Theorem cubic 22129

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3920 to convert the existential quantifier to a triple disjunction. (Contributed by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
cubic.r	|- R = { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) }
cubic.a	|- ( ph -> A e. CC )
cubic.z	|- ( ph -> A =/= 0 )
cubic.b	|- ( ph -> B e. CC )
cubic.c	|- ( ph -> C e. CC )
cubic.d	|- ( ph -> D e. CC )
cubic.x	|- ( ph -> X e. CC )
cubic.t	|- ( ph -> T = ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) )
cubic.g	|- ( ph -> G = ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) )
cubic.m	|- ( ph -> M = ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) )
cubic.n	|- ( ph -> N = ( ( ( 2 x. ( B ^ 3 ) ) - ( ( 9 x. A ) x. ( B x. C ) ) ) + (; 2 7 x. ( ( A ^ 2 ) x. D ) ) ) )
cubic.0	|- ( ph -> M =/= 0 )

Assertion

Ref	Expression
cubic	|- ( ph -> ( ( ( ( A x. ( X ^ 3 ) ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) = 0 <-> E. r e. R X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) )

Distinct variable groups:   A, r   B, r   M, r   N, r   ph, r   T, r   X, r

Allowed substitution hints:   C( r)   D( r)   R( r)   G( r)

Proof of Theorem cubic

Step	Hyp	Ref	Expression
1		cubic.a	. . 3 |- ( ph -> A e. CC )
2		cubic.z	. . 3 |- ( ph -> A =/= 0 )
3		cubic.b	. . 3 |- ( ph -> B e. CC )
4		cubic.c	. . 3 |- ( ph -> C e. CC )
5		cubic.d	. . 3 |- ( ph -> D e. CC )
6		cubic.x	. . 3 |- ( ph -> X e. CC )
7		cubic.t	. . . 4 |- ( ph -> T = ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) )
8		cubic.n	. . . . . . . 8 |- ( ph -> N = ( ( ( 2 x. ( B ^ 3 ) ) - ( ( 9 x. A ) x. ( B x. C ) ) ) + (; 2 7 x. ( ( A ^ 2 ) x. D ) ) ) )
9		2cn 10380	. . . . . . . . . . 11 |- 2 e. CC
10		3nn0 10585	. . . . . . . . . . . 12 |- 3 e. NN0
11		expcl 11867	. . . . . . . . . . . 12 |- ( ( B e. CC /\ 3 e. NN0 ) -> ( B ^ 3 ) e. CC )
12	3, 10, 11	sylancl 655	. . . . . . . . . . 11 |- ( ph -> ( B ^ 3 ) e. CC )
13		mulcl 9354	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ ( B ^ 3 ) e. CC ) -> ( 2 x. ( B ^ 3 ) ) e. CC )
14	9, 12, 13	sylancr 656	. . . . . . . . . 10 |- ( ph -> ( 2 x. ( B ^ 3 ) ) e. CC )
15		9cn 10397	. . . . . . . . . . . 12 |- 9 e. CC
16		mulcl 9354	. . . . . . . . . . . 12 |- ( ( 9 e. CC /\ A e. CC ) -> ( 9 x. A ) e. CC )
17	15, 1, 16	sylancr 656	. . . . . . . . . . 11 |- ( ph -> ( 9 x. A ) e. CC )
18	3, 4	mulcld 9394	. . . . . . . . . . 11 |- ( ph -> ( B x. C ) e. CC )
19	17, 18	mulcld 9394	. . . . . . . . . 10 |- ( ph -> ( ( 9 x. A ) x. ( B x. C ) ) e. CC )
20	14, 19	subcld 9707	. . . . . . . . 9 |- ( ph -> ( ( 2 x. ( B ^ 3 ) ) - ( ( 9 x. A ) x. ( B x. C ) ) ) e. CC )
21		2nn0 10584	. . . . . . . . . . . 12 |- 2 e. NN0
22		7nn 10472	. . . . . . . . . . . 12 |- 7 e. NN
23	21, 22	decnncl 10756	. . . . . . . . . . 11 |- ; 2 7 e. NN
24	23	nncni 10320	. . . . . . . . . 10 |- ; 2 7 e. CC
25	1	sqcld 11990	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
26	25, 5	mulcld 9394	. . . . . . . . . 10 |- ( ph -> ( ( A ^ 2 ) x. D ) e. CC )
27		mulcl 9354	. . . . . . . . . 10 |- ( (; 2 7 e. CC /\ ( ( A ^ 2 ) x. D ) e. CC ) -> (; 2 7 x. ( ( A ^ 2 ) x. D ) ) e. CC )
28	24, 26, 27	sylancr 656	. . . . . . . . 9 |- ( ph -> (; 2 7 x. ( ( A ^ 2 ) x. D ) ) e. CC )
29	20, 28	addcld 9393	. . . . . . . 8 |- ( ph -> ( ( ( 2 x. ( B ^ 3 ) ) - ( ( 9 x. A ) x. ( B x. C ) ) ) + (; 2 7 x. ( ( A ^ 2 ) x. D ) ) ) e. CC )
