Theorem quart 24588

Description: The quartic equation, writing out all roots using square and cube root functions so that only direct substitutions remain, and we can actually claim to have a "quartic equation". Naturally, this theorem is ridiculously long (see quartfull 31147) if all the substitutions are performed. This is Metamath 100 proof #46. (Contributed by Mario Carneiro, 6-May-2015.)

Hypotheses

Ref	Expression
quart.a	|- ( ph -> A e. CC )
quart.b	|- ( ph -> B e. CC )
quart.c	|- ( ph -> C e. CC )
quart.d	|- ( ph -> D e. CC )
quart.x	|- ( ph -> X e. CC )
quart.e	|- ( ph -> E = -u ( A / 4 ) )
quart.p	|- ( ph -> P = ( B - ( ( 3 / 8 ) x. ( A ^ 2 ) ) ) )
quart.q	|- ( ph -> Q = ( ( C - ( ( A x. B ) / 2 ) ) + ( ( A ^ 3 ) / 8 ) ) )
quart.r	|- ( ph -> R = ( ( D - ( ( C x. A ) / 4 ) ) + ( ( ( ( A ^ 2 ) x. B ) / ; 1 6 ) - ( ( 3 / ;; 2 5 6 ) x. ( A ^ 4 ) ) ) ) )
quart.u	|- ( ph -> U = ( ( P ^ 2 ) + (; 1 2 x. R ) ) )
quart.v	|- ( ph -> V = ( ( -u ( 2 x. ( P ^ 3 ) ) - (; 2 7 x. ( Q ^ 2 ) ) ) + (; 7 2 x. ( P x. R ) ) ) )
quart.w	|- ( ph -> W = ( sqr ` ( ( V ^ 2 ) - ( 4 x. ( U ^ 3 ) ) ) ) )
quart.s	|- ( ph -> S = ( ( sqr ` M ) / 2 ) )
quart.m	|- ( ph -> M = -u ( ( ( ( 2 x. P ) + T ) + ( U / T ) ) / 3 ) )
quart.t	|- ( ph -> T = ( ( ( V + W ) / 2 ) ^c ( 1 / 3 ) ) )
quart.t0	|- ( ph -> T =/= 0 )
quart.m0	|- ( ph -> M =/= 0 )
quart.i	|- ( ph -> I = ( sqr ` ( ( -u ( S ^ 2 ) - ( P / 2 ) ) + ( ( Q / 4 ) / S ) ) ) )
quart.j	|- ( ph -> J = ( sqr ` ( ( -u ( S ^ 2 ) - ( P / 2 ) ) - ( ( Q / 4 ) / S ) ) ) )

Assertion

Ref	Expression
quart	|- ( ph -> ( ( ( ( X ^ 4 ) + ( A x. ( X ^ 3 ) ) ) + ( ( B x. ( X ^ 2 ) ) + ( ( C x. X ) + D ) ) ) = 0 <-> ( ( X = ( ( E - S ) + I ) \/ X = ( ( E - S ) - I ) ) \/ ( X = ( ( E + S ) + J ) \/ X = ( ( E + S ) - J ) ) ) ) )

Proof of Theorem quart

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		quart.a	. . . 4 |- ( ph -> A e. CC )
2		quart.b	. . . 4 |- ( ph -> B e. CC )
3		quart.c	. . . 4 |- ( ph -> C e. CC )
4		quart.d	. . . 4 |- ( ph -> D e. CC )
5		quart.p	. . . 4 |- ( ph -> P = ( B - ( ( 3 / 8 ) x. ( A ^ 2 ) ) ) )
6		quart.q	. . . 4 |- ( ph -> Q = ( ( C - ( ( A x. B ) / 2 ) ) + ( ( A ^ 3 ) / 8 ) ) )
