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Theorem cubic 22376
Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 4039 to convert the existential quantifier to a triple disjunction. This is Metamath 100 proof #37. (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 10502 . . . . . . . . . . 11  |-  2  e.  CC
10 3nn0 10707 . . . . . . . . . . . 12  |-  3  e.  NN0
11 expcl 11999 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  3  e.  NN0 )  -> 
( B ^ 3 )  e.  CC )
123, 10, 11sylancl 662 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 3 )  e.  CC )
13 mulcl 9476 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  ( B ^ 3 )  e.  CC )  -> 
( 2  x.  ( B ^ 3 ) )  e.  CC )
149, 12, 13sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  ( B ^ 3 ) )  e.  CC )
15 9cn 10519 . . . . . . . . . . . 12  |-  9  e.  CC
16 mulcl 9476 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  A  e.  CC )  ->  ( 9  x.  A
)  e.  CC )
1715, 1, 16sylancr 663 . . . . . . . . . . 11  |-  ( ph  ->  ( 9  x.  A
)  e.  CC )
183, 4mulcld 9516 . . . . . . . . . . 11  |-  ( ph  ->  ( B  x.  C
)  e.  CC )
1917, 18mulcld 9516 . . . . . . . . . 10  |-  ( ph  ->  ( ( 9  x.  A )  x.  ( B  x.  C )
)  e.  CC )
2014, 19subcld 9829 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  ( B ^ 3 ) )  -  (
( 9  x.  A
)  x.  ( B  x.  C ) ) )  e.  CC )
21 2nn0 10706 . . . . . . . . . . . 12  |-  2  e.  NN0
22 7nn 10594 . . . . . . . . . . . 12  |-  7  e.  NN
2321, 22decnncl 10878 . . . . . . . . . . 11  |- ; 2 7  e.  NN
2423nncni 10442 . . . . . . . . . 10  |- ; 2 7  e.  CC
251sqcld 12122 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
2625, 5mulcld 9516 . . . . . . . . . 10  |-  ( ph  ->  ( ( A ^
2 )  x.  D
)  e.  CC )
27 mulcl 9476 . . . . . . . . . 10  |-  ( (; 2
7  e.  CC  /\  ( ( A ^
2 )  x.  D
)  e.  CC )  ->  (; 2 7  x.  (
( A ^ 2 )  x.  D ) )  e.  CC )
2824, 26, 27sylancr 663 . . . . . . . . 9  |-  ( ph  ->  (; 2 7  x.  (
( A ^ 2 )  x.  D ) )  e.  CC )
2920, 28addcld 9515 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  ( B ^
3 ) )  -  ( ( 9  x.  A )  x.  ( B  x.  C )
) )  +  (; 2
7  x.  ( ( A ^ 2 )  x.  D ) ) )  e.  CC )
308, 29eqeltrd 2542 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
31 cubic.g . . . . . . . . 9  |-  ( ph  ->  G  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )
3230sqcld 12122 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
33 4cn 10509 . . . . . . . . . . 11  |-  4  e.  CC
34 cubic.m . . . . . . . . . . . . 13  |-  ( ph  ->  M  =  ( ( B ^ 2 )  -  ( 3  x.  ( A  x.  C
) ) ) )
353sqcld 12122 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
36 3cn 10506 . . . . . . . . . . . . . . 15  |-  3  e.  CC
371, 4mulcld 9516 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
38 mulcl 9476 . . . . . . . . . . . . . . 15  |-  ( ( 3  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( 3  x.  ( A  x.  C
) )  e.  CC )
3936, 37, 38sylancr 663 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 3  x.  ( A  x.  C )
)  e.  CC )
4035, 39subcld 9829 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B ^
2 )  -  (
3  x.  ( A  x.  C ) ) )  e.  CC )
4134, 40eqeltrd 2542 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  CC )
42 expcl 11999 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
4341, 10, 42sylancl 662 . . . . . . . . . . 11  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
44 mulcl 9476 . . . . . . . . . . 11  |-  ( ( 4  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( 4  x.  ( M ^ 3 ) )  e.  CC )
4533, 43, 44sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  ( M ^ 3 ) )  e.  CC )
4632, 45subcld 9829 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
4  x.  ( M ^ 3 ) ) )  e.  CC )
4731, 46eqeltrd 2542 . . . . . . . 8  |-  ( ph  ->  G  e.  CC )
4847sqrcld 13040 . . . . . . 7  |-  ( ph  ->  ( sqr `  G
)  e.  CC )
4930, 48addcld 9515 . . . . . 6  |-  ( ph  ->  ( N  +  ( sqr `  G ) )  e.  CC )
5049halfcld 10679 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC )
51 3ne0 10526 . . . . . 6  |-  3  =/=  0
5236, 51reccli 10171 . . . . 5  |-  ( 1  /  3 )  e.  CC
53 cxpcl 22251 . . . . 5  |-  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC  /\  (
1  /  3 )  e.  CC )  -> 
( ( ( N  +  ( sqr `  G
) )  /  2
)  ^c  ( 1  /  3 ) )  e.  CC )
5450, 52, 53sylancl 662 . . . 4  |-  ( ph  ->  ( ( ( N  +  ( sqr `  G
) )  /  2
)  ^c  ( 1  /  3 ) )  e.  CC )
557, 54eqeltrd 2542 . . 3  |-  ( ph  ->  T  e.  CC )
567oveq1d 6214 . . . 4  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  ^c  ( 1  /  3 ) ) ^ 3 ) )