30	8, 29	eqeltrd 2507	. . . . . . 7 |- ( ph -> N e. CC )
31		cubic.g	. . . . . . . . 9 |- ( ph -> G = ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) )
32	30	sqcld 11990	. . . . . . . . . 10 |- ( ph -> ( N ^ 2 ) e. CC )
33		4cn 10387	. . . . . . . . . . 11 |- 4 e. CC
34		cubic.m	. . . . . . . . . . . . 13 |- ( ph -> M = ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) )
35	3	sqcld 11990	. . . . . . . . . . . . . 14 |- ( ph -> ( B ^ 2 ) e. CC )
36		3cn 10384	. . . . . . . . . . . . . . 15 |- 3 e. CC
37	1, 4	mulcld 9394	. . . . . . . . . . . . . . 15 |- ( ph -> ( A x. C ) e. CC )
38		mulcl 9354	. . . . . . . . . . . . . . 15 |- ( ( 3 e. CC /\ ( A x. C ) e. CC ) -> ( 3 x. ( A x. C ) ) e. CC )
39	36, 37, 38	sylancr 656	. . . . . . . . . . . . . 14 |- ( ph -> ( 3 x. ( A x. C ) ) e. CC )
40	35, 39	subcld 9707	. . . . . . . . . . . . 13 |- ( ph -> ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) e. CC )
41	34, 40	eqeltrd 2507	. . . . . . . . . . . 12 |- ( ph -> M e. CC )
42		expcl 11867	. . . . . . . . . . . 12 |- ( ( M e. CC /\ 3 e. NN0 ) -> ( M ^ 3 ) e. CC )
43	41, 10, 42	sylancl 655	. . . . . . . . . . 11 |- ( ph -> ( M ^ 3 ) e. CC )
44		mulcl 9354	. . . . . . . . . . 11 |- ( ( 4 e. CC /\ ( M ^ 3 ) e. CC ) -> ( 4 x. ( M ^ 3 ) ) e. CC )
45	33, 43, 44	sylancr 656	. . . . . . . . . 10 |- ( ph -> ( 4 x. ( M ^ 3 ) ) e. CC )
46	32, 45	subcld 9707	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) e. CC )
47	31, 46	eqeltrd 2507	. . . . . . . 8 |- ( ph -> G e. CC )
48	47	sqrcld 12907	. . . . . . 7 |- ( ph -> ( sqr ` G ) e. CC )
49	30, 48	addcld 9393	. . . . . 6 |- ( ph -> ( N + ( sqr ` G ) ) e. CC )
50	49	halfcld 10557	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) e. CC )
51		3ne0 10404	. . . . . 6 |- 3 =/= 0
52	36, 51	reccli 10049	. . . . 5 |- ( 1 / 3 ) e. CC
53		cxpcl 22004	. . . . 5 |- ( ( ( ( N + ( sqr ` G ) ) / 2 ) e. CC /\ ( 1 / 3 ) e. CC ) -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) e. CC )
54	50, 52, 53	sylancl 655	. . . 4 |- ( ph -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) e. CC )
55	7, 54	eqeltrd 2507	. . 3 |- ( ph -> T e. CC )
56	7	oveq1d 6095	. . . 4 |- ( ph -> ( T ^ 3 ) = ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) )
57		3nn 10468	. . . . 5 |- 3 e. NN
58		cxproot 22020	. . . . 5 |- ( ( ( ( N + ( sqr ` G ) ) / 2 ) e. CC /\ 3 e. NN ) -> ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
59	50, 57, 58	sylancl 655	. . . 4 |- ( ph -> ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
60	56, 59	eqtrd 2465	. . 3 |- ( ph -> ( T ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
61	47	sqsqrd 12909	. . . 4 |- ( ph -> ( ( sqr ` G ) ^ 2 ) = G )
62	61, 31	eqtrd 2465	. . 3 |- ( ph -> ( ( sqr ` G ) ^ 2 ) = ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) )
63	9	a1i 11	. . . . . 6 |- ( ph -> 2 e. CC )
64	33	a1i 11	. . . . . . . . 9 |- ( ph -> 4 e. CC )
65		4ne0 10406	. . . . . . . . . 10 |- 4 =/= 0
66	65	a1i 11	. . . . . . . . 9 |- ( ph -> 4 =/= 0 )
67		cubic.0	. . . . . . . . . 10 |- ( ph -> M =/= 0 )
68		3z 10667	. . . . . . . . . . 11 |- 3 e. ZZ
69	68	a1i 11	. . . . . . . . . 10 |- ( ph -> 3 e. ZZ )
70	41, 67, 69	expne0d 11998	. . . . . . . . 9 |- ( ph -> ( M ^ 3 ) =/= 0 )
71	64, 43, 66, 70	mulne0d 9976	. . . . . . . 8 |- ( ph -> ( 4 x. ( M ^ 3 ) ) =/= 0 )
72	62	oveq2d 6096	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) )
73		subsq 11957	. . . . . . . . . 10 |- ( ( N e. CC /\ ( sqr ` G ) e. CC ) -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) )
