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Theorem cubic 20140
Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3689 to convert the existential quantifier to a triple disjunction. (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
cubic.r  |-  R  =  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) }
cubic.a  |-  ( ph  ->  A  e.  CC )
cubic.z  |-  ( ph  ->  A  =/=  0 )
cubic.b  |-  ( ph  ->  B  e.  CC )
cubic.c  |-  ( ph  ->  C  e.  CC )
cubic.d  |-  ( ph  ->  D  e.  CC )
cubic.x  |-  ( ph  ->  X  e.  CC )
cubic.t  |-  ( ph  ->  T  =  ( ( ( N  +  ( sqr `  G ) )  /  2 )  ^ c  ( 1  /  3 ) ) )
cubic.g  |-  ( ph  ->  G  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )
cubic.m  |-  ( ph  ->  M  =  ( ( B ^ 2 )  -  ( 3  x.  ( A  x.  C
) ) ) )
cubic.n  |-  ( ph  ->  N  =  ( ( ( 2  x.  ( B ^ 3 ) )  -  ( ( 9  x.  A )  x.  ( B  x.  C
) ) )  +  (; 2 7  x.  (
( A ^ 2 )  x.  D ) ) ) )
cubic.0  |-  ( ph  ->  M  =/=  0 )
Assertion
Ref Expression
cubic  |-  ( ph  ->  ( ( ( ( A  x.  ( X ^ 3 ) )  +  ( B  x.  ( X ^ 2 ) ) )  +  ( ( C  x.  X
)  +  D ) )  =  0  <->  E. r  e.  R  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  / 
( r  x.  T
) ) )  / 
( 3  x.  A
) ) ) )
Distinct variable groups:    A, r    B, r    M, r    N, r    ph, r    T, r    X, r
Allowed substitution hints:    C( r)    D( r)    R( r)    G( r)

Proof of Theorem cubic
StepHypRef Expression
1 cubic.a . . 3  |-  ( ph  ->  A  e.  CC )
2 cubic.z . . 3  |-  ( ph  ->  A  =/=  0 )
3 cubic.b . . 3  |-  ( ph  ->  B  e.  CC )
4 cubic.c . . 3  |-  ( ph  ->  C  e.  CC )
5 cubic.d . . 3  |-  ( ph  ->  D  e.  CC )
6 cubic.x . . 3  |-  ( ph  ->  X  e.  CC )
7 cubic.t . . . 4  |-  ( ph  ->  T  =  ( ( ( N  +  ( sqr `  G ) )  /  2 )  ^ c  ( 1  /  3 ) ) )
8 cubic.n . . . . . . . 8  |-  ( ph  ->  N  =  ( ( ( 2  x.  ( B ^ 3 ) )  -  ( ( 9  x.  A )  x.  ( B  x.  C
) ) )  +  (; 2 7  x.  (
( A ^ 2 )  x.  D ) ) ) )
9 2cn 9811 . . . . . . . . . . 11  |-  2  e.  CC
10 3nn0 9978 . . . . . . . . . . . 12  |-  3  e.  NN0
11 expcl 11116 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  3  e.  NN0 )  -> 
( B ^ 3 )  e.  CC )
123, 10, 11sylancl 643 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 3 )  e.  CC )
13 mulcl 8816 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  ( B ^ 3 )  e.  CC )  -> 
( 2  x.  ( B ^ 3 ) )  e.  CC )
149, 12, 13sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  ( B ^ 3 ) )  e.  CC )
15 9nn 9879 . . . . . . . . . . . . 13  |-  9  e.  NN
1615nncni 9751 . . . . . . . . . . . 12  |-  9  e.  CC
17 mulcl 8816 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  A  e.  CC )  ->  ( 9  x.  A
)  e.  CC )
1816, 1, 17sylancr 644 . . . . . . . . . . 11  |-  ( ph  ->  ( 9  x.  A
)  e.  CC )
193, 4mulcld 8850 . . . . . . . . . . 11  |-  ( ph  ->  ( B  x.  C
)  e.  CC )
2018, 19mulcld 8850 . . . . . . . . . 10  |-  ( ph  ->  ( ( 9  x.  A )  x.  ( B  x.  C )
)  e.  CC )
2114, 20subcld 9152 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  ( B ^ 3 ) )  -  (
( 9  x.  A
)  x.  ( B  x.  C ) ) )  e.  CC )
22 2nn0 9977 . . . . . . . . . . . 12  |-  2  e.  NN0
23 7nn 9877 . . . . . . . . . . . 12  |-  7  e.  NN
2422, 23decnncl 10132 . . . . . . . . . . 11  |- ; 2 7  e.  NN
2524nncni 9751 . . . . . . . . . 10  |- ; 2 7  e.  CC
261sqcld 11238 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
