Theorem dcubic1 24572

Description: Forward direction of dcubic 24573: the claimed formula produces solutions to the cubic equation. (Contributed by Mario Carneiro, 25-Apr-2015.)

Hypotheses

Ref	Expression
dcubic.c	|- ( ph -> P e. CC )
dcubic.d	|- ( ph -> Q e. CC )
dcubic.x	|- ( ph -> X e. CC )
dcubic.t	|- ( ph -> T e. CC )
dcubic.3	|- ( ph -> ( T ^ 3 ) = ( G - N ) )
dcubic.g	|- ( ph -> G e. CC )
dcubic.2	|- ( ph -> ( G ^ 2 ) = ( ( N ^ 2 ) + ( M ^ 3 ) ) )
dcubic.m	|- ( ph -> M = ( P / 3 ) )
dcubic.n	|- ( ph -> N = ( Q / 2 ) )
dcubic.0	|- ( ph -> T =/= 0 )
dcubic1.x	|- ( ph -> X = ( T - ( M / T ) ) )

Assertion

Ref	Expression
dcubic1	|- ( ph -> ( ( X ^ 3 ) + ( ( P x. X ) + Q ) ) = 0 )

Proof of Theorem dcubic1

Step	Hyp	Ref	Expression
1		dcubic.3	. . . . . . 7 |- ( ph -> ( T ^ 3 ) = ( G - N ) )
2	1	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( T ^ 3 ) ^ 2 ) = ( ( G - N ) ^ 2 ) )
3		dcubic.g	. . . . . . 7 |- ( ph -> G e. CC )
4		dcubic.n	. . . . . . . 8 |- ( ph -> N = ( Q / 2 ) )
5		dcubic.d	. . . . . . . . 9 |- ( ph -> Q e. CC )
6	5	halfcld 11277	. . . . . . . 8 |- ( ph -> ( Q / 2 ) e. CC )
7	4, 6	eqeltrd 2701	. . . . . . 7 |- ( ph -> N e. CC )
8		binom2sub 12981	. . . . . . 7 |- ( ( G e. CC /\ N e. CC ) -> ( ( G - N ) ^ 2 ) = ( ( ( G ^ 2 ) - ( 2 x. ( G x. N ) ) ) + ( N ^ 2 ) ) )
9	3, 7, 8	syl2anc 693	. . . . . 6 |- ( ph -> ( ( G - N ) ^ 2 ) = ( ( ( G ^ 2 ) - ( 2 x. ( G x. N ) ) ) + ( N ^ 2 ) ) )
10		dcubic.2	. . . . . . . 8 |- ( ph -> ( G ^ 2 ) = ( ( N ^ 2 ) + ( M ^ 3 ) ) )
11		2cnd 11093	. . . . . . . . . 10 |- ( ph -> 2 e. CC )
12	11, 3, 7	mul12d 10245	. . . . . . . . 9 |- ( ph -> ( 2 x. ( G x. N ) ) = ( G x. ( 2 x. N ) ) )
13	4	oveq2d 6666	. . . . . . . . . . 11 |- ( ph -> ( 2 x. N ) = ( 2 x. ( Q / 2 ) ) )
14		2ne0 11113	. . . . . . . . . . . . 13 |- 2 =/= 0
15	14	a1i 11	. . . . . . . . . . . 12 |- ( ph -> 2 =/= 0 )
16	5, 11, 15	divcan2d 10803	. . . . . . . . . . 11 |- ( ph -> ( 2 x. ( Q / 2 ) ) = Q )
17	13, 16	eqtrd 2656	. . . . . . . . . 10 |- ( ph -> ( 2 x. N ) = Q )
18	17	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( G x. ( 2 x. N ) ) = ( G x. Q ) )
19	3, 5	mulcomd 10061	. . . . . . . . 9 |- ( ph -> ( G x. Q ) = ( Q x. G ) )
20	12, 18, 19	3eqtrd 2660	. . . . . . . 8 |- ( ph -> ( 2 x. ( G x. N ) ) = ( Q x. G ) )
21	10, 20	oveq12d 6668	. . . . . . 7 |- ( ph -> ( ( G ^ 2 ) - ( 2 x. ( G x. N ) ) ) = ( ( ( N ^ 2 ) + ( M ^ 3 ) ) - ( Q x. G ) ) )
22	21	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( G ^ 2 ) - ( 2 x. ( G x. N ) ) ) + ( N ^ 2 ) ) = ( ( ( ( N ^ 2 ) + ( M ^ 3 ) ) - ( Q x. G ) ) + ( N ^ 2 ) ) )
