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Theorem dcubic1 22358
Description: Forward direction of dcubic 22359: 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 6207 . . . . . 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 10672 . . . . . . . 8  |-  ( ph  ->  ( Q  /  2
)  e.  CC )
74, 6eqeltrd 2539 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
8 binom2sub 12086 . . . . . . 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 10497 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
1211, 3, 7mul12d 9681 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( G  x.  ( 2  x.  N ) ) )
134oveq2d 6208 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  N
)  =  ( 2  x.  ( Q  / 
2 ) ) )
14 2ne0 10517 . . . . . . . . . . . . 13  |-  2  =/=  0
1514a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  =/=  0 )
165, 11, 15divcan2d 10212 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( Q  /  2 ) )  =  Q )
1713, 16eqtrd 2492 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  N
)  =  Q )
1817oveq2d 6208 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  (
2  x.  N ) )  =  ( G  x.  Q ) )
193, 5mulcomd 9510 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  Q
)  =  ( Q  x.  G ) )
2012, 18, 193eqtrd 2496 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( Q  x.  G ) )
2110, 20oveq12d 6210 . . . . . . 7  |-  ( ph  ->  ( ( G ^
2 )  -  (
2  x.  ( G  x.  N ) ) )  =  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) ) )
2221oveq1d 6207 . . . . . 6  |-  ( ph  ->  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N )
) )  +  ( N ^ 2 ) )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
232, 9, 223eqtrd 2496 . . . . 5  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
247sqcld 12109 . . . . . . 7  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
25 dcubic.m . . . . . . . . 9  |-  ( ph  ->  M  =  ( P  /  3 ) )
26 dcubic.c . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
27 3cn 10499 . . . . . . . . . . 11  |-  3  e.  CC
2827a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  CC )
29 3ne0 10519 . . . . . . . . . . 11  |-  3  =/=  0
3029a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  =/=  0 )
3126, 28, 30divcld 10210 . . . . . . . . 9  |-  ( ph  ->  ( P  /  3
)  e.  CC )
3225, 31eqeltrd 2539 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
33 3nn0 10700 . . . . . . . 8  |-  3  e.  NN0
34 expcl 11986 . . . . . . . 8  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
3532, 33, 34sylancl 662 . . . . . . 7  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
3624, 35addcld 9508 . . . . . 6  |-  ( ph  ->  ( ( N ^
2 )  +  ( M ^ 3 ) )  e.  CC )
375, 3mulcld 9509 . . . . . 6  |-  ( ph  ->  ( Q  x.  G
)  e.  CC )
3836, 24, 37addsubd 9843 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
3924, 35, 24add32d 9695 . . . . . . 7  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
40242timesd 10670 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( ( N ^ 2 )  +  ( N ^ 2 ) ) )
4140oveq1d 6207 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4239, 41eqtr4d 2495 . . . . . 6  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4342oveq1d 6207 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
4423, 38, 433eqtr2d 2498 . . . 4  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
455, 3, 7subdid 9903 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( G  -  N )
)  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
461oveq2d 6208 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( Q  x.  ( G  -  N
) ) )
477sqvald 12108 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  =  ( N  x.  N ) )
4847oveq2d 6208 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( 2  x.  ( N  x.  N
) ) )
4911, 7, 7mulassd 9512 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( 2  x.  ( N  x.  N ) ) )
5017oveq1d 6207 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( Q  x.  N ) )
5148, 49, 503eqtr2d 2498 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( Q  x.  N ) )
5251oveq2d 6208 . . . . . . 7  |-  ( ph  ->  ( ( Q  x.  G )  -  (
2  x.  ( N ^ 2 ) ) )  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
5345, 46, 523eqtr4d 2502 . . . . . 6  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) ) )
5453oveq1d 6207 . . . . 5  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) ) )
55 2cn 10495 . . . . . . 7  |-  2  e.  CC
56 mulcl 9469 . . . . . . 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 9853 . . . . 5  |-  ( ph  ->  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
5954, 58eqtrd 2492 . . . 4  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
6044, 59oveq12d 6210 . . 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 9508 . . . 4  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  e.  CC )
62 npncan2 9739 . . . 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 2492 . 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 22356 . 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 1370    e. wcel 1758    =/= wne 2644  (class class class)co 6192   CCcc 9383   0cc0 9385    + caddc 9388    x. cmul 9390    - cmin 9698    / cdiv 10096   2c2 10474   3c3 10475   NN0cn0 10682   ^cexp 11968
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 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-fz 11541  df-seq 11910  df-exp 11969  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-dvds 13640
This theorem is referenced by:  dcubic  22359
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