7		quart.r	. . . 4 |- ( ph -> R = ( ( D - ( ( C x. A ) / 4 ) ) + ( ( ( ( A ^ 2 ) x. B ) / ; 1 6 ) - ( ( 3 / ;; 2 5 6 ) x. ( A ^ 4 ) ) ) ) )
8		quart.x	. . . 4 |- ( ph -> X e. CC )
9		quart.e	. . . . . 6 |- ( ph -> E = -u ( A / 4 ) )
10	9	oveq2d 6666	. . . . 5 |- ( ph -> ( X - E ) = ( X - -u ( A / 4 ) ) )
11		4cn 11098	. . . . . . . 8 |- 4 e. CC
12	11	a1i 11	. . . . . . 7 |- ( ph -> 4 e. CC )
13		4ne0 11117	. . . . . . . 8 |- 4 =/= 0
14	13	a1i 11	. . . . . . 7 |- ( ph -> 4 =/= 0 )
15	1, 12, 14	divcld 10801	. . . . . 6 |- ( ph -> ( A / 4 ) e. CC )
16	8, 15	subnegd 10399	. . . . 5 |- ( ph -> ( X - -u ( A / 4 ) ) = ( X + ( A / 4 ) ) )
17	10, 16	eqtrd 2656	. . . 4 |- ( ph -> ( X - E ) = ( X + ( A / 4 ) ) )
18	1, 2, 3, 4, 5, 6, 7, 8, 17	quart1 24583	. . 3 |- ( ph -> ( ( ( X ^ 4 ) + ( A x. ( X ^ 3 ) ) ) + ( ( B x. ( X ^ 2 ) ) + ( ( C x. X ) + D ) ) ) = ( ( ( ( X - E ) ^ 4 ) + ( P x. ( ( X - E ) ^ 2 ) ) ) + ( ( Q x. ( X - E ) ) + R ) ) )
19	18	eqeq1d 2624	. 2 |- ( ph -> ( ( ( ( X ^ 4 ) + ( A x. ( X ^ 3 ) ) ) + ( ( B x. ( X ^ 2 ) ) + ( ( C x. X ) + D ) ) ) = 0 <-> ( ( ( ( X - E ) ^ 4 ) + ( P x. ( ( X - E ) ^ 2 ) ) ) + ( ( Q x. ( X - E ) ) + R ) ) = 0 ) )
20	1, 2, 3, 4, 5, 6, 7	quart1cl 24581	. . . 4 |- ( ph -> ( P e. CC /\ Q e. CC /\ R e. CC ) )
21	20	simp1d 1073	. . 3 |- ( ph -> P e. CC )
22	20	simp2d 1074	. . 3 |- ( ph -> Q e. CC )
23	15	negcld 10379	. . . . 5 |- ( ph -> -u ( A / 4 ) e. CC )
24	9, 23	eqeltrd 2701	. . . 4 |- ( ph -> E e. CC )
25	8, 24	subcld 10392	. . 3 |- ( ph -> ( X - E ) e. CC )
26		quart.u	. . . . 5 |- ( ph -> U = ( ( P ^ 2 ) + (; 1 2 x. R ) ) )
27		quart.v	. . . . 5 |- ( ph -> V = ( ( -u ( 2 x. ( P ^ 3 ) ) - (; 2 7 x. ( Q ^ 2 ) ) ) + (; 7 2 x. ( P x. R ) ) ) )
28		quart.w	. . . . 5 |- ( ph -> W = ( sqr ` ( ( V ^ 2 ) - ( 4 x. ( U ^ 3 ) ) ) ) )
29		quart.s	. . . . 5 |- ( ph -> S = ( ( sqr ` M ) / 2 ) )
30		quart.m	. . . . 5 |- ( ph -> M = -u ( ( ( ( 2 x. P ) + T ) + ( U / T ) ) / 3 ) )
31		quart.t	. . . . 5 |- ( ph -> T = ( ( ( V + W ) / 2 ) ^c ( 1 / 3 ) ) )
32		quart.t0	. . . . 5 |- ( ph -> T =/= 0 )
33	1, 2, 3, 4, 1, 9, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32	quartlem3 24586	. . . 4 |- ( ph -> ( S e. CC /\ M e. CC /\ T e. CC ) )
