Theorem cubic 20642

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3824 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 10026	. . . . . . . . . . 11 |- 2 e. CC
10		3nn0 10195	. . . . . . . . . . . 12 |- 3 e. NN0
11		expcl 11354	. . . . . . . . . . . 12 |- ( ( B e. CC /\ 3 e. NN0 ) -> ( B ^ 3 ) e. CC )
12	3, 10, 11	sylancl 644	. . . . . . . . . . 11 |- ( ph -> ( B ^ 3 ) e. CC )
13		mulcl 9030	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ ( B ^ 3 ) e. CC ) -> ( 2 x. ( B ^ 3 ) ) e. CC )
14	9, 12, 13	sylancr 645	. . . . . . . . . 10 |- ( ph -> ( 2 x. ( B ^ 3 ) ) e. CC )
15		9nn 10096	. . . . . . . . . . . . 13 |- 9 e. NN
16	15	nncni 9966	. . . . . . . . . . . 12 |- 9 e. CC
17		mulcl 9030	. . . . . . . . . . . 12 |- ( ( 9 e. CC /\ A e. CC ) -> ( 9 x. A ) e. CC )
18	16, 1, 17	sylancr 645	. . . . . . . . . . 11 |- ( ph -> ( 9 x. A ) e. CC )
19	3, 4	mulcld 9064	. . . . . . . . . . 11 |- ( ph -> ( B x. C ) e. CC )
20	18, 19	mulcld 9064	. . . . . . . . . 10 |- ( ph -> ( ( 9 x. A ) x. ( B x. C ) ) e. CC )
21	14, 20	subcld 9367	. . . . . . . . 9 |- ( ph -> ( ( 2 x. ( B ^ 3 ) ) - ( ( 9 x. A ) x. ( B x. C ) ) ) e. CC )
22		2nn0 10194	. . . . . . . . . . . 12 |- 2 e. NN0
23		7nn 10094	. . . . . . . . . . . 12 |- 7 e. NN
24	22, 23	decnncl 10351	. . . . . . . . . . 11 |- ; 2 7 e. NN
25	24	nncni 9966	. . . . . . . . . 10 |- ; 2 7 e. CC
26	1	sqcld 11476	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
27	26, 5	mulcld 9064	. . . . . . . . . 10 |- ( ph -> ( ( A ^ 2 ) x. D ) e. CC )
28		mulcl 9030	. . . . . . . . . 10 |- ( (; 2 7 e. CC /\ ( ( A ^ 2 ) x. D ) e. CC ) -> (; 2 7 x. ( ( A ^ 2 ) x. D ) ) e. CC )
29	25, 27, 28	sylancr 645	. . . . . . . . 9 |- ( ph -> (; 2 7 x. ( ( A ^ 2 ) x. D ) ) e. CC )
30	21, 29	addcld 9063	. . . . . . . 8 |- ( ph -> ( ( ( 2 x. ( B ^ 3 ) ) - ( ( 9 x. A ) x. ( B x. C ) ) ) + (; 2 7 x. ( ( A ^ 2 ) x. D ) ) ) e. CC )
31	8, 30	eqeltrd 2478	. . . . . . 7 |- ( ph -> N e. CC )
32		cubic.g	. . . . . . . . 9 |- ( ph -> G = ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) )
33	31	sqcld 11476	. . . . . . . . . 10 |- ( ph -> ( N ^ 2 ) e. CC )
34		4cn 10030	. . . . . . . . . . 11 |- 4 e. CC
35		cubic.m	. . . . . . . . . . . . 13 |- ( ph -> M = ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) )
36	3	sqcld 11476	. . . . . . . . . . . . . 14 |- ( ph -> ( B ^ 2 ) e. CC )
37		3cn 10028	. . . . . . . . . . . . . . 15 |- 3 e. CC
38	1, 4	mulcld 9064	. . . . . . . . . . . . . . 15 |- ( ph -> ( A x. C ) e. CC )
39		mulcl 9030	. . . . . . . . . . . . . . 15 |- ( ( 3 e. CC /\ ( A x. C ) e. CC ) -> ( 3 x. ( A x. C ) ) e. CC )
40	37, 38, 39	sylancr 645	. . . . . . . . . . . . . 14 |- ( ph -> ( 3 x. ( A x. C ) ) e. CC )
41	36, 40	subcld 9367	. . . . . . . . . . . . 13 |- ( ph -> ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) e. CC )
42	35, 41	eqeltrd 2478	. . . . . . . . . . . 12 |- ( ph -> M e. CC )
43		expcl 11354	. . . . . . . . . . . 12 |- ( ( M e. CC /\ 3 e. NN0 ) -> ( M ^ 3 ) e. CC )
44	42, 10, 43	sylancl 644	. . . . . . . . . . 11 |- ( ph -> ( M ^ 3 ) e. CC )
45		mulcl 9030	. . . . . . . . . . 11 |- ( ( 4 e. CC /\ ( M ^ 3 ) e. CC ) -> ( 4 x. ( M ^ 3 ) ) e. CC )
46	34, 44, 45	sylancr 645	. . . . . . . . . 10 |- ( ph -> ( 4 x. ( M ^ 3 ) ) e. CC )
47	33, 46	subcld 9367	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) e. CC )
48	32, 47	eqeltrd 2478	. . . . . . . 8 |- ( ph -> G e. CC )
49	48	sqrcld 12194	. . . . . . 7 |- ( ph -> ( sqr ` G ) e. CC )
50	31, 49	addcld 9063	. . . . . 6 |- ( ph -> ( N + ( sqr ` G ) ) e. CC )
51	50	halfcld 10168	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) e. CC )
52		3ne0 10041	. . . . . 6 |- 3 =/= 0
53	37, 52	reccli 9700	. . . . 5 |- ( 1 / 3 ) e. CC
54		cxpcl 20518	. . . . 5 |- ( ( ( ( N + ( sqr ` G ) ) / 2 ) e. CC /\ ( 1 / 3 ) e. CC ) -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^ c ( 1 / 3 ) ) e. CC )
55	51, 53, 54	sylancl 644	. . . 4 |- ( ph -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^ c ( 1 / 3 ) ) e. CC )
56	7, 55	eqeltrd 2478	. . 3 |- ( ph -> T e. CC )