23	2, 9, 22	3eqtrd 2660	. . . . 5 |- ( ph -> ( ( T ^ 3 ) ^ 2 ) = ( ( ( ( N ^ 2 ) + ( M ^ 3 ) ) - ( Q x. G ) ) + ( N ^ 2 ) ) )
24	7	sqcld 13006	. . . . . . 7 |- ( ph -> ( N ^ 2 ) e. CC )
25		dcubic.m	. . . . . . . . 9 |- ( ph -> M = ( P / 3 ) )
26		dcubic.c	. . . . . . . . . 10 |- ( ph -> P e. CC )
27		3cn 11095	. . . . . . . . . . 11 |- 3 e. CC
28	27	a1i 11	. . . . . . . . . 10 |- ( ph -> 3 e. CC )
29		3ne0 11115	. . . . . . . . . . 11 |- 3 =/= 0
30	29	a1i 11	. . . . . . . . . 10 |- ( ph -> 3 =/= 0 )
31	26, 28, 30	divcld 10801	. . . . . . . . 9 |- ( ph -> ( P / 3 ) e. CC )
32	25, 31	eqeltrd 2701	. . . . . . . 8 |- ( ph -> M e. CC )
33		3nn0 11310	. . . . . . . 8 |- 3 e. NN0
34		expcl 12878	. . . . . . . 8 |- ( ( M e. CC /\ 3 e. NN0 ) -> ( M ^ 3 ) e. CC )
35	32, 33, 34	sylancl 694	. . . . . . 7 |- ( ph -> ( M ^ 3 ) e. CC )
36	24, 35	addcld 10059	. . . . . 6 |- ( ph -> ( ( N ^ 2 ) + ( M ^ 3 ) ) e. CC )
37	5, 3	mulcld 10060	. . . . . 6 |- ( ph -> ( Q x. G ) e. CC )
38	36, 24, 37	addsubd 10413	. . . . 5 |- ( ph -> ( ( ( ( N ^ 2 ) + ( M ^ 3 ) ) + ( N ^ 2 ) ) - ( Q x. G ) ) = ( ( ( ( N ^ 2 ) + ( M ^ 3 ) ) - ( Q x. G ) ) + ( N ^ 2 ) ) )
39	24, 35, 24	add32d 10263	. . . . . . 7 |- ( ph -> ( ( ( N ^ 2 ) + ( M ^ 3 ) ) + ( N ^ 2 ) ) = ( ( ( N ^ 2 ) + ( N ^ 2 ) ) + ( M ^ 3 ) ) )
40	24	2timesd 11275	. . . . . . . 8 |- ( ph -> ( 2 x. ( N ^ 2 ) ) = ( ( N ^ 2 ) + ( N ^ 2 ) ) )
41	40	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) = ( ( ( N ^ 2 ) + ( N ^ 2 ) ) + ( M ^ 3 ) ) )
42	39, 41	eqtr4d 2659	. . . . . 6 |- ( ph -> ( ( ( N ^ 2 ) + ( M ^ 3 ) ) + ( N ^ 2 ) ) = ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) )
43	42	oveq1d 6665	. . . . 5 |- ( ph -> ( ( ( ( N ^ 2 ) + ( M ^ 3 ) ) + ( N ^ 2 ) ) - ( Q x. G ) ) = ( ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) - ( Q x. G ) ) )
44	23, 38, 43	3eqtr2d 2662	. . . 4 |- ( ph -> ( ( T ^ 3 ) ^ 2 ) = ( ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) - ( Q x. G ) ) )
45	5, 3, 7	subdid 10486	. . . . . . 7 |- ( ph -> ( Q x. ( G - N ) ) = ( ( Q x. G ) - ( Q x. N ) ) )
46	1	oveq2d 6666	. . . . . . 7 |- ( ph -> ( Q x. ( T ^ 3 ) ) = ( Q x. ( G - N ) ) )
47	7	sqvald 13005	. . . . . . . . . 10 |- ( ph -> ( N ^ 2 ) = ( N x. N ) )
48	47	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( 2 x. ( N ^ 2 ) ) = ( 2 x. ( N x. N ) ) )
49	11, 7, 7	mulassd 10063	. . . . . . . . 9 |- ( ph -> ( ( 2 x. N ) x. N ) = ( 2 x. ( N x. N ) ) )
50	17	oveq1d 6665	. . . . . . . . 9 |- ( ph -> ( ( 2 x. N ) x. N ) = ( Q x. N ) )
