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Theorem cubic 19977
Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3593 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 9696 . . . . . . . . . . 11  |-  2  e.  CC
10 3nn0 9862 . . . . . . . . . . . 12  |-  3  e.  NN0
11 expcl 10999 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  3  e.  NN0 )  -> 
( B ^ 3 )  e.  CC )
123, 10, 11sylancl 646 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 3 )  e.  CC )
13 mulcl 8701 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  ( B ^ 3 )  e.  CC )  -> 
( 2  x.  ( B ^ 3 ) )  e.  CC )
149, 12, 13sylancr 647 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  ( B ^ 3 ) )  e.  CC )
15 9nn 9763 . . . . . . . . . . . . 13  |-  9  e.  NN
1615nncni 9636 . . . . . . . . . . . 12  |-  9  e.  CC
17 mulcl 8701 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  A  e.  CC )  ->  ( 9  x.  A
)  e.  CC )
1816, 1, 17sylancr 647 . . . . . . . . . . 11  |-  ( ph  ->  ( 9  x.  A
)  e.  CC )
193, 4mulcld 8735 . . . . . . . . . . 11  |-  ( ph  ->  ( B  x.  C
)  e.  CC )
2018, 19mulcld 8735 . . . . . . . . . 10  |-  ( ph  ->  ( ( 9  x.  A )  x.  ( B  x.  C )
)  e.  CC )
2114, 20subcld 9037 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  ( B ^ 3 ) )  -  (
( 9  x.  A
)  x.  ( B  x.  C ) ) )  e.  CC )
22 2nn0 9861 . . . . . . . . . . . 12  |-  2  e.  NN0
23 7nn 9761 . . . . . . . . . . . 12  |-  7  e.  NN
2422, 23decnncl 10016 . . . . . . . . . . 11  |- ; 2 7  e.  NN
2524nncni 9636 . . . . . . . . . 10  |- ; 2 7  e.  CC
261sqcld 11121 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
2726, 5mulcld 8735 . . . . . . . . . 10  |-  ( ph  ->  ( ( A ^
2 )  x.  D
)  e.  CC )
28 mulcl 8701 . . . . . . . . . 10  |-  ( (; 2
7  e.  CC  /\  ( ( A ^
2 )  x.  D
)  e.  CC )  ->  (; 2 7  x.  (
( A ^ 2 )  x.  D ) )  e.  CC )
2925, 27, 28sylancr 647 . . . . . . . . 9  |-  ( ph  ->  (; 2 7  x.  (
( A ^ 2 )  x.  D ) )  e.  CC )
3021, 29addcld 8734 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  ( B ^
3 ) )  -  ( ( 9  x.  A )  x.  ( B  x.  C )
) )  +  (; 2
7  x.  ( ( A ^ 2 )  x.  D ) ) )  e.  CC )
318, 30eqeltrd 2327 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
32 cubic.g . . . . . . . . 9  |-  ( ph  ->  G  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )
3331sqcld 11121 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
34 4cn 9700 . . . . . . . . . . 11  |-  4  e.  CC
35 cubic.m . . . . . . . . . . . . 13  |-  ( ph  ->  M  =  ( ( B ^ 2 )  -  ( 3  x.  ( A  x.  C
) ) ) )
363sqcld 11121 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
37 3cn 9698 . . . . . . . . . . . . . . 15  |-  3  e.  CC
381, 4mulcld 8735 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
39 mulcl 8701 . . . . . . . . . . . . . . 15  |-  ( ( 3  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( 3  x.  ( A  x.  C
) )  e.  CC )
4037, 38, 39sylancr 647 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 3  x.  ( A  x.  C )
)  e.  CC )
4136, 40subcld 9037 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B ^
2 )  -  (
3  x.  ( A  x.  C ) ) )  e.  CC )
4235, 41eqeltrd 2327 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  CC )
43 expcl 10999 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
4442, 10, 43sylancl 646 . . . . . . . . . . 11  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