34	33	simp1d 1073	. . 3 |- ( ph -> S e. CC )
35	29	oveq2d 6666	. . . . . 6 |- ( ph -> ( 2 x. S ) = ( 2 x. ( ( sqr ` M ) / 2 ) ) )
36	33	simp2d 1074	. . . . . . . 8 |- ( ph -> M e. CC )
37	36	sqrtcld 14176	. . . . . . 7 |- ( ph -> ( sqr ` M ) e. CC )
38		2cnd 11093	. . . . . . 7 |- ( ph -> 2 e. CC )
39		2ne0 11113	. . . . . . . 8 |- 2 =/= 0
40	39	a1i 11	. . . . . . 7 |- ( ph -> 2 =/= 0 )
41	37, 38, 40	divcan2d 10803	. . . . . 6 |- ( ph -> ( 2 x. ( ( sqr ` M ) / 2 ) ) = ( sqr ` M ) )
42	35, 41	eqtrd 2656	. . . . 5 |- ( ph -> ( 2 x. S ) = ( sqr ` M ) )
43	42	oveq1d 6665	. . . 4 |- ( ph -> ( ( 2 x. S ) ^ 2 ) = ( ( sqr ` M ) ^ 2 ) )
44	36	sqsqrtd 14178	. . . 4 |- ( ph -> ( ( sqr ` M ) ^ 2 ) = M )
45	43, 44	eqtr2d 2657	. . 3 |- ( ph -> M = ( ( 2 x. S ) ^ 2 ) )
46		quart.m0	. . 3 |- ( ph -> M =/= 0 )
47		quart.i	. . . . 5 |- ( ph -> I = ( sqr ` ( ( -u ( S ^ 2 ) - ( P / 2 ) ) + ( ( Q / 4 ) / S ) ) ) )
48		quart.j	. . . . 5 |- ( ph -> J = ( sqr ` ( ( -u ( S ^ 2 ) - ( P / 2 ) ) - ( ( Q / 4 ) / S ) ) ) )
49	1, 2, 3, 4, 1, 9, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 46, 47, 48	quartlem4 24587	. . . 4 |- ( ph -> ( S =/= 0 /\ I e. CC /\ J e. CC ) )
50	49	simp2d 1074	. . 3 |- ( ph -> I e. CC )
51	47	oveq1d 6665	. . . 4 |- ( ph -> ( I ^ 2 ) = ( ( sqr ` ( ( -u ( S ^ 2 ) - ( P / 2 ) ) + ( ( Q / 4 ) / S ) ) ) ^ 2 ) )
52	34	sqcld 13006	. . . . . . . 8 |- ( ph -> ( S ^ 2 ) e. CC )
53	52	negcld 10379	. . . . . . 7 |- ( ph -> -u ( S ^ 2 ) e. CC )
54	21	halfcld 11277	. . . . . . 7 |- ( ph -> ( P / 2 ) e. CC )
55	53, 54	subcld 10392	. . . . . 6 |- ( ph -> ( -u ( S ^ 2 ) - ( P / 2 ) ) e. CC )
56	22, 12, 14	divcld 10801	. . . . . . 7 |- ( ph -> ( Q / 4 ) e. CC )
57	49	simp1d 1073	. . . . . . 7 |- ( ph -> S =/= 0 )
58	56, 34, 57	divcld 10801	. . . . . 6 |- ( ph -> ( ( Q / 4 ) / S ) e. CC )
59	55, 58	addcld 10059	. . . . 5 |- ( ph -> ( ( -u ( S ^ 2 ) - ( P / 2 ) ) + ( ( Q / 4 ) / S ) ) e. CC )
60	59	sqsqrtd 14178	. . . 4 |- ( ph -> ( ( sqr ` ( ( -u ( S ^ 2 ) - ( P / 2 ) ) + ( ( Q / 4 ) / S ) ) ) ^ 2 ) = ( ( -u ( S ^ 2 ) - ( P / 2 ) ) + ( ( Q / 4 ) / S ) ) )
61	51, 60	eqtrd 2656	. . 3 |- ( ph -> ( I ^ 2 ) = ( ( -u ( S ^ 2 ) - ( P / 2 ) ) + ( ( Q / 4 ) / S ) ) )