2726, 5mulcld 8850 . . . . . . . . . 10  |-  ( ph  ->  ( ( A ^
2 )  x.  D
)  e.  CC )
28 mulcl 8816 . . . . . . . . . 10  |-  ( (; 2
7  e.  CC  /\  ( ( A ^
2 )  x.  D
)  e.  CC )  ->  (; 2 7  x.  (
( A ^ 2 )  x.  D ) )  e.  CC )
2925, 27, 28sylancr 644 . . . . . . . . 9  |-  ( ph  ->  (; 2 7  x.  (
( A ^ 2 )  x.  D ) )  e.  CC )
3021, 29addcld 8849 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  ( B ^
3 ) )  -  ( ( 9  x.  A )  x.  ( B  x.  C )
) )  +  (; 2
7  x.  ( ( A ^ 2 )  x.  D ) ) )  e.  CC )
318, 30eqeltrd 2357 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
32 cubic.g . . . . . . . . 9  |-  ( ph  ->  G  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )
3331sqcld 11238 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
34 4cn 9815 . . . . . . . . . . 11  |-  4  e.  CC
35 cubic.m . . . . . . . . . . . . 13  |-  ( ph  ->  M  =  ( ( B ^ 2 )  -  ( 3  x.  ( A  x.  C
) ) ) )
363sqcld 11238 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
37 3cn 9813 . . . . . . . . . . . . . . 15  |-  3  e.  CC
381, 4mulcld 8850 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
39 mulcl 8816 . . . . . . . . . . . . . . 15  |-  ( ( 3  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( 3  x.  ( A  x.  C
) )  e.  CC )
4037, 38, 39sylancr 644 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 3  x.  ( A  x.  C )
)  e.  CC )
4136, 40subcld 9152 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B ^
2 )  -  (
3  x.  ( A  x.  C ) ) )  e.  CC )
4235, 41eqeltrd 2357 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  CC )
43 expcl 11116 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
4442, 10, 43sylancl 643 . . . . . . . . . . 11  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
45 mulcl 8816 . . . . . . . . . . 11  |-  ( ( 4  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( 4  x.  ( M ^ 3 ) )  e.  CC )
4634, 44, 45sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  ( M ^ 3 ) )  e.  CC )
4733, 46subcld 9152 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
4  x.  ( M ^ 3 ) ) )  e.  CC )
4832, 47eqeltrd 2357 . . . . . . . 8  |-  ( ph  ->  G  e.  CC )
4948sqrcld 11914 . . . . . . 7  |-  ( ph  ->  ( sqr `  G
)  e.  CC )
5031, 49addcld 8849 . . . . . 6  |-  ( ph  ->  ( N  +  ( sqr `  G ) )  e.  CC )
5150halfcld 9951 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC )
52 3ne0 9826 . . . . . 6  |-  3  =/=  0
5337, 52reccli 9485 . . . . 5  |-  ( 1  /  3 )  e.  CC
54 cxpcl 20016 . . . . 5  |-  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC  /\  (
1  /  3 )  e.  CC )  -> 
( ( ( N  +  ( sqr `  G
) )  /  2
)  ^ c  ( 1  /  3 ) )  e.  CC )
5551, 53, 54sylancl 643 . . . 4  |-  ( ph  ->  ( ( ( N  +  ( sqr `  G
) )  /  2
)  ^ c  ( 1  /  3 ) )  e.  CC )
567, 55eqeltrd 2357 . . 3  |-  ( ph  ->  T  e.  CC )
577oveq1d 5834 . . . 4  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  ^ c  ( 1  /  3 ) ) ^ 3 ) )
58 3nn 9873 . . . . 5  |-  3  e.  NN
59 cxproot 20032 . . . . 5  |-  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC  /\  3  e.  NN )  ->  (
( ( ( N  +  ( sqr `  G
) )  /  2
)  ^ c  ( 1  /  3 ) ) ^ 3 )  =  ( ( N  +  ( sqr `  G
) )  /  2
) )
6051, 58, 59sylancl 643 . . . 4  |-  ( ph  ->  ( ( ( ( N  +  ( sqr `  G ) )  / 
2 )  ^ c 
( 1  /  3
) ) ^ 3 )  =  ( ( N  +  ( sqr `  G ) )  / 
2 ) )
6157, 60eqtrd 2315 . . 3  |-  ( ph  ->  ( T ^ 3 )  =  ( ( N  +  ( sqr `  G ) )  / 
2 ) )