74	30, 48, 73	syl2anc 654	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) )
75	32, 45	nncand 9712	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) = ( 4 x. ( M ^ 3 ) ) )
76	72, 74, 75	3eqtr3d 2473	. . . . . . . 8 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 4 x. ( M ^ 3 ) ) )
77	30, 48	subcld 9707	. . . . . . . . 9 |- ( ph -> ( N - ( sqr ` G ) ) e. CC )
78	77	mul02d 9555	. . . . . . . 8 |- ( ph -> ( 0 x. ( N - ( sqr ` G ) ) ) = 0 )
79	71, 76, 78	3netr4d 2625	. . . . . . 7 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) =/= ( 0 x. ( N - ( sqr ` G ) ) ) )
80		oveq1 6087	. . . . . . . 8 |- ( ( N + ( sqr ` G ) ) = 0 -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 0 x. ( N - ( sqr ` G ) ) ) )
81	80	necon3i 2640	. . . . . . 7 |- ( ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) =/= ( 0 x. ( N - ( sqr ` G ) ) ) -> ( N + ( sqr ` G ) ) =/= 0 )
82	79, 81	syl 16	. . . . . 6 |- ( ph -> ( N + ( sqr ` G ) ) =/= 0 )
83		2ne0 10402	. . . . . . 7 |- 2 =/= 0
84	83	a1i 11	. . . . . 6 |- ( ph -> 2 =/= 0 )
85	49, 63, 82, 84	divne0d 10111	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) =/= 0 )
86	52	a1i 11	. . . . 5 |- ( ph -> ( 1 / 3 ) e. CC )
87	50, 85, 86	cxpne0d 22043	. . . 4 |- ( ph -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) =/= 0 )
88	7, 87	eqnetrd 2616	. . 3 |- ( ph -> T =/= 0 )
89	1, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88	cubic2 22128	. 2 |- ( ph -> ( ( ( ( A x. ( X ^ 3 ) ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) = 0 <-> E. r e. CC ( ( r ^ 3 ) = 1 /\ X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) ) )
90		cubic.r	. . . . . 6 |- R = { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) }
91	90	1cubr 22122	. . . . 5 |- ( r e. R <-> ( r e. CC /\ ( r ^ 3 ) = 1 ) )
92	91	anbi1i 688	. . . 4 |- ( ( r e. R /\ X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) <-> ( ( r e. CC /\ ( r ^ 3 ) = 1 ) /\ X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) )
93		anass 642	. . . 4 |- ( ( ( r e. CC /\ ( r ^ 3 ) = 1 ) /\ X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) <-> ( r e. CC /\ ( ( r ^ 3 ) = 1 /\ X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) ) )
94	92, 93	bitri 249	. . 3 |- ( ( r e. R /\ X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) <-> ( r e. CC /\ ( ( r ^ 3 ) = 1 /\ X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) ) )
95	94	rexbii2 2734	. 2 |- ( E. r e. R X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) <-> E. r e. CC ( ( r ^ 3 ) = 1 /\ X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) )
96	89, 95	syl6bbr 263	1 |- ( ph -> ( ( ( ( A x. ( X ^ 3 ) ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) = 0 <-> E. r e. R X = -u ( ( ( B + ( r x. T ) ) + ( M / ( r x. T ) ) ) / ( 3 x. A ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1362   e. wcel 1755   =/= wne 2596   E.wrex 2706   {ctp 3869   ` cfv 5406  (class class class)co 6080   CCcc 9268   0cc0 9270   1c1 9271   _ici 9272   + caddc 9273   x. cmul 9275   - cmin 9583   -ucneg 9584   / cdiv 9981   NNcn 10310   2c2 10359   3c3 10360   4c4 10361   7c7 10364   9c9 10366   NN0cn0 10567   ZZcz 10634  ;cdc 10743   ^cexp 11849   sqrcsqr 12706   ^c ccxp 21892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-dvds 13519  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894

This theorem is referenced by: (None)