45 mulcl 8701 . . . . . . . . . . 11  |-  ( ( 4  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( 4  x.  ( M ^ 3 ) )  e.  CC )
4634, 44, 45sylancr 647 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  ( M ^ 3 ) )  e.  CC )
4733, 46subcld 9037 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
4  x.  ( M ^ 3 ) ) )  e.  CC )
4832, 47eqeltrd 2327 . . . . . . . 8  |-  ( ph  ->  G  e.  CC )
4948sqrcld 11796 . . . . . . 7  |-  ( ph  ->  ( sqr `  G
)  e.  CC )
5031, 49addcld 8734 . . . . . 6  |-  ( ph  ->  ( N  +  ( sqr `  G ) )  e.  CC )
5150halfcld 9835 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC )
52 3ne0 9711 . . . . . 6  |-  3  =/=  0
5337, 52reccli 9370 . . . . 5  |-  ( 1  /  3 )  e.  CC
54 cxpcl 19889 . . . . 5  |-  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC  /\  (
1  /  3 )  e.  CC )  -> 
( ( ( N  +  ( sqr `  G
) )  /  2
)  ^ c  ( 1  /  3 ) )  e.  CC )
5551, 53, 54sylancl 646 . . . 4  |-  ( ph  ->  ( ( ( N  +  ( sqr `  G
) )  /  2
)  ^ c  ( 1  /  3 ) )  e.  CC )
567, 55eqeltrd 2327 . . 3  |-  ( ph  ->  T  e.  CC )
577oveq1d 5725 . . . 4  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  ^ c  ( 1  /  3 ) ) ^ 3 ) )
58 3nn 9757 . . . . 5  |-  3  e.  NN
59 cxproot 19905 . . . . 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 646 . . . 4  |-  ( ph  ->  ( ( ( ( N  +  ( sqr `  G ) )  / 
2 )  ^ c 
( 1  /  3
) ) ^ 3 )  =  ( ( N  +  ( sqr `  G ) )  / 
2 ) )
6157, 60eqtrd 2285 . . 3  |-  ( ph  ->  ( T ^ 3 )  =  ( ( N  +  ( sqr `  G ) )  / 
2 ) )
6248sqsqrd 11798 . . . 4  |-  ( ph  ->  ( ( sqr `  G
) ^ 2 )  =  G )
6362, 32eqtrd 2285 . . 3  |-  ( ph  ->  ( ( sqr `  G
) ^ 2 )  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )
649a1i 12 . . . . . 6  |-  ( ph  ->  2  e.  CC )
6534a1i 12 . . . . . . . . 9  |-  ( ph  ->  4  e.  CC )
66 4nn 9758 . . . . . . . . . . 11  |-  4  e.  NN
6766nnne0i 9660 . . . . . . . . . 10  |-  4  =/=  0
6867a1i 12 . . . . . . . . 9  |-  ( ph  ->  4  =/=  0 )
69 cubic.0 . . . . . . . . . 10  |-  ( ph  ->  M  =/=  0 )
7010nn0zi 9927 . . . . . . . . . . 11  |-  3  e.  ZZ
7170a1i 12 . . . . . . . . . 10  |-  ( ph  ->  3  e.  ZZ )
7242, 69, 71expne0d 11129 . . . . . . . . 9  |-  ( ph  ->  ( M ^ 3 )  =/=  0 )
7365, 44, 68, 72mulne0d 9300 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( M ^ 3 ) )  =/=  0 )
7463oveq2d 5726 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N ^ 2 )  -  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) ) )
75 subsq 11088 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  ( sqr `  G )  e.  CC )  -> 
( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) ) )
7631, 49, 75syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) ) )
7733, 46nncand 9042 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( N ^ 2 )  -  ( 4  x.  ( M ^
3 ) ) ) )  =  ( 4  x.  ( M ^
3 ) ) )
7874, 76, 773eqtr3d 2293 . . . . . . . 8  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =  ( 4  x.  ( M ^ 3 ) ) )
7931, 49subcld 9037 . . . . . . . . 9  |-  ( ph  ->  ( N  -  ( sqr `  G ) )  e.  CC )
8079mul02d 8890 . . . . . . . 8  |-  ( ph  ->  ( 0  x.  ( N  -  ( sqr `  G ) ) )  =  0 )
8173, 78, 803netr4d 2439 . . . . . . 7  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =/=  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
