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Theorem cubic 22129
Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3920 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 10380 . . . . . . . . . . 11  |-  2  e.  CC
10 3nn0 10585 . . . . . . . . . . . 12  |-  3  e.  NN0
11 expcl 11867 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  3  e.  NN0 )  -> 
( B ^ 3 )  e.  CC )
123, 10, 11sylancl 655 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 3 )  e.  CC )
13 mulcl 9354 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  ( B ^ 3 )  e.  CC )  -> 
( 2  x.  ( B ^ 3 ) )  e.  CC )
149, 12, 13sylancr 656 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  ( B ^ 3 ) )  e.  CC )
15 9cn 10397 . . . . . . . . . . . 12  |-  9  e.  CC
16 mulcl 9354 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  A  e.  CC )  ->  ( 9  x.  A
)  e.  CC )
1715, 1, 16sylancr 656 . . . . . . . . . . 11  |-  ( ph  ->  ( 9  x.  A
)  e.  CC )
183, 4mulcld 9394 . . . . . . . . . . 11  |-  ( ph  ->  ( B  x.  C
)  e.  CC )
1917, 18mulcld 9394 . . . . . . . . . 10  |-  ( ph  ->  ( ( 9  x.  A )  x.  ( B  x.  C )
)  e.  CC )
2014, 19subcld 9707 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  ( B ^ 3 ) )  -  (
( 9  x.  A
)  x.  ( B  x.  C ) ) )  e.  CC )
21 2nn0 10584 . . . . . . . . . . . 12  |-  2  e.  NN0
22 7nn 10472 . . . . . . . . . . . 12  |-  7  e.  NN
2321, 22decnncl 10756 . . . . . . . . . . 11  |- ; 2 7  e.  NN
2423nncni 10320 . . . . . . . . . 10  |- ; 2 7  e.  CC
251sqcld 11990 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
2625, 5mulcld 9394 . . . . . . . . . 10  |-  ( ph  ->  ( ( A ^
2 )  x.  D
)  e.  CC )
27 mulcl 9354 . . . . . . . . . 10  |-  ( (; 2
7  e.  CC  /\  ( ( A ^
2 )  x.  D
)  e.  CC )  ->  (; 2 7  x.  (
( A ^ 2 )  x.  D ) )  e.  CC )
2824, 26, 27sylancr 656 . . . . . . . . 9  |-  ( ph  ->  (; 2 7  x.  (
( A ^ 2 )  x.  D ) )  e.  CC )
2920, 28addcld 9393 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  ( B ^
3 ) )  -  ( ( 9  x.  A )  x.  ( B  x.  C )
) )  +  (; 2
7  x.  ( ( A ^ 2 )  x.  D ) ) )  e.  CC )
308, 29eqeltrd 2507 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
31 cubic.g . . . . . . . . 9  |-  ( ph  ->  G  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )
3230sqcld 11990 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
33 4cn 10387 . . . . . . . . . . 11  |-  4  e.  CC
34 cubic.m . . . . . . . . . . . . 13  |-  ( ph  ->  M  =  ( ( B ^ 2 )  -  ( 3  x.  ( A  x.  C
) ) ) )
353sqcld 11990 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
36 3cn 10384 . . . . . . . . . . . . . . 15  |-  3  e.  CC
371, 4mulcld 9394 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
38 mulcl 9354 . . . . . . . . . . . . . . 15  |-  ( ( 3  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( 3  x.  ( A  x.  C
) )  e.  CC )
3936, 37, 38sylancr 656 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 3  x.  ( A  x.  C )
)  e.  CC )
4035, 39subcld 9707 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B ^
2 )  -  (
3  x.  ( A  x.  C ) ) )  e.  CC )
4134, 40eqeltrd 2507 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  CC )
42 expcl 11867 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
4341, 10, 42sylancl 655 . . . . . . . . . . 11  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
44 mulcl 9354 . . . . . . . . . . 11  |-  ( ( 4  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( 4  x.  ( M ^ 3 ) )  e.  CC )
4533, 43, 44sylancr 656 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  ( M ^ 3 ) )  e.  CC )
4632, 45subcld 9707 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
4  x.  ( M ^ 3 ) ) )  e.  CC )
4731, 46eqeltrd 2507 . . . . . . . 8  |-  ( ph  ->  G  e.  CC )
4847sqrcld 12907 . . . . . . 7  |-  ( ph  ->  ( sqr `  G
)  e.  CC )
4930, 48addcld 9393 . . . . . 6  |-  ( ph  ->  ( N  +  ( sqr `  G ) )  e.  CC )
5049halfcld 10557 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC )
51 3ne0 10404 . . . . . 6  |-  3  =/=  0
5236, 51reccli 10049 . . . . 5  |-  ( 1  /  3 )  e.  CC
53 cxpcl 22004 . . . . 5  |-  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  e.  CC  /\  (