62	20	simp3d 1075	. . 3 |- ( ph -> R e. CC )
63		1cnd 10056	. . . . 5 |- ( ph -> 1 e. CC )
64		3z 11410	. . . . . 6 |- 3 e. ZZ
65		1exp 12889	. . . . . 6 |- ( 3 e. ZZ -> ( 1 ^ 3 ) = 1 )
66	64, 65	mp1i 13	. . . . 5 |- ( ph -> ( 1 ^ 3 ) = 1 )
67	33	simp3d 1075	. . . . . . . . . . 11 |- ( ph -> T e. CC )
68	67	mulid2d 10058	. . . . . . . . . 10 |- ( ph -> ( 1 x. T ) = T )
69	68	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( ( 2 x. P ) + ( 1 x. T ) ) = ( ( 2 x. P ) + T ) )
70	68	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( U / ( 1 x. T ) ) = ( U / T ) )
71	69, 70	oveq12d 6668	. . . . . . . 8 |- ( ph -> ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) = ( ( ( 2 x. P ) + T ) + ( U / T ) ) )
72	71	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) / 3 ) = ( ( ( ( 2 x. P ) + T ) + ( U / T ) ) / 3 ) )
73	72	negeqd 10275	. . . . . 6 |- ( ph -> -u ( ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) / 3 ) = -u ( ( ( ( 2 x. P ) + T ) + ( U / T ) ) / 3 ) )
74	30, 73	eqtr4d 2659	. . . . 5 |- ( ph -> M = -u ( ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) / 3 ) )
75		oveq1 6657	. . . . . . . 8 |- ( x = 1 -> ( x ^ 3 ) = ( 1 ^ 3 ) )
76	75	eqeq1d 2624	. . . . . . 7 |- ( x = 1 -> ( ( x ^ 3 ) = 1 <-> ( 1 ^ 3 ) = 1 ) )
77		oveq1 6657	. . . . . . . . . . . 12 |- ( x = 1 -> ( x x. T ) = ( 1 x. T ) )
78	77	oveq2d 6666	. . . . . . . . . . 11 |- ( x = 1 -> ( ( 2 x. P ) + ( x x. T ) ) = ( ( 2 x. P ) + ( 1 x. T ) ) )
79	77	oveq2d 6666	. . . . . . . . . . 11 |- ( x = 1 -> ( U / ( x x. T ) ) = ( U / ( 1 x. T ) ) )
80	78, 79	oveq12d 6668	. . . . . . . . . 10 |- ( x = 1 -> ( ( ( 2 x. P ) + ( x x. T ) ) + ( U / ( x x. T ) ) ) = ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) )
81	80	oveq1d 6665	. . . . . . . . 9 |- ( x = 1 -> ( ( ( ( 2 x. P ) + ( x x. T ) ) + ( U / ( x x. T ) ) ) / 3 ) = ( ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) / 3 ) )
82	81	negeqd 10275	. . . . . . . 8 |- ( x = 1 -> -u ( ( ( ( 2 x. P ) + ( x x. T ) ) + ( U / ( x x. T ) ) ) / 3 ) = -u ( ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) / 3 ) )
