Theorem cubic 22376

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 4039 to convert the existential quantifier to a triple disjunction. This is Metamath 100 proof #37. (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 10502	. . . . . . . . . . 11 |- 2 e. CC
10		3nn0 10707	. . . . . . . . . . . 12 |- 3 e. NN0
11		expcl 11999	. . . . . . . . . . . 12 |- ( ( B e. CC /\ 3 e. NN0 ) -> ( B ^ 3 ) e. CC )
12	3, 10, 11	sylancl 662	. . . . . . . . . . 11 |- ( ph -> ( B ^ 3 ) e. CC )
13		mulcl 9476	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ ( B ^ 3 ) e. CC ) -> ( 2 x. ( B ^ 3 ) ) e. CC )
14	9, 12, 13	sylancr 663	. . . . . . . . . 10 |- ( ph -> ( 2 x. ( B ^ 3 ) ) e. CC )
15		9cn 10519	. . . . . . . . . . . 12 |- 9 e. CC
16		mulcl 9476	. . . . . . . . . . . 12 |- ( ( 9 e. CC /\ A e. CC ) -> ( 9 x. A ) e. CC )
17	15, 1, 16	sylancr 663	. . . . . . . . . . 11 |- ( ph -> ( 9 x. A ) e. CC )
18	3, 4	mulcld 9516	. . . . . . . . . . 11 |- ( ph -> ( B x. C ) e. CC )
19	17, 18	mulcld 9516	. . . . . . . . . 10 |- ( ph -> ( ( 9 x. A ) x. ( B x. C ) ) e. CC )
20	14, 19	subcld 9829	. . . . . . . . 9 |- ( ph -> ( ( 2 x. ( B ^ 3 ) ) - ( ( 9 x. A ) x. ( B x. C ) ) ) e. CC )
21		2nn0 10706	. . . . . . . . . . . 12 |- 2 e. NN0
22		7nn 10594	. . . . . . . . . . . 12 |- 7 e. NN
23	21, 22	decnncl 10878	. . . . . . . . . . 11 |- ; 2 7 e. NN
24	23	nncni 10442	. . . . . . . . . 10 |- ; 2 7 e. CC
25	1	sqcld 12122	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
26	25, 5	mulcld 9516	. . . . . . . . . 10 |- ( ph -> ( ( A ^ 2 ) x. D ) e. CC )
27		mulcl 9476	. . . . . . . . . 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 663	. . . . . . . . 9 |- ( ph -> (; 2 7 x. ( ( A ^ 2 ) x. D ) ) e. CC )
29	20, 28	addcld 9515	. . . . . . . 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 2542	. . . . . . 7 |- ( ph -> N e. CC )
31		cubic.g	. . . . . . . . 9 |- ( ph -> G = ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) )
32	30	sqcld 12122	. . . . . . . . . 10 |- ( ph -> ( N ^ 2 ) e. CC )
33		4cn 10509	. . . . . . . . . . 11 |- 4 e. CC
34		cubic.m	. . . . . . . . . . . . 13 |- ( ph -> M = ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) )
35	3	sqcld 12122	. . . . . . . . . . . . . 14 |- ( ph -> ( B ^ 2 ) e. CC )
36		3cn 10506	. . . . . . . . . . . . . . 15 |- 3 e. CC
37	1, 4	mulcld 9516	. . . . . . . . . . . . . . 15 |- ( ph -> ( A x. C ) e. CC )
38		mulcl 9476	. . . . . . . . . . . . . . 15 |- ( ( 3 e. CC /\ ( A x. C ) e. CC ) -> ( 3 x. ( A x. C ) ) e. CC )
39	36, 37, 38	sylancr 663	. . . . . . . . . . . . . 14 |- ( ph -> ( 3 x. ( A x. C ) ) e. CC )
40	35, 39	subcld 9829	. . . . . . . . . . . . 13 |- ( ph -> ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) e. CC )
41	34, 40	eqeltrd 2542	. . . . . . . . . . . 12 |- ( ph -> M e. CC )
42		expcl 11999	. . . . . . . . . . . 12 |- ( ( M e. CC /\ 3 e. NN0 ) -> ( M ^ 3 ) e. CC )
43	41, 10, 42	sylancl 662	. . . . . . . . . . 11 |- ( ph -> ( M ^ 3 ) e. CC )
44		mulcl 9476	. . . . . . . . . . 11 |- ( ( 4 e. CC /\ ( M ^ 3 ) e. CC ) -> ( 4 x. ( M ^ 3 ) ) e. CC )
45	33, 43, 44	sylancr 663	. . . . . . . . . 10 |- ( ph -> ( 4 x. ( M ^ 3 ) ) e. CC )
46	32, 45	subcld 9829	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) e. CC )
47	31, 46	eqeltrd 2542	. . . . . . . 8 |- ( ph -> G e. CC )
48	47	sqrcld 13040	. . . . . . 7 |- ( ph -> ( sqr ` G ) e. CC )
49	30, 48	addcld 9515	. . . . . 6 |- ( ph -> ( N + ( sqr ` G ) ) e. CC )
50	49	halfcld 10679	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) e. CC )
51		3ne0 10526	. . . . . 6 |- 3 =/= 0
52	36, 51	reccli 10171	. . . . 5 |- ( 1 / 3 ) e. CC