82 oveq1 5717 . . . . . . . 8  |-  ( ( N  +  ( sqr `  G ) )  =  0  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) )  =  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
8382necon3i 2451 . . . . . . 7  |-  ( ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =/=  ( 0  x.  ( N  -  ( sqr `  G ) ) )  ->  ( N  +  ( sqr `  G ) )  =/=  0 )
8481, 83syl 17 . . . . . 6  |-  ( ph  ->  ( N  +  ( sqr `  G ) )  =/=  0 )
85 2ne0 9709 . . . . . . 7  |-  2  =/=  0
8685a1i 12 . . . . . 6  |-  ( ph  ->  2  =/=  0 )
8750, 64, 84, 86divne0d 9432 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  =/=  0 )
8853a1i 12 . . . . 5  |-  ( ph  ->  ( 1  /  3
)  e.  CC )
8951, 87, 88cxpne0d 19928 . . . 4  |-  ( ph  ->  ( ( ( N  +  ( sqr `  G
) )  /  2
)  ^ c  ( 1  /  3 ) )  =/=  0 )
907, 89eqnetrd 2430 . . 3  |-  ( ph  ->  T  =/=  0 )
911, 2, 3, 4, 5, 6, 56, 61, 49, 63, 35, 8, 90cubic2 19976 . 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 19970 . . . . 5  |-  ( r  e.  R  <->  ( r  e.  CC  /\  ( r ^ 3 )  =  1 ) )
9493anbi1i 679 . . . 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 633 . . . 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 242 . . 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 2536 . 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 256 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   {ctp 3546   ` cfv 4592  (class class class)co 5710   CCcc 8615   0cc0 8617   1c1 8618   _ici 8619    + caddc 8620    x. cmul 8622    - cmin 8917   -ucneg 8918    / cdiv 9303   NNcn 9626   2c2 9675   3c3 9676   4c4 9677   7c7 9680   9c9 9682   NN0cn0 9844   ZZcz 9903  ;cdc 10003   ^cexp 10982   sqrcsqr 11595    ^ c ccxp 19745
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-map 6660  df-pm 6661  df-ixp 6704  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-fi 7049  df-sup 7078  df-oi 7109  df-card 7456  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-q 10196  df-rp 10234  df-xneg 10331  df-xadd 10332  df-xmul 10333  df-ioo 10538  df-ioc 10539  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-fl 10803  df-mod 10852  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-shft 11439  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-limsup 11822  df-clim 11839  df-rlim 11840  df-sum 12036  df-ef 12223  df-sin 12225  df-cos 12226  df-pi 12228  df-divides 12406  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-starv 13097  df-sca 13098  df-vsca 13099  df-tset 13101  df-ple 13102  df-ds 13104  df-hom 13106  df-cco 13107  df-rest 13201  df-topn 13202  df-topgen 13218  df-pt 13219  df-prds 13222  df-xrs 13277  df-0g 13278  df-gsum 13279  df-qtop 13284  df-imas 13285  df-xps 13287  df-mre 13361  df-mrc 13362  df-acs 13363  df-mnd 14202  df-submnd 14251  df-mulg 14327  df-cntz 14628  df-cmn 14926  df-xmet 16205  df-met 16206  df-bl 16207  df-mopn 16208  df-cnfld 16210  df-top 16468  df-bases 16470  df-topon 16471  df-topsp 16472  df-cld 16588  df-ntr 16589  df-cls 16590  df-nei 16667  df-lp 16700  df-perf 16701  df-cn 16789  df-cnp 16790  df-haus 16875  df-tx 17089  df-hmeo 17278  df-fbas 17352  df-fg 17353  df-fil 17373  df-fm 17465  df-flim 17466  df-flf 17467  df-xms 17717  df-ms 17718  df-tms 17719  df-cncf 18214  df-limc 19048  df-dv 19049  df-log 19746  df-cxp 19747
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