83	82	eqeq2d 2632	. . . . . . 7 |- ( x = 1 -> ( M = -u ( ( ( ( 2 x. P ) + ( x x. T ) ) + ( U / ( x x. T ) ) ) / 3 ) <-> M = -u ( ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) / 3 ) ) )
84	76, 83	anbi12d 747	. . . . . 6 |- ( x = 1 -> ( ( ( x ^ 3 ) = 1 /\ M = -u ( ( ( ( 2 x. P ) + ( x x. T ) ) + ( U / ( x x. T ) ) ) / 3 ) ) <-> ( ( 1 ^ 3 ) = 1 /\ M = -u ( ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) / 3 ) ) ) )
85	84	rspcev 3309	. . . . 5 |- ( ( 1 e. CC /\ ( ( 1 ^ 3 ) = 1 /\ M = -u ( ( ( ( 2 x. P ) + ( 1 x. T ) ) + ( U / ( 1 x. T ) ) ) / 3 ) ) ) -> E. x e. CC ( ( x ^ 3 ) = 1 /\ M = -u ( ( ( ( 2 x. P ) + ( x x. T ) ) + ( U / ( x x. T ) ) ) / 3 ) ) )
86	63, 66, 74, 85	syl12anc 1324	. . . 4 |- ( ph -> E. x e. CC ( ( x ^ 3 ) = 1 /\ M = -u ( ( ( ( 2 x. P ) + ( x x. T ) ) + ( U / ( x x. T ) ) ) / 3 ) ) )
87		2cn 11091	. . . . . 6 |- 2 e. CC
88		mulcl 10020	. . . . . 6 |- ( ( 2 e. CC /\ P e. CC ) -> ( 2 x. P ) e. CC )
89	87, 21, 88	sylancr 695	. . . . 5 |- ( ph -> ( 2 x. P ) e. CC )
90	21	sqcld 13006	. . . . . 6 |- ( ph -> ( P ^ 2 ) e. CC )
91		mulcl 10020	. . . . . . 7 |- ( ( 4 e. CC /\ R e. CC ) -> ( 4 x. R ) e. CC )
92	11, 62, 91	sylancr 695	. . . . . 6 |- ( ph -> ( 4 x. R ) e. CC )
93	90, 92	subcld 10392	. . . . 5 |- ( ph -> ( ( P ^ 2 ) - ( 4 x. R ) ) e. CC )
94	22	sqcld 13006	. . . . . 6 |- ( ph -> ( Q ^ 2 ) e. CC )
95	94	negcld 10379	. . . . 5 |- ( ph -> -u ( Q ^ 2 ) e. CC )
96	31	oveq1d 6665	. . . . . 6 |- ( ph -> ( T ^ 3 ) = ( ( ( ( V + W ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) )
97	1, 2, 3, 4, 1, 9, 5, 6, 7, 26, 27, 28	quartlem2 24585	. . . . . . . . . 10 |- ( ph -> ( U e. CC /\ V e. CC /\ W e. CC ) )
98	97	simp2d 1074	. . . . . . . . 9 |- ( ph -> V e. CC )
99	97	simp3d 1075	. . . . . . . . 9 |- ( ph -> W e. CC )
100	98, 99	addcld 10059	. . . . . . . 8 |- ( ph -> ( V + W ) e. CC )
101	100	halfcld 11277	. . . . . . 7 |- ( ph -> ( ( V + W ) / 2 ) e. CC )
102		3nn 11186	. . . . . . 7 |- 3 e. NN
103		cxproot 24436	. . . . . . 7 |- ( ( ( ( V + W ) / 2 ) e. CC /\ 3 e. NN ) -> ( ( ( ( V + W ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) = ( ( V + W ) / 2 ) )
104	101, 102, 103	sylancl 694	. . . . . 6 |- ( ph -> ( ( ( ( V + W ) / 2 ) ^c ( 1 / 3 ) ) ^ 3 ) = ( ( V + W ) / 2 ) )
