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Theorem dcubic1 23154
Description: Forward direction of dcubic 23155: the claimed formula produces solutions to the cubic equation. (Contributed by Mario Carneiro, 25-Apr-2015.)
Hypotheses
Ref Expression
dcubic.c  |-  ( ph  ->  P  e.  CC )
dcubic.d  |-  ( ph  ->  Q  e.  CC )
dcubic.x  |-  ( ph  ->  X  e.  CC )
dcubic.t  |-  ( ph  ->  T  e.  CC )
dcubic.3  |-  ( ph  ->  ( T ^ 3 )  =  ( G  -  N ) )
dcubic.g  |-  ( ph  ->  G  e.  CC )
dcubic.2  |-  ( ph  ->  ( G ^ 2 )  =  ( ( N ^ 2 )  +  ( M ^
3 ) ) )
dcubic.m  |-  ( ph  ->  M  =  ( P  /  3 ) )
dcubic.n  |-  ( ph  ->  N  =  ( Q  /  2 ) )
dcubic.0  |-  ( ph  ->  T  =/=  0 )
dcubic1.x  |-  ( ph  ->  X  =  ( T  -  ( M  /  T ) ) )
Assertion
Ref Expression
dcubic1  |-  ( ph  ->  ( ( X ^
3 )  +  ( ( P  x.  X
)  +  Q ) )  =  0 )

Proof of Theorem dcubic1
StepHypRef Expression
1 dcubic.3 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  =  ( G  -  N ) )
21oveq1d 6296 . . . . . 6  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( G  -  N ) ^ 2 ) )
3 dcubic.g . . . . . . 7  |-  ( ph  ->  G  e.  CC )
4 dcubic.n . . . . . . . 8  |-  ( ph  ->  N  =  ( Q  /  2 ) )
5 dcubic.d . . . . . . . . 9  |-  ( ph  ->  Q  e.  CC )
65halfcld 10790 . . . . . . . 8  |-  ( ph  ->  ( Q  /  2
)  e.  CC )
74, 6eqeltrd 2531 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
8 binom2sub 12267 . . . . . . 7  |-  ( ( G  e.  CC  /\  N  e.  CC )  ->  ( ( G  -  N ) ^ 2 )  =  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N ) ) )  +  ( N ^
2 ) ) )
93, 7, 8syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( G  -  N ) ^ 2 )  =  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N ) ) )  +  ( N ^
2 ) ) )
10 dcubic.2 . . . . . . . 8  |-  ( ph  ->  ( G ^ 2 )  =  ( ( N ^ 2 )  +  ( M ^
3 ) ) )
11 2cnd 10615 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
1211, 3, 7mul12d 9792 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( G  x.  ( 2  x.  N ) ) )
134oveq2d 6297 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  N
)  =  ( 2  x.  ( Q  / 
2 ) ) )
14 2ne0 10635 . . . . . . . . . . . . 13  |-  2  =/=  0
1514a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  =/=  0 )
165, 11, 15divcan2d 10329 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( Q  /  2 ) )  =  Q )
1713, 16eqtrd 2484 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  N
)  =  Q )
1817oveq2d 6297 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  (
2  x.  N ) )  =  ( G  x.  Q ) )
193, 5mulcomd 9620 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  Q
)  =  ( Q  x.  G ) )
2012, 18, 193eqtrd 2488 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( Q  x.  G ) )
2110, 20oveq12d 6299 . . . . . . 7  |-  ( ph  ->  ( ( G ^
2 )  -  (
2  x.  ( G  x.  N ) ) )  =  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) ) )
2221oveq1d 6296 . . . . . 6  |-  ( ph  ->  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N )
) )  +  ( N ^ 2 ) )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
232, 9, 223eqtrd 2488 . . . . 5  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
247sqcld 12290 . . . . . . 7  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
25 dcubic.m . . . . . . . . 9  |-  ( ph  ->  M  =  ( P  /  3 ) )
26 dcubic.c . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
27 3cn 10617 . . . . . . . . . . 11  |-  3  e.  CC
2827a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  CC )
29 3ne0 10637 . . . . . . . . . . 11  |-  3  =/=  0
3029a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  =/=  0 )
3126, 28, 30divcld 10327 . . . . . . . . 9  |-  ( ph  ->  ( P  /  3
)  e.  CC )
3225, 31eqeltrd 2531 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
33 3nn0 10820 . . . . . . . 8  |-  3  e.  NN0
34 expcl 12166 . . . . . . . 8  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
3532, 33, 34sylancl 662 . . . . . . 7  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
3624, 35addcld 9618 . . . . . 6  |-  ( ph  ->  ( ( N ^
2 )  +  ( M ^ 3 ) )  e.  CC )
375, 3mulcld 9619 . . . . . 6  |-  ( ph  ->  ( Q  x.  G
