Theorem cubic 22219

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3927 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 10384	. . . . . . . . . . 11 |- 2 e. CC
10		3nn0 10589	. . . . . . . . . . . 12 |- 3 e. NN0
11		expcl 11875	. . . . . . . . . . . 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 9358	. . . . . . . . . . 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 10401	. . . . . . . . . . . 12 |- 9 e. CC
16		mulcl 9358	. . . . . . . . . . . 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 9398	. . . . . . . . . . 11 |- ( ph -> ( B x. C ) e. CC )
19	17, 18	mulcld 9398	. . . . . . . . . 10 |- ( ph -> ( ( 9 x. A ) x. ( B x. C ) ) e. CC )
20	14, 19	subcld 9711	. . . . . . . . 9 |- ( ph -> ( ( 2 x. ( B ^ 3 ) ) - ( ( 9 x. A ) x. ( B x. C ) ) ) e. CC )
21		2nn0 10588	. . . . . . . . . . . 12 |- 2 e. NN0
22		7nn 10476	. . . . . . . . . . . 12 |- 7 e. NN
23	21, 22	decnncl 10760	. . . . . . . . . . 11 |- ; 2 7 e. NN
24	23	nncni 10324	. . . . . . . . . 10 |- ; 2 7 e. CC
25	1	sqcld 11998	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
26	25, 5	mulcld 9398	. . . . . . . . . 10 |- ( ph -> ( ( A ^ 2 ) x. D ) e. CC )
27		mulcl 9358	. . . . . . . . . 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 9397	. . . . . . . 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 2512	. . . . . . 7 |- ( ph -> N e. CC )
31		cubic.g	. . . . . . . . 9 |- ( ph -> G = ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) )
32	30	sqcld 11998	. . . . . . . . . 10 |- ( ph -> ( N ^ 2 ) e. CC )
33		4cn 10391	. . . . . . . . . . 11 |- 4 e. CC
34		cubic.m	. . . . . . . . . . . . 13 |- ( ph -> M = ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) )
35	3	sqcld 11998	. . . . . . . . . . . . . 14 |- ( ph -> ( B ^ 2 ) e. CC )
36		3cn 10388	. . . . . . . . . . . . . . 15 |- 3 e. CC
37	1, 4	mulcld 9398	. . . . . . . . . . . . . . 15 |- ( ph -> ( A x. C ) e. CC )
38		mulcl 9358	. . . . . . . . . . . . . . 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 9711	. . . . . . . . . . . . 13 |- ( ph -> ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) e. CC )
41	34, 40	eqeltrd 2512	. . . . . . . . . . . 12 |- ( ph -> M e. CC )
42		expcl 11875	. . . . . . . . . . . 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 9358	. . . . . . . . . . 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 9711	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) e. CC )
47	31, 46	eqeltrd 2512	. . . . . . . 8 |- ( ph -> G e. CC )
48	47	sqrcld 12915	. . . . . . 7 |- ( ph -> ( sqr ` G ) e. CC )
49	30, 48	addcld 9397	. . . . . 6 |- ( ph -> ( N + ( sqr ` G ) ) e. CC )
50	49	halfcld 10561	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) e. CC )
51		3ne0 10408	. . . . . 6 |- 3 =/= 0
52	36, 51	reccli 10053	. . . . 5 |- ( 1 / 3 ) e. CC
53		cxpcl 22094	. . . . 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 2512	. . 3 |- ( ph -> T e. CC )
56	7	oveq1d 6101	. . . 4 |- ( ph -> ( T ^ 3 ) = ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) )
57		3nn 10472	. . . . 5 |- 3 e. NN
58		cxproot 22110	. . . . 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 2470	. . 3 |- ( ph -> ( T ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
61	47	sqsqrd 12917	. . . 4 |- ( ph -> ( ( sqr ` G ) ^ 2 ) = G )
62	61, 31	eqtrd 2470	. . 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 10410	. . . . . . . . . 10 |- 4 =/= 0
66	65	a1i 11	. . . . . . . . 9 |- ( ph -> 4 =/= 0 )
67		cubic.0	. . . . . . . . . 10 |- ( ph -> M =/= 0 )
68		3z 10671	. . . . . . . . . . 11 |- 3 e. ZZ
69	68	a1i 11	. . . . . . . . . 10 |- ( ph -> 3 e. ZZ )
70	41, 67, 69	expne0d 12006	. . . . . . . . 9 |- ( ph -> ( M ^ 3 ) =/= 0 )
71	64, 43, 66, 70	mulne0d 9980	. . . . . . . 8 |- ( ph -> ( 4 x. ( M ^ 3 ) ) =/= 0 )
72	62	oveq2d 6102	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) )
73		subsq 11965	. . . . . . . . . 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 9716	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) = ( 4 x. ( M ^ 3 ) ) )
76	72, 74, 75	3eqtr3d 2478	. . . . . . . 8 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 4 x. ( M ^ 3 ) ) )
77	30, 48	subcld 9711	. . . . . . . . 9 |- ( ph -> ( N - ( sqr ` G ) ) e. CC )
78	77	mul02d 9559	. . . . . . . 8 |- ( ph -> ( 0 x. ( N - ( sqr ` G ) ) ) = 0 )
79	71, 76, 78	3netr4d 2630	. . . . . . 7 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) =/= ( 0 x. ( N - ( sqr ` G ) ) ) )
80		oveq1 6093	. . . . . . . 8 |- ( ( N + ( sqr ` G ) ) = 0 -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 0 x. ( N - ( sqr ` G ) ) ) )
81	80	necon3i 2645	. . . . . . 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 10406	. . . . . . 7 |- 2 =/= 0
84	83	a1i 11	. . . . . 6 |- ( ph -> 2 =/= 0 )
85	49, 63, 82, 84	divne0d 10115	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) =/= 0 )
86	52	a1i 11	. . . . 5 |- ( ph -> ( 1 / 3 ) e. CC )
87	50, 85, 86	cxpne0d 22133	. . . 4 |- ( ph -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) =/= 0 )
88	7, 87	eqnetrd 2621	. . 3 |- ( ph -> T =/= 0 )
89	1, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88	cubic2 22218	. 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 22212	. . . . 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 2739	. 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 1369   e. wcel 1756   =/= wne 2601   E.wrex 2711   {ctp 3876   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275   _ici 9276   + caddc 9277   x. cmul 9279   - cmin 9587   -ucneg 9588   / cdiv 9985   NNcn 10314   2c2 10363   3c3 10364   4c4 10365   7c7 10368   9c9 10370   NN0cn0 10571   ZZcz 10638  ;cdc 10747   ^cexp 11857   sqrcsqr 12714   ^c ccxp 21982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983  df-cxp 21984

This theorem is referenced by: (None)