105	96, 104	eqtrd 2656	. . . . 5 |- ( ph -> ( T ^ 3 ) = ( ( V + W ) / 2 ) )
106	28	oveq1d 6665	. . . . . 6 |- ( ph -> ( W ^ 2 ) = ( ( sqr ` ( ( V ^ 2 ) - ( 4 x. ( U ^ 3 ) ) ) ) ^ 2 ) )
107	98	sqcld 13006	. . . . . . . 8 |- ( ph -> ( V ^ 2 ) e. CC )
108	97	simp1d 1073	. . . . . . . . . 10 |- ( ph -> U e. CC )
109		3nn0 11310	. . . . . . . . . 10 |- 3 e. NN0
110		expcl 12878	. . . . . . . . . 10 |- ( ( U e. CC /\ 3 e. NN0 ) -> ( U ^ 3 ) e. CC )
111	108, 109, 110	sylancl 694	. . . . . . . . 9 |- ( ph -> ( U ^ 3 ) e. CC )
112		mulcl 10020	. . . . . . . . 9 |- ( ( 4 e. CC /\ ( U ^ 3 ) e. CC ) -> ( 4 x. ( U ^ 3 ) ) e. CC )
113	11, 111, 112	sylancr 695	. . . . . . . 8 |- ( ph -> ( 4 x. ( U ^ 3 ) ) e. CC )
114	107, 113	subcld 10392	. . . . . . 7 |- ( ph -> ( ( V ^ 2 ) - ( 4 x. ( U ^ 3 ) ) ) e. CC )
115	114	sqsqrtd 14178	. . . . . 6 |- ( ph -> ( ( sqr ` ( ( V ^ 2 ) - ( 4 x. ( U ^ 3 ) ) ) ) ^ 2 ) = ( ( V ^ 2 ) - ( 4 x. ( U ^ 3 ) ) ) )
116	106, 115	eqtrd 2656	. . . . 5 |- ( ph -> ( W ^ 2 ) = ( ( V ^ 2 ) - ( 4 x. ( U ^ 3 ) ) ) )
117	21, 22, 62, 26, 27	quartlem1 24584	. . . . . 6 |- ( ph -> ( U = ( ( ( 2 x. P ) ^ 2 ) - ( 3 x. ( ( P ^ 2 ) - ( 4 x. R ) ) ) ) /\ V = ( ( ( 2 x. ( ( 2 x. P ) ^ 3 ) ) - ( 9 x. ( ( 2 x. P ) x. ( ( P ^ 2 ) - ( 4 x. R ) ) ) ) ) + (; 2 7 x. -u ( Q ^ 2 ) ) ) ) )
118	117	simpld 475	. . . . 5 |- ( ph -> U = ( ( ( 2 x. P ) ^ 2 ) - ( 3 x. ( ( P ^ 2 ) - ( 4 x. R ) ) ) ) )
119	117	simprd 479	. . . . 5 |- ( ph -> V = ( ( ( 2 x. ( ( 2 x. P ) ^ 3 ) ) - ( 9 x. ( ( 2 x. P ) x. ( ( P ^ 2 ) - ( 4 x. R ) ) ) ) ) + (; 2 7 x. -u ( Q ^ 2 ) ) ) )
120	89, 93, 95, 36, 67, 105, 99, 116, 118, 119, 32	mcubic 24574	. . . 4 |- ( ph -> ( ( ( ( M ^ 3 ) + ( ( 2 x. P ) x. ( M ^ 2 ) ) ) + ( ( ( ( P ^ 2 ) - ( 4 x. R ) ) x. M ) + -u ( Q ^ 2 ) ) ) = 0 <-> E. x e. CC ( ( x ^ 3 ) = 1 /\ M = -u ( ( ( ( 2 x. P ) + ( x x. T ) ) + ( U / ( x x. T ) ) ) / 3 ) ) ) )
121	86, 120	mpbird 247	. . 3 |- ( ph -> ( ( ( M ^ 3 ) + ( ( 2 x. P ) x. ( M ^ 2 ) ) ) + ( ( ( ( P ^ 2 ) - ( 4 x. R ) ) x. M ) + -u ( Q ^ 2 ) ) ) = 0 )
122	49	simp3d 1075	. . 3 |- ( ph -> J e. CC )
123	48	oveq1d 6665	. . . 4 |- ( ph -> ( J ^ 2 ) = ( ( sqr ` ( ( -u ( S ^ 2 ) - ( P / 2 ) ) - ( ( Q / 4 ) / S ) ) ) ^ 2 ) )
