550 . . . . . . . 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 9684 . . . . . . . 8  |-  ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  e.  CC
13 halfcl 10550 . . . . . . . 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 1166 . . . . . 6  |-  ( 1  e.  CC  /\  (
( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC  /\  (
( -u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e.  CC )
162elexi 2982 . . . . . . 7  |-  1  e.  _V
17 ovex 6116 . . . . . . 7  |-  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )  e.  _V
18 ovex 6116 . . . . . . 7  |-  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )  e. 
_V
1916, 17, 18tpss 4038 . . . . . 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 208 . . . . 5  |-  { 1 ,  ( ( -u
1  +  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) ,  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) }  C_  CC
211, 20eqsstri 3386 . . . 4  |-  R  C_  CC
2221sseli 3352 . . 3  |-  ( A  e.  R  ->  A  e.  CC )
2322pm4.71ri 633 . 2  |-  ( A  e.  R  <->  ( A  e.  CC  /\  A  e.  R ) )
24 3nn 10480 . . . . 5  |-  3  e.  NN
25 cxpeq 22195 . . . . 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 1306 . . . 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 3918 . . . . 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 2507 . . . . 5  |-  ( A  e.  R  <->  A  e.  { 1 ,  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) } )
29 3m1e2 10438 . . . . . . . . . 10  |-  ( 3  -  1 )  =  2
30 2cn 10392 . . . . . . . . . . 11  |-  2  e.  CC
3130addid2i 9557 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
3229, 31eqtr4i 2466 . . . . . . . . 9  |-  ( 3  -  1 )  =  ( 0  +  2 )
3332oveq2i 6102 . . . . . . . 8  |-  ( 0 ... ( 3  -  1 ) )  =  ( 0 ... (
0  +  2 ) )
34 0z 10657 . . . . . . . . 9  |-  0  e.  ZZ
35 fztp 11512 . . . . . . . . 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 2463 . . . . . . 7  |-  ( 0 ... ( 3  -  1 ) )  =  { 0 ,  ( 0  +  1 ) ,  ( 0  +  2 ) }
3837rexeqi 2922 . . . . . 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 10416 . . . . . . . . . . 11  |-  3  =/=  0
405, 39reccli 10061 . . . . . . . . . 10  |-  ( 1  /  3 )  e.  CC
41 1cxp 22117 . . . . . . . . . 10  |-  ( ( 1  /  3 )  e.  CC  ->  (
1  ^c  ( 1  /  3 ) )  =  1 )
4240, 41ax-mp 5 . . . . . . . . 9  |-  ( 1  ^c  ( 1  /  3 ) )  =  1
4342oveq1i 6101 . . . . . . . 8  |-  ( ( 1  ^c  ( 1  /  3 ) )  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )
4443eqeq2i 2453 . . . . . . 7  |-  ( A  =  ( ( 1  ^c  ( 1  /  3 ) )  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ n ) )  <->  A  =  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) ) )
4544rexbii 2740 . . . . . 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 2982 . . . . . . 7  |-  0  e.  _V
47 ovex 6116 . . . . . . 7  |-  ( 0  +  1 )  e. 
_V
48 ovex 6116 . . . . . . 7  |-  ( 0  +  2 )  e. 
_V
49 oveq2 6099 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 0 ) )
5030, 5, 39divcli 10073 . . . . . . . . . . . . 13  |-  ( 2  /  3 )  e.  CC
51 cxpcl 22119 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  ( 2  /  3
)  e.  CC )  ->  ( -u 1  ^c  ( 2  /  3 ) )  e.  CC )
523, 50, 51mp2an 672 . . . . . . . . . . . 12  |-  ( -u
1  ^c  ( 2  /  3 ) )  e.  CC
53 exp0 11869 . . . . . . . . . . . 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 2491 . . . . . . . . . 10  |-  ( n  =  0  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  1 )
5655oveq2d 6107 . . . . . . . . 9  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  1 ) )
57 1t1e1 10469 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
5856, 57syl6eq 2491 . . . . . . . 8  |-  ( n  =  0  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  1 )
5958eqeq2d 2454 . . . . . . 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 9557 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
6260, 61syl6eq 2491 . . . . . . . . . . . 12  |-  ( n  =  ( 0  +  1 )  ->  n  =  1 )
6362oveq2d 6107 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 1 ) )
64 exp1 11871 . . . . . . . . . . . 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 2491 . . . . . . . . . 10  |-  ( n  =  ( 0  +  1 )  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  ( -u
1  ^c  ( 2  /  3 ) ) )
6766oveq2d 6107 . . . . . . . . 9  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( -u
1  ^c  ( 2  /  3 ) ) ) )
6852mulid2i 9389 . . . . . . . . . 10  |-  ( 1  x.  ( -u 1  ^c  ( 2  /  3 ) ) )  =  ( -u
1  ^c  ( 2  /  3 ) )
69 1cubrlem 22236 . . . . . . . . . . 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 458 . . . . . . . . . 10  |-  ( -u
1  ^c  ( 2  /  3 ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7168, 70eqtri 2463 . . . . . . . . 9  |-  ( 1  x.  ( -u 1  ^c  ( 2  /  3 ) ) )  =  ( (
-u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 )
7267, 71syl6eq 2491 . . . . . . . 8  |-  ( n  =  ( 0  +  1 )  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
7372eqeq2d 2454 . . . . . . 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 2491 . . . . . . . . . . 11  |-  ( n  =  ( 0  +  2 )  ->  n  =  2 )
7675oveq2d 6107 . . . . . . . . . 10  |-  ( n  =  ( 0  +  2 )  ->  (
( -u 1  ^c 
( 2  /  3
) ) ^ n
)  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 ) )
7776oveq2d 6107 . . . . . . . . 9  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( 1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 ) ) )
7852sqcli 11946 . . . . . . . . . . 11  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )  e.  CC
7978mulid2i 9389 . . . . . . . . . 10  |-  ( 1  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )
8069simpri 462 . . . . . . . . . 10  |-  ( (
-u 1  ^c 
( 2  /  3
) ) ^ 2 )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8179, 80eqtri 2463 . . . . . . . . 9  |-  ( 1  x.  ( ( -u
1  ^c  ( 2  /  3 ) ) ^ 2 ) )  =  ( (
-u 1  -  (
_i  x.  ( sqr `  3 ) ) )  /  2 )
8277, 81syl6eq 2491 . . . . . . . 8  |-  ( n  =  ( 0  +  2 )  ->  (
1  x.  ( (
-u 1  ^c 
( 2  /  3
) ) ^ n
) )  =  ( ( -u 1  -  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) )
8382eqeq2d 2454 . . . . . . 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 3932 . . . . . 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 271 . . . . 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 288 . . . 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 256 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 3 )  =  1  <->  A  e.  R ) )
8887pm5.32i 637 . 2  |-  ( ( A  e.  CC  /\  ( A ^ 3 )  =  1 )  <->  ( A  e.  CC  /\  A  e.  R ) )
8923, 88bitr4i 252 1  |-  ( A  e.  R  <->  ( A  e.  CC  /\  ( A ^ 3 )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2716    C_ wss 3328   {ctp 3881   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283   _ici 9284    + caddc 9285    x. cmul 9287    - cmin 9595   -ucneg 9596    / cdiv 9993   NNcn 10322   2c2 10371   3c3 10372   ZZcz 10646   ...cfz 11437   ^cexp 11865   sqrcsqr 12722    ^c ccxp 22007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008  df-cxp 22009
This theorem is referenced by:  cubic  22244
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