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Theorem dcubic1 23763
Description: Forward direction of dcubic 23764: 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 6318 . . . . . 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 10859 . . . . . . . 8  |-  ( ph  ->  ( Q  /  2
)  e.  CC )
74, 6eqeltrd 2511 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
8 binom2sub 12392 . . . . . . 7  |-  ( ( G  e.  CC  /\  N  e.  CC )  ->  ( ( G  -  N ) ^ 2 )  =  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N ) ) )  +  ( N ^
2 ) ) )
93, 7, 8syl2anc 666 . . . . . 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 10684 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
1211, 3, 7mul12d 9844 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( G  x.  ( 2  x.  N ) ) )
134oveq2d 6319 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  N
)  =  ( 2  x.  ( Q  / 
2 ) ) )
14 2ne0 10704 . . . . . . . . . . . . 13  |-  2  =/=  0
1514a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  =/=  0 )
165, 11, 15divcan2d 10387 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( Q  /  2 ) )  =  Q )
1713, 16eqtrd 2464 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  N
)  =  Q )
1817oveq2d 6319 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  (
2  x.  N ) )  =  ( G  x.  Q ) )
193, 5mulcomd 9666 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  Q
)  =  ( Q  x.  G ) )
2012, 18, 193eqtrd 2468 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( Q  x.  G ) )
2110, 20oveq12d 6321 . . . . . . 7  |-  ( ph  ->  ( ( G ^
2 )  -  (
2  x.  ( G  x.  N ) ) )  =  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) ) )
2221oveq1d 6318 . . . . . 6  |-  ( ph  ->  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N )
) )  +  ( N ^ 2 ) )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
232, 9, 223eqtrd 2468 . . . . 5  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
247sqcld 12415 . . . . . . 7  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
25 dcubic.m . . . . . . . . 9  |-  ( ph  ->  M  =  ( P  /  3 ) )
26 dcubic.c . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
27 3cn 10686 . . . . . . . . . . 11  |-  3  e.  CC
2827a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  CC )
29 3ne0 10706 . . . . . . . . . . 11  |-  3  =/=  0
3029a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  =/=  0 )
3126, 28, 30divcld 10385 . . . . . . . . 9  |-  ( ph  ->  ( P  /  3
)  e.  CC )
3225, 31eqeltrd 2511 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
33 3nn0 10889 . . . . . . . 8  |-  3  e.  NN0
34 expcl 12291 . . . . . . . 8  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
3532, 33, 34sylancl 667 . . . . . . 7  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
3624, 35addcld 9664 . . . . . 6  |-  ( ph  ->  ( ( N ^
2 )  +  ( M ^ 3 ) )  e.  CC )
375, 3mulcld 9665 . . . . . 6  |-  ( ph  ->  ( Q  x.  G
)  e.  CC )
3836, 24, 37addsubd 10009 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
3924, 35, 24add32d 9859 . . . . . . 7  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
40242timesd 10857 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( ( N ^ 2 )  +  ( N ^ 2 ) ) )
4140oveq1d 6318 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4239, 41eqtr4d 2467 . . . . . 6  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4342oveq1d 6318 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
4423, 38, 433eqtr2d 2470 . . . 4  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
455, 3, 7subdid 10076 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( G  -  N )
)  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
461oveq2d 6319 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( Q  x.  ( G  -  N
) ) )
477sqvald 12414 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  =  ( N  x.  N ) )
4847oveq2d 6319 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( 2  x.  ( N  x.  N
) ) )
4911, 7, 7mulassd 9668 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( 2  x.  ( N  x.  N ) ) )
5017oveq1d 6318 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( Q  x.  N ) )
5148, 49, 503eqtr2d 2470 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( Q  x.  N ) )
5251oveq2d 6319 . . . . . . 7  |-  ( ph  ->  ( ( Q  x.  G )  -  (
2  x.  ( N ^ 2 ) ) )  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
5345, 46, 523eqtr4d 2474 . . . . . 6  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) ) )
5453oveq1d 6318 . . . . 5  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) ) )
55 2cn 10682 . . . . . . 7  |-  2  e.  CC
56 mulcl 9625 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( N ^ 2 )  e.  CC )  -> 
( 2  x.  ( N ^ 2 ) )  e.  CC )
5755, 24, 56sylancr 668 . . . . . 6  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  e.  CC )
5837, 57, 35subsub4d 10019 . . . . 5  |-  ( ph  ->  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
5954, 58eqtrd 2464 . . . 4  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
6044, 59oveq12d 6321 . . 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 9664 . . . 4  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  e.  CC )
62 npncan2 9903 . . . 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 666 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) )  +  ( ( Q  x.  G
)  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) ) )  =  0 )
6460, 63eqtrd 2464 . 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 23761 . 2  |-  ( ph  ->  ( ( ( X ^ 3 )  +  ( ( P  x.  X )  +  Q
) )  =  0  <-> 
( ( ( T ^ 3 ) ^
2 )  +  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^
3 ) ) )  =  0 ) )
7064, 69mpbird 236 1  |-  ( ph  ->  ( ( X ^
3 )  +  ( ( P  x.  X
)  +  Q ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438    e. wcel 1869    =/= wne 2619  (class class class)co 6303   CCcc 9539   0cc0 9541    + caddc 9544    x. cmul 9546    - cmin 9862    / cdiv 10271   2c2 10661   3c3 10662   NN0cn0 10871   ^cexp 12273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-sup 7960  df-inf 7961  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-rp 11305  df-fz 11787  df-seq 12215  df-exp 12274  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-dvds 14299
This theorem is referenced by:  dcubic  23764
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