124	55, 58	subcld 10392	. . . . 5 |- ( ph -> ( ( -u ( S ^ 2 ) - ( P / 2 ) ) - ( ( Q / 4 ) / S ) ) e. CC )
125	124	sqsqrtd 14178	. . . 4 |- ( ph -> ( ( sqr ` ( ( -u ( S ^ 2 ) - ( P / 2 ) ) - ( ( Q / 4 ) / S ) ) ) ^ 2 ) = ( ( -u ( S ^ 2 ) - ( P / 2 ) ) - ( ( Q / 4 ) / S ) ) )
126	123, 125	eqtrd 2656	. . 3 |- ( ph -> ( J ^ 2 ) = ( ( -u ( S ^ 2 ) - ( P / 2 ) ) - ( ( Q / 4 ) / S ) ) )
127	21, 22, 25, 34, 45, 46, 50, 61, 62, 121, 122, 126	dquart 24580	. 2 |- ( ph -> ( ( ( ( ( X - E ) ^ 4 ) + ( P x. ( ( X - E ) ^ 2 ) ) ) + ( ( Q x. ( X - E ) ) + R ) ) = 0 <-> ( ( ( X - E ) = ( -u S + I ) \/ ( X - E ) = ( -u S - I ) ) \/ ( ( X - E ) = ( S + J ) \/ ( X - E ) = ( S - J ) ) ) ) )
128	34	negcld 10379	. . . . . . . 8 |- ( ph -> -u S e. CC )
129	128, 50	addcld 10059	. . . . . . 7 |- ( ph -> ( -u S + I ) e. CC )
130	8, 24, 129	subaddd 10410	. . . . . 6 |- ( ph -> ( ( X - E ) = ( -u S + I ) <-> ( E + ( -u S + I ) ) = X ) )
131	24, 34	negsubd 10398	. . . . . . . . 9 |- ( ph -> ( E + -u S ) = ( E - S ) )
132	131	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( E + -u S ) + I ) = ( ( E - S ) + I ) )
133	24, 128, 50	addassd 10062	. . . . . . . 8 |- ( ph -> ( ( E + -u S ) + I ) = ( E + ( -u S + I ) ) )
134	132, 133	eqtr3d 2658	. . . . . . 7 |- ( ph -> ( ( E - S ) + I ) = ( E + ( -u S + I ) ) )
135	134	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( ( E - S ) + I ) = X <-> ( E + ( -u S + I ) ) = X ) )
136	130, 135	bitr4d 271	. . . . 5 |- ( ph -> ( ( X - E ) = ( -u S + I ) <-> ( ( E - S ) + I ) = X ) )
137		eqcom 2629	. . . . 5 |- ( ( ( E - S ) + I ) = X <-> X = ( ( E - S ) + I ) )
138	136, 137	syl6bb 276	. . . 4 |- ( ph -> ( ( X - E ) = ( -u S + I ) <-> X = ( ( E - S ) + I ) ) )
139	128, 50	subcld 10392	. . . . . . 7 |- ( ph -> ( -u S - I ) e. CC )
140	8, 24, 139	subaddd 10410	. . . . . 6 |- ( ph -> ( ( X - E ) = ( -u S - I ) <-> ( E + ( -u S - I ) ) = X ) )
141	131	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( E + -u S ) - I ) = ( ( E - S ) - I ) )
142	24, 128, 50	addsubassd 10412	. . . . . . . 8 |- ( ph -> ( ( E + -u S ) - I ) = ( E + ( -u S - I ) ) )
143	141, 142	eqtr3d 2658	. . . . . . 7 |- ( ph -> ( ( E - S ) - I ) = ( E + ( -u S - I ) ) )