53		cxpcl 22251	. . . . 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 662	. . . 4 |- ( ph -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) e. CC )
55	7, 54	eqeltrd 2542	. . 3 |- ( ph -> T e. CC )
56	7	oveq1d 6214	. . . 4 |- ( ph -> ( T ^ 3 ) = ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) )
57		3nn 10590	. . . . 5 |- 3 e. NN
58		cxproot 22267	. . . . 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 662	. . . 4 |- ( ph -> ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
60	56, 59	eqtrd 2495	. . 3 |- ( ph -> ( T ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
61	47	sqsqrd 13042	. . . 4 |- ( ph -> ( ( sqr ` G ) ^ 2 ) = G )
62	61, 31	eqtrd 2495	. . 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 10528	. . . . . . . . . 10 |- 4 =/= 0
66	65	a1i 11	. . . . . . . . 9 |- ( ph -> 4 =/= 0 )
67		cubic.0	. . . . . . . . . 10 |- ( ph -> M =/= 0 )
68		3z 10789	. . . . . . . . . . 11 |- 3 e. ZZ
69	68	a1i 11	. . . . . . . . . 10 |- ( ph -> 3 e. ZZ )
70	41, 67, 69	expne0d 12130	. . . . . . . . 9 |- ( ph -> ( M ^ 3 ) =/= 0 )
71	64, 43, 66, 70	mulne0d 10098	. . . . . . . 8 |- ( ph -> ( 4 x. ( M ^ 3 ) ) =/= 0 )
72	62	oveq2d 6215	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) )
73		subsq 12089	. . . . . . . . . 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 661	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) )
75	32, 45	nncand 9834	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) = ( 4 x. ( M ^ 3 ) ) )
76	72, 74, 75	3eqtr3d 2503	. . . . . . . 8 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 4 x. ( M ^ 3 ) ) )
77	30, 48	subcld 9829	. . . . . . . . 9 |- ( ph -> ( N - ( sqr ` G ) ) e. CC )
78	77	mul02d 9677	. . . . . . . 8 |- ( ph -> ( 0 x. ( N - ( sqr ` G ) ) ) = 0 )
79	71, 76, 78	3netr4d 2756	. . . . . . 7 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) =/= ( 0 x. ( N - ( sqr ` G ) ) ) )
80		oveq1 6206	. . . . . . . 8 |- ( ( N + ( sqr ` G ) ) = 0 -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 0 x. ( N - ( sqr ` G ) ) ) )
81	80	necon3i 2691	. . . . . . 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 10524	. . . . . . 7 |- 2 =/= 0
84	83	a1i 11	. . . . . 6 |- ( ph -> 2 =/= 0 )
85	49, 63, 82, 84	divne0d 10233	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) =/= 0 )
86	52	a1i 11	. . . . 5 |- ( ph -> ( 1 / 3 ) e. CC )
87	50, 85, 86	cxpne0d 22290	. . . 4 |- ( ph -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) =/= 0 )
88	7, 87	eqnetrd 2744	. . 3 |- ( ph -> T =/= 0 )
89	1, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88	cubic2 22375	. 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 22369	. . . . 5 |- ( r e. R <-> ( r e. CC /\ ( r ^ 3 ) = 1 ) )
92	91	anbi1i 695	. . . 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 649	. . . 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 2857	. 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 1370   e. wcel 1758   =/= wne 2647   E.wrex 2799   {ctp 3988   ` cfv 5525  (class class class)co 6199   CCcc 9390   0cc0 9392   1c1 9393   _ici 9394   + caddc 9395   x. cmul 9397   - cmin 9705   -ucneg 9706   / cdiv 10103   NNcn 10432   2c2 10481   3c3 10482   4c4 10483   7c7 10486   9c9 10488   NN0cn0 10689   ZZcz 10756  ;cdc 10865   ^cexp 11981   sqrcsqr 12839   ^c ccxp 22139

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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-shft 12673  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-limsup 13066  df-clim 13083  df-rlim 13084  df-sum 13281  df-ef 13470  df-sin 13472  df-cos 13473  df-pi 13475  df-dvds 13653  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474  df-log 22140  df-cxp 22141

This theorem is referenced by: (None)