6248sqsqrd 11916 . . . 4  |-  ( ph  ->  ( ( sqr `  G
) ^ 2 )  =  G )
6362, 32eqtrd 2315 . . 3  |-  ( ph  ->  ( ( sqr `  G
) ^ 2 )  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )
649a1i 10 . . . . . 6  |-  ( ph  ->  2  e.  CC )
6534a1i 10 . . . . . . . . 9  |-  ( ph  ->  4  e.  CC )
66 4nn 9874 . . . . . . . . . . 11  |-  4  e.  NN
6766nnne0i 9775 . . . . . . . . . 10  |-  4  =/=  0
6867a1i 10 . . . . . . . . 9  |-  ( ph  ->  4  =/=  0 )
69 cubic.0 . . . . . . . . . 10  |-  ( ph  ->  M  =/=  0 )
7010nn0zi 10043 . . . . . . . . . . 11  |-  3  e.  ZZ
7170a1i 10 . . . . . . . . . 10  |-  ( ph  ->  3  e.  ZZ )
7242, 69, 71expne0d 11246 . . . . . . . . 9  |-  ( ph  ->  ( M ^ 3 )  =/=  0 )
7365, 44, 68, 72mulne0d 9415 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( M ^ 3 ) )  =/=  0 )
7463oveq2d 5835 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N ^ 2 )  -  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) ) )
75 subsq 11205 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  ( sqr `  G )  e.  CC )  -> 
( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) ) )
7631, 49, 75syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) ) )
7733, 46nncand 9157 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( N ^ 2 )  -  ( 4  x.  ( M ^
3 ) ) ) )  =  ( 4  x.  ( M ^
3 ) ) )
7874, 76, 773eqtr3d 2323 . . . . . . . 8  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =  ( 4  x.  ( M ^ 3 ) ) )
7931, 49subcld 9152 . . . . . . . . 9  |-  ( ph  ->  ( N  -  ( sqr `  G ) )  e.  CC )
8079mul02d 9005 . . . . . . . 8  |-  ( ph  ->  ( 0  x.  ( N  -  ( sqr `  G ) ) )  =  0 )
8173, 78, 803netr4d 2473 . . . . . . 7  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =/=  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
82 oveq1 5826 . . . . . . . 8  |-  ( ( N  +  ( sqr `  G ) )  =  0  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) )  =  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
8382necon3i 2485 . . . . . . 7  |-  ( ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =/=  ( 0  x.  ( N  -  ( sqr `  G ) ) )  ->  ( N  +  ( sqr `  G ) )  =/=  0 )
8481, 83syl 15 . . . . . 6  |-  ( ph  ->  ( N  +  ( sqr `  G ) )  =/=  0 )
85 2ne0 9824 . . . . . . 7  |-  2  =/=  0
8685a1i 10 . . . . . 6  |-  ( ph  ->  2  =/=  0 )
8750, 64, 84, 86divne0d 9547 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  =/=  0 )
8853a1i 10 . . . . 5  |-  ( ph  ->  ( 1  /  3
)  e.  CC )
8951, 87, 88cxpne0d 20055 . . . 4  |-  ( ph  ->  ( ( ( N  +  ( sqr `  G
) )  /  2
)  ^ c  ( 1  /  3 ) )  =/=  0 )
907, 89eqnetrd 2464 . . 3  |-  ( ph  ->  T  =/=  0 )
911, 2, 3, 4, 5, 6, 56, 61, 49, 63, 35, 8, 90cubic2 20139 . 2  |-  ( ph  ->  ( ( ( ( A  x.  ( X ^ 3 ) )  +  ( B  x.  ( X ^ 2 ) ) )  +  ( ( C  x.  X
)  +  D ) )  =  0  <->  E. r  e.  CC  (
( r ^ 3 )  =  1  /\  X  =  -u (
( ( B  +  ( r  x.  T
) )  +  ( M  /  ( r  x.  T ) ) )  /  ( 3  x.  A ) ) ) ) )
92 cubic.r . . . . . 6  |-  R  =  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) }
93921cubr 20133 . . . . 5  |-  ( r  e.  R  <->  ( r  e.  CC  /\  ( r ^ 3 )  =  1 ) )
9493anbi1i 676 . . . 4  |-  ( ( r  e.  R  /\  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  /  ( r  x.  T ) ) )  /  ( 3  x.  A ) ) )  <-> 
( ( r  e.  CC  /\  ( r ^ 3 )  =  1 )  /\  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  / 
( r  x.  T
) ) )  / 
( 3  x.  A
) ) ) )