1  /  3 )  e.  CC )  -> 
( ( ( N  +  ( sqr `  G
) )  /  2
)  ^c  ( 1  /  3 ) )  e.  CC )
5450, 52, 53sylancl 655 . . . 4  |-  ( ph  ->  ( ( ( N  +  ( sqr `  G
) )  /  2
)  ^c  ( 1  /  3 ) )  e.  CC )
557, 54eqeltrd 2507 . . 3  |-  ( ph  ->  T  e.  CC )
567oveq1d 6095 . . . 4  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( N  +  ( sqr `  G ) )  /  2 )  ^c  ( 1  /  3 ) ) ^ 3 ) )
57 3nn 10468 . . . . 5  |-  3  e.  NN
58 cxproot 22020 . . . . 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 655 . . . 4  |-  ( ph  ->  ( ( ( ( N  +  ( sqr `  G ) )  / 
2 )  ^c 
( 1  /  3
) ) ^ 3 )  =  ( ( N  +  ( sqr `  G ) )  / 
2 ) )
6056, 59eqtrd 2465 . . 3  |-  ( ph  ->  ( T ^ 3 )  =  ( ( N  +  ( sqr `  G ) )  / 
2 ) )
6147sqsqrd 12909 . . . 4  |-  ( ph  ->  ( ( sqr `  G
) ^ 2 )  =  G )
6261, 31eqtrd 2465 . . 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 10406 . . . . . . . . . 10  |-  4  =/=  0
6665a1i 11 . . . . . . . . 9  |-  ( ph  ->  4  =/=  0 )
67 cubic.0 . . . . . . . . . 10  |-  ( ph  ->  M  =/=  0 )
68 3z 10667 . . . . . . . . . . 11  |-  3  e.  ZZ
6968a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  ZZ )
7041, 67, 69expne0d 11998 . . . . . . . . 9  |-  ( ph  ->  ( M ^ 3 )  =/=  0 )
7164, 43, 66, 70mulne0d 9976 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( M ^ 3 ) )  =/=  0 )
7262oveq2d 6096 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N ^ 2 )  -  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) ) )
73 subsq 11957 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  ( sqr `  G )  e.  CC )  -> 
( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) ) )
7430, 48, 73syl2anc 654 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( sqr `  G
) ^ 2 ) )  =  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) ) )
7532, 45nncand 9712 . . . . . . . . 9  |-  ( ph  ->  ( ( N ^
2 )  -  (
( N ^ 2 )  -  ( 4  x.  ( M ^
3 ) ) ) )  =  ( 4  x.  ( M ^
3 ) ) )
7672, 74, 753eqtr3d 2473 . . . . . . . 8  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =  ( 4  x.  ( M ^ 3 ) ) )
7730, 48subcld 9707 . . . . . . . . 9  |-  ( ph  ->  ( N  -  ( sqr `  G ) )  e.  CC )
7877mul02d 9555 . . . . . . . 8  |-  ( ph  ->  ( 0  x.  ( N  -  ( sqr `  G ) ) )  =  0 )
7971, 76, 783netr4d 2625 . . . . . . 7  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G
) ) )  =/=  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
80 oveq1 6087 . . . . . . . 8  |-  ( ( N  +  ( sqr `  G ) )  =  0  ->  ( ( N  +  ( sqr `  G ) )  x.  ( N  -  ( sqr `  G ) ) )  =  ( 0  x.  ( N  -  ( sqr `  G ) ) ) )
8180necon3i 2640 . . . . . . 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 10402 . . . . . . 7  |-  2  =/=  0
8483a1i 11 . . . . . 6  |-  ( ph  ->  2  =/=  0 )
8549, 63, 82, 84divne0d 10111 . . . . 5  |-  ( ph  ->  ( ( N  +  ( sqr `  G ) )  /  2 )  =/=  0 )
8652a1i 11 . . . . 5  |-  ( ph  ->  ( 1  /  3
)  e.  CC )
8750, 85, 86cxpne0d 22043 . . . 4  |-  ( ph  ->  ( ( ( N  +  ( sqr `  G
) )  /  2
)  ^c  ( 1  /  3 ) )  =/=  0 )
887, 87eqnetrd 2616 . . 3  |-  ( ph  ->  T  =/=  0 )
891, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88cubic2 22128 . 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 22122 . . . . 5  |-  ( r  e.  R  <->  ( r  e.  CC  /\  ( r ^ 3 )  =  1 ) )
9291anbi1i 688 . . . 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 642 . . . 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 2734 . 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 1362    e. wcel 1755    =/= wne 2596   E.wrex 2706   {ctp 3869   ` cfv 5406  (class class class)co 6080   CCcc 9268   0cc0 9270   1c1 9271   _ici 9272    + caddc 9273    x. cmul 9275    - cmin 9583   -ucneg 9584    / cdiv 9981   NNcn 10310   2c2 10359   3c3 10360   4c4 10361   7c7 10364   9c9 10366   NN0cn0 10567   ZZcz 10634  ;cdc 10743   ^cexp 11849   sqrcsqr 12706    ^c ccxp 21892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-dvds 13519  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894
This theorem is referenced by: (None)
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