)  e.  CC )
3836, 24, 37addsubd 9957 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
3924, 35, 24add32d 9807 . . . . . . 7  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
40242timesd 10788 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( ( N ^ 2 )  +  ( N ^ 2 ) ) )
4140oveq1d 6296 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4239, 41eqtr4d 2487 . . . . . 6  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4342oveq1d 6296 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
4423, 38, 433eqtr2d 2490 . . . 4  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
455, 3, 7subdid 10019 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( G  -  N )
)  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
461oveq2d 6297 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( Q  x.  ( G  -  N
) ) )
477sqvald 12289 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  =  ( N  x.  N ) )
4847oveq2d 6297 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( 2  x.  ( N  x.  N
) ) )
4911, 7, 7mulassd 9622 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( 2  x.  ( N  x.  N ) ) )
5017oveq1d 6296 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( Q  x.  N ) )
5148, 49, 503eqtr2d 2490 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( Q  x.  N ) )
5251oveq2d 6297 . . . . . . 7  |-  ( ph  ->  ( ( Q  x.  G )  -  (
2  x.  ( N ^ 2 ) ) )  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
5345, 46, 523eqtr4d 2494 . . . . . 6  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) ) )
5453oveq1d 6296 . . . . 5  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) ) )
55 2cn 10613 . . . . . . 7  |-  2  e.  CC
56 mulcl 9579 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( N ^ 2 )  e.  CC )  -> 
( 2  x.  ( N ^ 2 ) )  e.  CC )
5755, 24, 56sylancr 663 . . . . . 6  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  e.  CC )
5837, 57, 35subsub4d 9967 . . . . 5  |-  ( ph  ->  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
5954, 58eqtrd 2484 . . . 4  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
6044, 59oveq12d 6299 . . 3  |-  ( ph  ->  ( ( ( T ^ 3 ) ^
2 )  +  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^
3 ) ) )  =  ( ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) )  +  ( ( Q  x.  G
)  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) ) ) )
6157, 35addcld 9618 . . . 4  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  e.  CC )
62 npncan2 9851 . . . 4  |-  ( ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  e.  CC  /\  ( Q  x.  G
)  e.  CC )  ->  ( ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) )  +  ( ( Q  x.  G
)  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) ) )  =  0 )
6361, 37, 62syl2anc 661 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) )  +  ( ( Q  x.  G
)  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) ) )  =  0 )
6460, 63eqtrd 2484 . 2  |-  ( ph  ->  ( ( ( T ^ 3 ) ^
2 )  +  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^
3 ) ) )  =  0 )
65 dcubic.x . . 3  |-  ( ph  ->  X  e.  CC )
66 dcubic.t . . 3  |-  ( ph  ->  T  e.  CC )
67 dcubic.0 . . 3  |-  ( ph  ->  T  =/=  0 )
68 dcubic1.x . . 3  |-  ( ph  ->  X  =  ( T  -  ( M  /  T ) ) )
6926, 5, 65, 66, 1, 3, 10, 25, 4, 67, 66, 67, 68dcubic1lem 23152 . 2  |-  ( ph  ->  ( ( ( X ^ 3 )  +  ( ( P  x.  X )  +  Q
) )  =  0  <-> 
( ( ( T ^ 3 ) ^
2 )  +  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^
3 ) ) )  =  0 ) )
7064, 69mpbird 232 1  |-  ( ph  ->  ( ( X ^
3 )  +  ( ( P  x.  X
)  +  Q ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804    =/= wne 2638  (class class class)co 6281   CCcc 9493   0cc0 9495    + caddc 9498    x. cmul 9500    - cmin 9810    / cdiv 10213   2c2 10592   3c3 10593   NN0cn0 10802   ^cexp 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-fz 11684  df-seq 12090  df-exp 12149  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-dvds 13969
This theorem is referenced by:  dcubic  23155
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