57 3nn 10590 . . . . 5  |-  3  e.  NN
58 cxproot 22267 . . . . 5  |-  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC  /\  3  e.  NN )  ->  (
( ( ( N  +  ( sqr `  G
) )  /  2
)  ^c  ( 1  /  3 ) ) ^ 3 )  =  ( ( N  +  ( sqr `  G
) )  /  2
) )
5950, 57, 58sylancl 662 . . . 4  |-  ( ph  ->  ( ( ( ( N  +  ( sqr `  G ) )  / 
2 )  ^c 
( 1  /  3
) ) ^ 3 )  =  ( ( N  +  ( sqr `  G ) )  / 
2 ) )
6056, 59eqtrd 2495 . . 3  |-  ( ph  ->  ( T ^ 3 )  =  ( ( N  +  ( sqr `  G ) )  / 
2 ) )
6147sqsqrd 13042 . . . 4  |-  ( ph  ->  ( ( sqr `  G
) ^ 2 )  =  G )
6261, 31eqtrd 2495 . . 3  |-  ( ph  ->  ( ( sqr `  G
) ^ 2 )  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )
639a1i 11 . . . . . 6  |-  ( ph  ->  2  e.  CC )
6433a1i 11 . . . . . . . . 9  |-  ( ph  ->  4  e.  CC )
65 4ne0 10528 . . . . . . . . . 10  |-  4  =/=  0
6665a1i 11 . . . . . . . . 9  |-  ( ph  ->  4  =/=  0 )
67 cubic.0 . . . . . . . . . 10  |-  ( ph  ->  M  =/=  0 )
68 3z 10789 . . . . . . . . . . 11  |-  3  e.  ZZ
6968a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  ZZ )
7041, 67, 69expne0d 12130 . . . . . . . . 9  |-  ( ph  ->  ( M ^ 3 )  =/=  0 )
7164, 43, 66, 70mulne0d 10098 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( M ^ 3 ) )  =/=  0 )
7262oveq2d 6215 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N ^ 2 )  -  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) ) )
73 subsq 12089 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  ( sqr `  G )  e.  CC )  -> 
( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) ) )
7430, 48, 73syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) ) )
7532, 45nncand 9834 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( N ^ 2 )  -  ( 4  x.  ( M ^
3 ) ) ) )  =  ( 4  x.  ( M ^
3 ) ) )
7672, 74, 753eqtr3d 2503 . . . . . . . 8  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =  ( 4  x.  ( M ^ 3 ) ) )
7730, 48subcld 9829 . . . . . . . . 9  |-  ( ph  ->  ( N  -  ( sqr `  G ) )  e.  CC )
7877mul02d 9677 . . . . . . . 8  |-  ( ph  ->  ( 0  x.  ( N  -  ( sqr `  G ) ) )  =  0 )
7971, 76, 783netr4d 2756 . . . . . . 7  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =/=  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
80 oveq1 6206 . . . . . . . 8  |-  ( ( N  +  ( sqr `  G ) )  =  0  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) )  =  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
8180necon3i 2691 . . . . . . 7  |-  ( ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =/=  ( 0  x.  ( N  -  ( sqr `  G ) ) )  ->  ( N  +  ( sqr `  G ) )  =/=  0 )
8279, 81syl 16 . . . . . 6  |-  ( ph  ->  ( N  +  ( sqr `  G ) )  =/=  0 )
83 2ne0 10524 . . . . . . 7  |-  2  =/=  0
8483a1i 11 . . . . . 6  |-  ( ph  ->  2  =/=  0 )
8549, 63, 82, 84divne0d 10233 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  =/=  0 )
8652a1i 11 . . . . 5  |-  ( ph  ->  ( 1  /  3
)  e.  CC )
8750, 85, 86cxpne0d 22290 . . . 4  |-  ( ph  ->  ( ( ( N  +  ( sqr `  G
) )  /  2
)  ^c  ( 1  /  3 ) )  =/=  0 )
887, 87eqnetrd 2744 . . 3  |-  ( ph  ->  T  =/=  0 )
891, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88cubic2 22375 . 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 ) ) ) ) )
90 cubic.r . . . . . 6  |-  R  =  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3 ) ) )  /  2 ) ,  ( ( -u
1  -  ( _i  x.  ( sqr `  3
) ) )  / 
2 ) }
91901cubr 22369 . . . . 5  |-  ( r  e.  R  <->  ( r  e.  CC  /\  ( r ^ 3 )  =  1 ) )
9291anbi1i 695 . . . 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
) ) ) )
93 anass 649 . . . 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
) ) ) ) )
9492, 93bitri 249 . . 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 ) ) ) ) )
9594rexbii2 2857 . 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
) ) ) )
9689, 95syl6bbr 263 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799   {ctp 3988   ` cfv 5525  (class class class)co 6199   CCcc 9390   0cc0 9392   1c1 9393   _ici 9394    + caddc 9395    x. cmul 9397    - cmin 9705   -ucneg 9706    / cdiv 10103   NNcn 10432   2c2 10481   3c3 10482   4c4 10483   7c7 10486   9c9 10488   NN0cn0 10689   ZZcz 10756  ;cdc 10865   ^cexp 11981   sqrcsqr 12839    ^c ccxp 22139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-shft 12673  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-limsup 13066  df-clim 13083  df-rlim 13084  df-sum 13281  df-ef 13470  df-sin 13472  df-cos 13473  df-pi 13475  df-dvds 13653  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474  df-log 22140  df-cxp 22141
This theorem is referenced by: (None)
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