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Theorem dcubic1 23391
Description: Forward direction of dcubic 23392: 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 6247 . . . . . 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 10742 . . . . . . . 8  |-  ( ph  ->  ( Q  /  2
)  e.  CC )
74, 6eqeltrd 2488 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
8 binom2sub 12237 . . . . . . 7  |-  ( ( G  e.  CC  /\  N  e.  CC )  ->  ( ( G  -  N ) ^ 2 )  =  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N ) ) )  +  ( N ^
2 ) ) )
93, 7, 8syl2anc 659 . . . . . 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 10567 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
1211, 3, 7mul12d 9741 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( G  x.  ( 2  x.  N ) ) )
134oveq2d 6248 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  N
)  =  ( 2  x.  ( Q  / 
2 ) ) )
14 2ne0 10587 . . . . . . . . . . . . 13  |-  2  =/=  0
1514a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  =/=  0 )
165, 11, 15divcan2d 10281 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( Q  /  2 ) )  =  Q )
1713, 16eqtrd 2441 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  N
)  =  Q )
1817oveq2d 6248 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  (
2  x.  N ) )  =  ( G  x.  Q ) )
193, 5mulcomd 9565 . . . . . . . . 9  |-  ( ph  ->  ( G  x.  Q
)  =  ( Q  x.  G ) )
2012, 18, 193eqtrd 2445 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( G  x.  N )
)  =  ( Q  x.  G ) )
2110, 20oveq12d 6250 . . . . . . 7  |-  ( ph  ->  ( ( G ^
2 )  -  (
2  x.  ( G  x.  N ) ) )  =  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) ) )
2221oveq1d 6247 . . . . . 6  |-  ( ph  ->  ( ( ( G ^ 2 )  -  ( 2  x.  ( G  x.  N )
) )  +  ( N ^ 2 ) )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
232, 9, 223eqtrd 2445 . . . . 5  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
247sqcld 12260 . . . . . . 7  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
25 dcubic.m . . . . . . . . 9  |-  ( ph  ->  M  =  ( P  /  3 ) )
26 dcubic.c . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
27 3cn 10569 . . . . . . . . . . 11  |-  3  e.  CC
2827a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  CC )
29 3ne0 10589 . . . . . . . . . . 11  |-  3  =/=  0
3029a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  =/=  0 )
3126, 28, 30divcld 10279 . . . . . . . . 9  |-  ( ph  ->  ( P  /  3
)  e.  CC )
3225, 31eqeltrd 2488 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
33 3nn0 10772 . . . . . . . 8  |-  3  e.  NN0
34 expcl 12136 . . . . . . . 8  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
3532, 33, 34sylancl 660 . . . . . . 7  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
3624, 35addcld 9563 . . . . . 6  |-  ( ph  ->  ( ( N ^
2 )  +  ( M ^ 3 ) )  e.  CC )
375, 3mulcld 9564 . . . . . 6  |-  ( ph  ->  ( Q  x.  G
)  e.  CC )
3836, 24, 37addsubd 9906 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( ( N ^
2 )  +  ( M ^ 3 ) )  -  ( Q  x.  G ) )  +  ( N ^
2 ) ) )
3924, 35, 24add32d 9756 . . . . . . 7  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
40242timesd 10740 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( ( N ^ 2 )  +  ( N ^ 2 ) ) )
4140oveq1d 6247 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  =  ( ( ( N ^ 2 )  +  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4239, 41eqtr4d 2444 . . . . . 6  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  +  ( N ^ 2 ) )  =  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) )
4342oveq1d 6247 . . . . 5  |-  ( ph  ->  ( ( ( ( N ^ 2 )  +  ( M ^
3 ) )  +  ( N ^ 2 ) )  -  ( Q  x.  G )
)  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
4423, 38, 433eqtr2d 2447 . . . 4  |-  ( ph  ->  ( ( T ^
3 ) ^ 2 )  =  ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) ) )
455, 3, 7subdid 9971 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( G  -  N )
)  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
461oveq2d 6248 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( Q  x.  ( G  -  N
) ) )
477sqvald 12259 . . . . . . . . . 10  |-  ( ph  ->  ( N ^ 2 )  =  ( N  x.  N ) )
4847oveq2d 6248 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( 2  x.  ( N  x.  N
) ) )
4911, 7, 7mulassd 9567 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( 2  x.  ( N  x.  N ) ) )
5017oveq1d 6247 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N )  x.  N
)  =  ( Q  x.  N ) )
5148, 49, 503eqtr2d 2447 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  =  ( Q  x.  N ) )
5251oveq2d 6248 . . . . . . 7  |-  ( ph  ->  ( ( Q  x.  G )  -  (
2  x.  ( N ^ 2 ) ) )  =  ( ( Q  x.  G )  -  ( Q  x.  N ) ) )
5345, 46, 523eqtr4d 2451 . . . . . 6  |-  ( ph  ->  ( Q  x.  ( T ^ 3 ) )  =  ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) ) )
5453oveq1d 6247 . . . . 5  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) ) )
55 2cn 10565 . . . . . . 7  |-  2  e.  CC
56 mulcl 9524 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( N ^ 2 )  e.  CC )  -> 
( 2  x.  ( N ^ 2 ) )  e.  CC )
5755, 24, 56sylancr 661 . . . . . 6  |-  ( ph  ->  ( 2  x.  ( N ^ 2 ) )  e.  CC )
5837, 57, 35subsub4d 9916 . . . . 5  |-  ( ph  ->  ( ( ( Q  x.  G )  -  ( 2  x.  ( N ^ 2 ) ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
5954, 58eqtrd 2441 . . . 4  |-  ( ph  ->  ( ( Q  x.  ( T ^ 3 ) )  -  ( M ^ 3 ) )  =  ( ( Q  x.  G )  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) ) ) )
6044, 59oveq12d 6250 . . 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 9563 . . . 4  |-  ( ph  ->  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^ 3 ) )  e.  CC )
62 npncan2 9800 . . . 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 659 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) )  -  ( Q  x.  G
) )  +  ( ( Q  x.  G
)  -  ( ( 2  x.  ( N ^ 2 ) )  +  ( M ^
3 ) ) ) )  =  0 )
6460, 63eqtrd 2441 . 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 23389 . 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 1403    e. wcel 1840    =/= wne 2596  (class class class)co 6232   CCcc 9438   0cc0 9440    + caddc 9443    x. cmul 9445    - cmin 9759    / cdiv 10165   2c2 10544   3c3 10545   NN0cn0 10754   ^cexp 12118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  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 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-fz 11642  df-seq 12060  df-exp 12119  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-dvds 14086
This theorem is referenced by:  dcubic  23392
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