Theorem cubic 22908

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 4083 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 10602	. . . . . . . . . . 11 |- 2 e. CC
10		3nn0 10809	. . . . . . . . . . . 12 |- 3 e. NN0
11		expcl 12148	. . . . . . . . . . . 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 9572	. . . . . . . . . . 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 10619	. . . . . . . . . . . 12 |- 9 e. CC
16		mulcl 9572	. . . . . . . . . . . 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 9612	. . . . . . . . . . 11 |- ( ph -> ( B x. C ) e. CC )
19	17, 18	mulcld 9612	. . . . . . . . . 10 |- ( ph -> ( ( 9 x. A ) x. ( B x. C ) ) e. CC )
20	14, 19	subcld 9926	. . . . . . . . 9 |- ( ph -> ( ( 2 x. ( B ^ 3 ) ) - ( ( 9 x. A ) x. ( B x. C ) ) ) e. CC )
21		2nn0 10808	. . . . . . . . . . . 12 |- 2 e. NN0
22		7nn 10694	. . . . . . . . . . . 12 |- 7 e. NN
23	21, 22	decnncl 10985	. . . . . . . . . . 11 |- ; 2 7 e. NN
24	23	nncni 10542	. . . . . . . . . 10 |- ; 2 7 e. CC
25	1	sqcld 12272	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
26	25, 5	mulcld 9612	. . . . . . . . . 10 |- ( ph -> ( ( A ^ 2 ) x. D ) e. CC )
27		mulcl 9572	. . . . . . . . . 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 9611	. . . . . . . 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 2555	. . . . . . 7 |- ( ph -> N e. CC )
31		cubic.g	. . . . . . . . 9 |- ( ph -> G = ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) )
32	30	sqcld 12272	. . . . . . . . . 10 |- ( ph -> ( N ^ 2 ) e. CC )
33		4cn 10609	. . . . . . . . . . 11 |- 4 e. CC
34		cubic.m	. . . . . . . . . . . . 13 |- ( ph -> M = ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) )
35	3	sqcld 12272	. . . . . . . . . . . . . 14 |- ( ph -> ( B ^ 2 ) e. CC )
36		3cn 10606	. . . . . . . . . . . . . . 15 |- 3 e. CC
37	1, 4	mulcld 9612	. . . . . . . . . . . . . . 15 |- ( ph -> ( A x. C ) e. CC )
38		mulcl 9572	. . . . . . . . . . . . . . 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 9926	. . . . . . . . . . . . 13 |- ( ph -> ( ( B ^ 2 ) - ( 3 x. ( A x. C ) ) ) e. CC )
41	34, 40	eqeltrd 2555	. . . . . . . . . . . 12 |- ( ph -> M e. CC )
42		expcl 12148	. . . . . . . . . . . 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 9572	. . . . . . . . . . 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 9926	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) e. CC )
47	31, 46	eqeltrd 2555	. . . . . . . 8 |- ( ph -> G e. CC )
48	47	sqrtcld 13227	. . . . . . 7 |- ( ph -> ( sqr ` G ) e. CC )
49	30, 48	addcld 9611	. . . . . 6 |- ( ph -> ( N + ( sqr ` G ) ) e. CC )
50	49	halfcld 10779	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) e. CC )
51		3ne0 10626	. . . . . 6 |- 3 =/= 0
52	36, 51	reccli 10270	. . . . 5 |- ( 1 / 3 ) e. CC
53		cxpcl 22783	. . . . 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 2555	. . 3 |- ( ph -> T e. CC )
56	7	oveq1d 6297	. . . 4 |- ( ph -> ( T ^ 3 ) = ( ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) )
57		3nn 10690	. . . . 5 |- 3 e. NN
58		cxproot 22799	. . . . 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 2508	. . 3 |- ( ph -> ( T ^ 3 ) = ( ( N + ( sqr ` G ) ) / 2 ) )
61	47	sqsqrtd 13229	. . . 4 |- ( ph -> ( ( sqr ` G ) ^ 2 ) = G )
62	61, 31	eqtrd 2508	. . 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 10628	. . . . . . . . . 10 |- 4 =/= 0
66	65	a1i 11	. . . . . . . . 9 |- ( ph -> 4 =/= 0 )
67		cubic.0	. . . . . . . . . 10 |- ( ph -> M =/= 0 )
68		3z 10893	. . . . . . . . . . 11 |- 3 e. ZZ
69	68	a1i 11	. . . . . . . . . 10 |- ( ph -> 3 e. ZZ )
70	41, 67, 69	expne0d 12280	. . . . . . . . 9 |- ( ph -> ( M ^ 3 ) =/= 0 )
71	64, 43, 66, 70	mulne0d 10197	. . . . . . . 8 |- ( ph -> ( 4 x. ( M ^ 3 ) ) =/= 0 )
72	62	oveq2d 6298	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( sqr ` G ) ^ 2 ) ) = ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) )
73		subsq 12239	. . . . . . . . . 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 9931	. . . . . . . . 9 |- ( ph -> ( ( N ^ 2 ) - ( ( N ^ 2 ) - ( 4 x. ( M ^ 3 ) ) ) ) = ( 4 x. ( M ^ 3 ) ) )
76	72, 74, 75	3eqtr3d 2516	. . . . . . . 8 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 4 x. ( M ^ 3 ) ) )
77	30, 48	subcld 9926	. . . . . . . . 9 |- ( ph -> ( N - ( sqr ` G ) ) e. CC )
78	77	mul02d 9773	. . . . . . . 8 |- ( ph -> ( 0 x. ( N - ( sqr ` G ) ) ) = 0 )
79	71, 76, 78	3netr4d 2772	. . . . . . 7 |- ( ph -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) =/= ( 0 x. ( N - ( sqr ` G ) ) ) )
80		oveq1 6289	. . . . . . . 8 |- ( ( N + ( sqr ` G ) ) = 0 -> ( ( N + ( sqr ` G ) ) x. ( N - ( sqr ` G ) ) ) = ( 0 x. ( N - ( sqr ` G ) ) ) )
81	80	necon3i 2707	. . . . . . 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 10624	. . . . . . 7 |- 2 =/= 0
84	83	a1i 11	. . . . . 6 |- ( ph -> 2 =/= 0 )
85	49, 63, 82, 84	divne0d 10332	. . . . 5 |- ( ph -> ( ( N + ( sqr ` G ) ) / 2 ) =/= 0 )
86	52	a1i 11	. . . . 5 |- ( ph -> ( 1 / 3 ) e. CC )
87	50, 85, 86	cxpne0d 22822	. . . 4 |- ( ph -> ( ( ( N + ( sqr ` G ) ) / 2 ) ^c ( 1 / 3 ) ) =/= 0 )
88	7, 87	eqnetrd 2760	. . 3 |- ( ph -> T =/= 0 )
89	1, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88	cubic2 22907	. 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 22901	. . . . 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 2963	. 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 1379   e. wcel 1767   =/= wne 2662   E.wrex 2815   {ctp 4031   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489   _ici 9490   + caddc 9491   x. cmul 9493   - cmin 9801   -ucneg 9802   / cdiv 10202   NNcn 10532   2c2 10581   3c3 10582   4c4 10583   7c7 10586   9c9 10588   NN0cn0 10791   ZZcz 10860  ;cdc 10972   ^cexp 12130   sqrcsqrt 13025   ^c ccxp 22671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-pi 13666  df-dvds 13844  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006  df-log 22672  df-cxp 22673

This theorem is referenced by: (None)