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Theorem dcubic1 20677
Description: Forward direction of dcubic 20678: 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 6088 . . . . . 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 10204 . . . . . . . 8  |-  ( ph  ->  ( Q  /  2
)  e.  CC )
74, 6eqeltrd 2509 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
8 binom2sub 11490 . . . . . . 7  |-  ( ( G  e.  CC  /\  N  e.  CC )  ->  ( ( G  -  N ) ^ 2 )  =  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N ) ) )  +  ( N ^
2 ) ) )
93, 7, 8syl2anc 643 . . . . . 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 2cn 10062 . . . . . . . . . . 11  |-  2  e.  CC
1211a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
1312, 3, 7mul12d 9267 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( G  x.  ( 2  x.  N ) ) )
144oveq2d 6089 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  N
)  =  ( 2  x.  ( Q  / 
2 ) ) )
15 2ne0 10075 . . . . . . . . . . . . 13  |-  2  =/=  0
1615a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  =/=  0 )
175, 12, 16divcan2d 9784 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( Q  /  2 ) )  =  Q )
1814, 17eqtrd 2467 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  N
)  =  Q )
1918oveq2d 6089 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  (
2  x.  N ) )  =  ( G  x.  Q ) )
203, 5mulcomd 9101 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  Q
)  =  ( Q  x.  G ) )
2113, 19, 203eqtrd 2471 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( Q  x.  G ) )
2210, 21oveq12d 6091 . . . . . . 7  |-  ( ph  ->  ( ( G ^
2 )  -  (
2  x.  ( G  x.  N ) ) )  =  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) ) )
2322oveq1d 6088 . . . . . 6  |-  ( ph  ->  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N )
) )  +  ( N ^ 2 ) )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
242, 9, 233eqtrd 2471 . . . . 5  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
257sqcld 11513 . . . . . . 7  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
26 dcubic.m . . . . . . . . 9  |-  ( ph  ->  M  =  ( P  /  3 ) )
27 dcubic.c . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
28 3cn 10064 . . . . . . . . . . 11  |-  3  e.  CC
2928a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  CC )
30 3ne0 10077 . . . . . . . . . . 11  |-  3  =/=  0
3130a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  =/=  0 )
3227, 29, 31divcld 9782 . . . . . . . . 9  |-  ( ph  ->  ( P  /  3
)  e.  CC )
3326, 32eqeltrd 2509 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
34 3nn0 10231 . . . . . . . 8  |-  3  e.  NN0
35 expcl 11391 . . . . . . . 8  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
3633, 34, 35sylancl 644 . . . . . . 7  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
3725, 36addcld 9099 . . . . . 6  |-  ( ph  ->  ( ( N ^
2 )  +  ( M ^ 3 ) )  e.  CC )
385, 3mulcld 9100 . . . . . 6  |-  ( ph  ->  ( Q  x.  G
)  e.  CC )
3937, 25, 38addsubd 9424 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
4025, 36, 25add32d 9280 . . . . . . 7  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
41252timesd 10202 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( ( N ^ 2 )  +  ( N ^ 2 ) ) )
4241oveq1d 6088 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4340, 42eqtr4d 2470 . . . . . 6  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4443oveq1d 6088 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
4524, 39, 443eqtr2d 2473 . . . 4  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
465, 3, 7subdid 9481 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( G  -  N )
)  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
471oveq2d 6089 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( Q  x.  ( G  -  N
) ) )
487sqvald 11512 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  =  ( N  x.  N ) )
4948oveq2d 6089 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( 2  x.  ( N  x.  N
) ) )
5012, 7, 7mulassd 9103 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( 2  x.  ( N  x.  N ) ) )
5118oveq1d 6088 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( Q  x.  N ) )
5249, 50, 513eqtr2d 2473 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( Q  x.  N ) )
5352oveq2d 6089 . . . . . . 7  |-  ( ph  ->  ( ( Q  x.  G )  -  (
2  x.  ( N ^ 2 ) ) )  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
5446, 47, 533eqtr4d 2477 . . . . . 6  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) ) )
5554oveq1d 6088 . . . . 5  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) ) )
56 mulcl 9066 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( N ^ 2 )  e.  CC )  -> 
( 2  x.  ( N ^ 2 ) )  e.  CC )
5711, 25, 56sylancr 645 . . . . . 6  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  e.  CC )
5838, 57, 36subsub4d 9434 . . . . 5  |-  ( ph  ->  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
5955, 58eqtrd 2467 . . . 4  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
6045, 59oveq12d 6091 . . 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, 36addcld 9099 . . . 4  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  e.  CC )
62 npncan2 9320 . . . 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, 38, 62syl2anc 643 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) )  +  ( ( Q  x.  G
)  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) ) )  =  0 )
6460, 63eqtrd 2467 . 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 ) ) )
6927, 5, 65, 66, 1, 3, 10, 26, 4, 67, 66, 67, 68dcubic1lem 20675 . 2  |-  ( ph  ->  ( ( ( X ^ 3 )  +  ( ( P  x.  X )  +  Q
) )  =  0  <-> 
( ( ( T ^ 3 ) ^
2 )  +  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^
3 ) ) )  =  0 ) )
7064, 69mpbird 224 1  |-  ( ph  ->  ( ( X ^
3 )  +  ( ( P  x.  X
)  +  Q ) )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6073   CCcc 8980   0cc0 8982    + caddc 8985    x. cmul 8987    - cmin 9283    / cdiv 9669   2c2 10041   3c3 10042   NN0cn0 10213   ^cexp 11374
This theorem is referenced by:  dcubic  20678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845
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