95 anass 630 . . . 4  |-  ( ( ( r  e.  CC  /\  ( r ^ 3 )  =  1 )  /\  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  /  (
r  x.  T ) ) )  /  (
3  x.  A ) ) )  <->  ( r  e.  CC  /\  ( ( r ^ 3 )  =  1  /\  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  / 
( r  x.  T
) ) )  / 
( 3  x.  A
) ) ) ) )
9694, 95bitri 240 . . 3  |-  ( ( r  e.  R  /\  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  /  ( r  x.  T ) ) )  /  ( 3  x.  A ) ) )  <-> 
( r  e.  CC  /\  ( ( r ^
3 )  =  1  /\  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  /  (
r  x.  T ) ) )  /  (
3  x.  A ) ) ) ) )
9796rexbii2 2572 . 2  |-  ( E. r  e.  R  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  / 
( r  x.  T
) ) )  / 
( 3  x.  A
) )  <->  E. r  e.  CC  ( ( r ^ 3 )  =  1  /\  X  = 
-u ( ( ( B  +  ( r  x.  T ) )  +  ( M  / 
( r  x.  T
) ) )  / 
( 3  x.  A
) ) ) )
9891, 97syl6bbr 254 1  |-  ( ph  ->  ( ( ( ( A  x.  ( X ^ 3 ) )  +  ( B  x.  ( X ^ 2 ) ) )  +  ( ( C  x.  X
)  +  D ) )  =  0  <->  E. r  e.  R  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  / 
( r  x.  T
) ) )  / 
( 3  x.  A
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {ctp 3642   ` cfv 5220  (class class class)co 5819   CCcc 8730   0cc0 8732   1c1 8733   _ici 8734    + caddc 8735    x. cmul 8737    - cmin 9032   -ucneg 9033    / cdiv 9418   NNcn 9741   2c2 9790   3c3 9791   4c4 9792   7c7 9795   9c9 9797   NN0cn0 9960   ZZcz 10019  ;cdc 10119   ^cexp 11099   sqrcsqr 11713    ^ c ccxp 19908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4186  ax-pr 4212  ax-un 4510  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4303  df-id 4307  df-po 4312  df-so 4313  df-fr 4350  df-se 4351  df-we 4352  df-ord 4393  df-on 4394  df-lim 4395  df-suc 4396  df-om 4655  df-xp 4693  df-rel 4694  df-cnv 4695  df-co 4696  df-dm 4697  df-rn 4698  df-res 4699  df-ima 4700  df-fun 5222  df-fn 5223  df-f 5224  df-f1 5225  df-fo 5226  df-f1o 5227  df-fv 5228  df-isom 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10655  df-ioc 10656  df-ico 10657  df-icc 10658  df-fz 10778  df-fzo 10866  df-fl 10920  df-mod 10969  df-seq 11042  df-exp 11100  df-fac 11284  df-bc 11311  df-hash 11333  df-shft 11557  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-limsup 11940  df-clim 11957  df-rlim 11958  df-sum 12154  df-ef 12344  df-sin 12346  df-cos 12347  df-pi 12349  df-dvds 12527  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-starv 13218  df-sca 13219  df-vsca 13220  df-tset 13222  df-ple 13223  df-ds 13225  df-hom 13227  df-cco 13228  df-rest 13322  df-topn 13323  df-topgen 13339  df-pt 13340  df-prds 13343  df-xrs 13398  df-0g 13399  df-gsum 13400  df-qtop 13405  df-imas 13406  df-xps 13408  df-mre 13483  df-mrc 13484  df-acs 13486  df-mnd 14362  df-submnd 14411  df-mulg 14487  df-cntz 14788  df-cmn 15086  df-xmet 16368  df-met 16369  df-bl 16370  df-mopn 16371  df-cnfld 16373  df-top 16631  df-bases 16633  df-topon 16634  df-topsp 16635  df-cld 16751  df-ntr 16752  df-cls 16753  df-nei 16830  df-lp 16863  df-perf 16864  df-cn 16952  df-cnp 16953  df-haus 17038  df-tx 17252  df-hmeo 17441  df-fbas 17515  df-fg 17516  df-fil 17536  df-fm 17628  df-flim 17629  df-flf 17630  df-xms 17880  df-ms 17881  df-tms 17882  df-cncf 18377  df-limc 19211  df-dv 19212  df-log 19909  df-cxp 19910
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