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Theorem cubic 20161
Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3702 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 9832 . . . . . . . . . . 11  |-  2  e.  CC
10 3nn0 9999 . . . . . . . . . . . 12  |-  3  e.  NN0
11 expcl 11137 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  3  e.  NN0 )  -> 
( B ^ 3 )  e.  CC )
123, 10, 11sylancl 643 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 3 )  e.  CC )
13 mulcl 8837 . . . . . . . . . . 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 9900 . . . . . . . . . . . . 13  |-  9  e.  NN
1615nncni 9772 . . . . . . . . . . . 12  |-  9  e.  CC
17 mulcl 8837 . . . . . . . . . . . 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 8871 . . . . . . . . . . 11  |-  ( ph  ->  ( B  x.  C
)  e.  CC )
2018, 19mulcld 8871 . . . . . . . . . 10  |-  ( ph  ->  ( ( 9  x.  A )  x.  ( B  x.  C )
)  e.  CC )
2114, 20subcld 9173 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  ( B ^ 3 ) )  -  (
( 9  x.  A
)  x.  ( B  x.  C ) ) )  e.  CC )
22 2nn0 9998 . . . . . . . . . . . 12  |-  2  e.  NN0
23 7nn 9898 . . . . . . . . . . . 12  |-  7  e.  NN
2422, 23decnncl 10153 . . . . . . . . . . 11  |- ; 2 7  e.  NN
2524nncni 9772 . . . . . . . . . 10  |- ; 2 7  e.  CC
261sqcld 11259 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
2726, 5mulcld 8871 . . . . . . . . . 10  |-  ( ph  ->  ( ( A ^
2 )  x.  D
)  e.  CC )
28 mulcl 8837 . . . . . . . . . 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 8870 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  ( B ^
3 ) )  -  ( ( 9  x.  A )  x.  ( B  x.  C )
) )  +  (; 2
7  x.  ( ( A ^ 2 )  x.  D ) ) )  e.  CC )
318, 30eqeltrd 2370 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
32 cubic.g . . . . . . . . 9  |-  ( ph  ->  G  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )
3331sqcld 11259 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
34 4cn 9836 . . . . . . . . . . 11  |-  4  e.  CC
35 cubic.m . . . . . . . . . . . . 13  |-  ( ph  ->  M  =  ( ( B ^ 2 )  -  ( 3  x.  ( A  x.  C
) ) ) )
363sqcld 11259 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
37 3cn 9834 . . . . . . . . . . . . . . 15  |-  3  e.  CC
381, 4mulcld 8871 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
39 mulcl 8837 . . . . . . . . . . . . . . 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 9173 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B ^
2 )  -  (
3  x.  ( A  x.  C ) ) )  e.  CC )
4235, 41eqeltrd 2370 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  CC )
43 expcl 11137 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
4442, 10, 43sylancl 643 . . . . . . . . . . 11  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
45 mulcl 8837 . . . . . . . . . . 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 9173 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
4  x.  ( M ^ 3 ) ) )  e.  CC )
4832, 47eqeltrd 2370 . . . . . . . 8  |-  ( ph  ->  G  e.  CC )
4948sqrcld 11935 . . . . . . 7  |-  ( ph  ->  ( sqr `  G
)  e.  CC )
5031, 49addcld 8870 . . . . . 6  |-  ( ph  ->  ( N  +  ( sqr `  G ) )  e.  CC )
5150halfcld 9972 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC )
52 3ne0 9847 . . . . . 6  |-  3  =/=  0
5337, 52reccli 9506 . . . . 5  |-  ( 1  /  3 )  e.  CC
54 cxpcl 20037 . . . . 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 2370 . . 3  |-  ( ph  ->  T  e.  CC )
577oveq1d 5889 . . . 4  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  ^ c  ( 1  /  3 ) ) ^ 3 ) )
58 3nn 9894 . . . . 5  |-  3  e.  NN
59 cxproot 20053 . . . . 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 2328 . . 3  |-  ( ph  ->  ( T ^ 3 )  =  ( ( N  +  ( sqr `  G ) )  / 
2 ) )
6248sqsqrd 11937 . . . 4  |-  ( ph  ->  ( ( sqr `  G
) ^ 2 )  =  G )
6362, 32eqtrd 2328 . . 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 9895 . . . . . . . . . . 11  |-  4  e.  NN
6766nnne0i 9796 . . . . . . . . . 10  |-  4  =/=  0
6867a1i 10 . . . . . . . . 9  |-  ( ph  ->  4  =/=  0 )
69 cubic.0 . . . . . . . . . 10  |-  ( ph  ->  M  =/=  0 )
7010nn0zi 10064 . . . . . . . . . . 11  |-  3  e.  ZZ
7170a1i 10 . . . . . . . . . 10  |-  ( ph  ->  3  e.  ZZ )
7242, 69, 71expne0d 11267 . . . . . . . . 9  |-  ( ph  ->  ( M ^ 3 )  =/=  0 )
7365, 44, 68, 72mulne0d 9436 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( M ^ 3 ) )  =/=  0 )
7463oveq2d 5890 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N ^ 2 )  -  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) ) )
75 subsq 11226 . . . . . . . . . 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 9178 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( N ^ 2 )  -  ( 4  x.  ( M ^
3 ) ) ) )  =  ( 4  x.  ( M ^
3 ) ) )
7874, 76, 773eqtr3d 2336 . . . . . . . 8  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =  ( 4  x.  ( M ^ 3 ) ) )
7931, 49subcld 9173 . . . . . . . . 9  |-  ( ph  ->  ( N  -  ( sqr `  G ) )  e.  CC )
8079mul02d 9026 . . . . . . . 8  |-  ( ph  ->  ( 0  x.  ( N  -  ( sqr `  G ) ) )  =  0 )
8173, 78, 803netr4d 2486 . . . . . . 7  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =/=  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
82 oveq1 5881 . . . . . . . 8  |-  ( ( N  +  ( sqr `  G ) )  =  0  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) )  =  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
8382necon3i 2498 . . . . . . 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 9845 . . . . . . 7  |-  2  =/=  0
8685a1i 10 . . . . . 6  |-  ( ph  ->  2  =/=  0 )
8750, 64, 84, 86divne0d 9568 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  =/=  0 )
8853a1i 10 . . . . 5  |-  ( ph  ->  ( 1  /  3
)  e.  CC )
8951, 87, 88cxpne0d 20076 . . . 4  |-  ( ph  ->  ( ( ( N  +  ( sqr `  G
) )  /  2
)  ^ c  ( 1  /  3 ) )  =/=  0 )
907, 89eqnetrd 2477 . . 3  |-  ( ph  ->  T  =/=  0 )
911, 2, 3, 4, 5, 6, 56, 61, 49, 63, 35, 8, 90cubic2 20160 . 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 20154 . . . . 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 2585 . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {ctp 3655   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   3c3 9812   4c4 9813   7c7 9816   9c9 9818   NN0cn0 9981   ZZcz 10040  ;cdc 10140   ^cexp 11120   sqrcsqr 11734    ^ c ccxp 19929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931
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