57	7	oveq1d 6055	. . . 4 |- ( ph -> ( T ^ 3 ) = ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^ c ( 1 / 3 ) ) ^ 3 ) )
58		3nn 10090	. . . . 5 |- 3 e. NN
59		cxproot 20534	. . . . 5 |- ( ( ( ( N + ( sqr ` G ) ) / 2 ) e. CC /\ 3 e. NN ) -> ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^ c ( 1 / 3 ) ) ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
60	51, 58, 59	sylancl 644	. . . 4 |- ( ph -> ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^ c ( 1 / 3 ) ) ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
61	57, 60	eqtrd 2436	. . 3 |- ( ph -> ( T ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
62	48	sqsqrd 12196	. . . 4 |- ( ph -> ( ( sqr ` G ) ^ 2 ) = G )
63	62, 32	eqtrd 2436	. . 3 |- ( ph -> ( ( sqr ` G ) ^ 2 ) = ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) )
64	9	a1i 11	. . . . . 6 |- ( ph -> 2 e. CC )
65	34	a1i 11	. . . . . . . . 9 |- ( ph -> 4 e. CC )
66		4nn 10091	. . . . . . . . . . 11 |- 4 e. NN
67	66	nnne0i 9990	. . . . . . . . . 10 |- 4 =/= 0
68	67	a1i 11	. . . . . . . . 9 |- ( ph -> 4 =/= 0 )
69		cubic.0	. . . . . . . . . 10 |- ( ph -> M =/= 0 )
70	10	nn0zi 10262	. . . . . . . . . . 11 |- 3 e. ZZ
71	70	a1i 11	. . . . . . . . . 10 |- ( ph -> 3 e. ZZ )
72	42, 69, 71	expne0d 11484	. . . . . . . . 9 |- ( ph -> ( M ^ 3 ) =/= 0 )
73	65, 44, 68, 72	mulne0d 9630	. . . . . . . 8 |- ( ph -> ( 4 x. ( M ^ 3 ) ) =/= 0 )
74	63	oveq2d 6056	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) )
75		subsq 11443	. . . . . . . . . 10 |- ( ( N e. CC /\ ( sqr ` G ) e. CC ) -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) )
76	31, 49, 75	syl2anc 643	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) )
77	33, 46	nncand 9372	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) = ( 4 x. ( M ^ 3 ) ) )
78	74, 76, 77	3eqtr3d 2444	. . . . . . . 8 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 4 x. ( M ^ 3 ) ) )
79	31, 49	subcld 9367	. . . . . . . . 9 |- ( ph -> ( N - ( sqr ` G ) ) e. CC )
80	79	mul02d 9220	. . . . . . . 8 |- ( ph -> ( 0 x. ( N - ( sqr ` G ) ) ) = 0 )
81	73, 78, 80	3netr4d 2594	. . . . . . 7 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) =/= ( 0 x. ( N - ( sqr ` G ) ) ) )
82		oveq1 6047	. . . . . . . 8 |- ( ( N + ( sqr ` G ) ) = 0 -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 0 x. ( N - ( sqr ` G ) ) ) )
83	82	necon3i 2606	. . . . . . 7 |- ( ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) =/= ( 0 x. ( N - ( sqr ` G ) ) ) -> ( N + ( sqr ` G ) ) =/= 0 )
84	81, 83	syl 16	. . . . . 6 |- ( ph -> ( N + ( sqr ` G ) ) =/= 0 )
85		2ne0 10039	. . . . . . 7 |- 2 =/= 0
86	85	a1i 11	. . . . . 6 |- ( ph -> 2 =/= 0 )
87	50, 64, 84, 86	divne0d 9762	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) =/= 0 )
88	53	a1i 11	. . . . 5 |- ( ph -> ( 1 / 3 ) e. CC )
89	51, 87, 88	cxpne0d 20557	. . . 4 |- ( ph -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^ c ( 1 / 3 ) ) =/= 0 )
90	7, 89	eqnetrd 2585	. . 3 |- ( ph -> T =/= 0 )
91	1, 2, 3, 4, 5, 6, 56, 61, 49, 63, 35, 8, 90	cubic2 20641	. 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 ) ) ) ) )
92		cubic.r	. . . . . 6 |- R = { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) }
93	92	1cubr 20635	. . . . 5 |- ( r e. R <-> ( r e. CC /\ ( r ^ 3 ) = 1 ) )
94	93	anbi1i 677	. . . 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 ) ) ) )
95		anass 631	. . . 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 ) ) ) ) )
96	94, 95	bitri 241	. . 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 ) ) ) ) )
97	96	rexbii2 2695	. 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 ) ) ) )
98	91, 97	syl6bbr 255	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 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2567   E.wrex 2667   {ctp 3776   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947   _ici 8948   + caddc 8949   x. cmul 8951   - cmin 9247   -ucneg 9248   / cdiv 9633   NNcn 9956   2c2 10005   3c3 10006   4c4 10007   7c7 10010   9c9 10012   NN0cn0 10177   ZZcz 10238  ;cdc 10338   ^cexp 11337   sqrcsqr 11993   ^ c ccxp 20406

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408