51	48, 49, 50	3eqtr2d 2662	. . . . . . . 8 |- ( ph -> ( 2 x. ( N ^ 2 ) ) = ( Q x. N ) )
52	51	oveq2d 6666	. . . . . . 7 |- ( ph -> ( ( Q x. G ) - ( 2 x. ( N ^ 2 ) ) ) = ( ( Q x. G ) - ( Q x. N ) ) )
53	45, 46, 52	3eqtr4d 2666	. . . . . 6 |- ( ph -> ( Q x. ( T ^ 3 ) ) = ( ( Q x. G ) - ( 2 x. ( N ^ 2 ) ) ) )
54	53	oveq1d 6665	. . . . 5 |- ( ph -> ( ( Q x. ( T ^ 3 ) ) - ( M ^ 3 ) ) = ( ( ( Q x. G ) - ( 2 x. ( N ^ 2 ) ) ) - ( M ^ 3 ) ) )
55		2cn 11091	. . . . . . 7 |- 2 e. CC
56		mulcl 10020	. . . . . . 7 |- ( ( 2 e. CC /\ ( N ^ 2 ) e. CC ) -> ( 2 x. ( N ^ 2 ) ) e. CC )
57	55, 24, 56	sylancr 695	. . . . . 6 |- ( ph -> ( 2 x. ( N ^ 2 ) ) e. CC )
58	37, 57, 35	subsub4d 10423	. . . . 5 |- ( ph -> ( ( ( Q x. G ) - ( 2 x. ( N ^ 2 ) ) ) - ( M ^ 3 ) ) = ( ( Q x. G ) - ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) ) )
59	54, 58	eqtrd 2656	. . . 4 |- ( ph -> ( ( Q x. ( T ^ 3 ) ) - ( M ^ 3 ) ) = ( ( Q x. G ) - ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) ) )
60	44, 59	oveq12d 6668	. . 3 |- ( ph -> ( ( ( T ^ 3 ) ^ 2 ) + ( ( Q x. ( T ^ 3 ) ) - ( M ^ 3 ) ) ) = ( ( ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) - ( Q x. G ) ) + ( ( Q x. G ) - ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) ) ) )
61	57, 35	addcld 10059	. . . 4 |- ( ph -> ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) e. CC )
62		npncan2 10308	. . . 4 |- ( ( ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) e. CC /\ ( Q x. G ) e. CC ) -> ( ( ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) - ( Q x. G ) ) + ( ( Q x. G ) - ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) ) ) = 0 )
63	61, 37, 62	syl2anc 693	. . 3 |- ( ph -> ( ( ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) - ( Q x. G ) ) + ( ( Q x. G ) - ( ( 2 x. ( N ^ 2 ) ) + ( M ^ 3 ) ) ) ) = 0 )
64	60, 63	eqtrd 2656	. 2 |- ( ph -> ( ( ( T ^ 3 ) ^ 2 ) + ( ( Q x. ( T ^ 3 ) ) - ( M ^ 3 ) ) ) = 0 )
65		dcubic.x	. . 3 |- ( ph -> X e. CC )
66		dcubic.t	. . 3 |- ( ph -> T e. CC )
67		dcubic.0	. . 3 |- ( ph -> T =/= 0 )
68		dcubic1.x	. . 3 |- ( ph -> X = ( T - ( M / T ) ) )
69	26, 5, 65, 66, 1, 3, 10, 25, 4, 67, 66, 67, 68	dcubic1lem 24570	. 2 |- ( ph -> ( ( ( X ^ 3 ) + ( ( P x. X ) + Q ) ) = 0 <-> ( ( ( T ^ 3 ) ^ 2 ) + ( ( Q x. ( T ^ 3 ) ) - ( M ^ 3 ) ) ) = 0 ) )
70	64, 69	mpbird 247	1 |- ( ph -> ( ( X ^ 3 ) + ( ( P x. X ) + Q ) ) = 0 )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   2c2 11070   3c3 11071   NN0cn0 11292   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984

This theorem is referenced by:  dcubic  24573