144	143	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( ( E - S ) - I ) = X <-> ( E + ( -u S - I ) ) = X ) )
145	140, 144	bitr4d 271	. . . . 5 |- ( ph -> ( ( X - E ) = ( -u S - I ) <-> ( ( E - S ) - I ) = X ) )
146		eqcom 2629	. . . . 5 |- ( ( ( E - S ) - I ) = X <-> X = ( ( E - S ) - I ) )
147	145, 146	syl6bb 276	. . . 4 |- ( ph -> ( ( X - E ) = ( -u S - I ) <-> X = ( ( E - S ) - I ) ) )
148	138, 147	orbi12d 746	. . 3 |- ( ph -> ( ( ( X - E ) = ( -u S + I ) \/ ( X - E ) = ( -u S - I ) ) <-> ( X = ( ( E - S ) + I ) \/ X = ( ( E - S ) - I ) ) ) )
149	34, 122	addcld 10059	. . . . . . 7 |- ( ph -> ( S + J ) e. CC )
150	8, 24, 149	subaddd 10410	. . . . . 6 |- ( ph -> ( ( X - E ) = ( S + J ) <-> ( E + ( S + J ) ) = X ) )
151	24, 34, 122	addassd 10062	. . . . . . 7 |- ( ph -> ( ( E + S ) + J ) = ( E + ( S + J ) ) )
152	151	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( ( E + S ) + J ) = X <-> ( E + ( S + J ) ) = X ) )
153	150, 152	bitr4d 271	. . . . 5 |- ( ph -> ( ( X - E ) = ( S + J ) <-> ( ( E + S ) + J ) = X ) )
154		eqcom 2629	. . . . 5 |- ( ( ( E + S ) + J ) = X <-> X = ( ( E + S ) + J ) )
155	153, 154	syl6bb 276	. . . 4 |- ( ph -> ( ( X - E ) = ( S + J ) <-> X = ( ( E + S ) + J ) ) )
156	34, 122	subcld 10392	. . . . . . 7 |- ( ph -> ( S - J ) e. CC )
157	8, 24, 156	subaddd 10410	. . . . . 6 |- ( ph -> ( ( X - E ) = ( S - J ) <-> ( E + ( S - J ) ) = X ) )
158	24, 34, 122	addsubassd 10412	. . . . . . 7 |- ( ph -> ( ( E + S ) - J ) = ( E + ( S - J ) ) )
159	158	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( ( E + S ) - J ) = X <-> ( E + ( S - J ) ) = X ) )
160	157, 159	bitr4d 271	. . . . 5 |- ( ph -> ( ( X - E ) = ( S - J ) <-> ( ( E + S ) - J ) = X ) )
161		eqcom 2629	. . . . 5 |- ( ( ( E + S ) - J ) = X <-> X = ( ( E + S ) - J ) )
162	160, 161	syl6bb 276	. . . 4 |- ( ph -> ( ( X - E ) = ( S - J ) <-> X = ( ( E + S ) - J ) ) )
163	155, 162	orbi12d 746	. . 3 |- ( ph -> ( ( ( X - E ) = ( S + J ) \/ ( X - E ) = ( S - J ) ) <-> ( X = ( ( E + S ) + J ) \/ X = ( ( E + S ) - J ) ) ) )
164	148, 163	orbi12d 746	. 2 |- ( ph -> ( ( ( ( X - E ) = ( -u S + I ) \/ ( X - E ) = ( -u S - I ) ) \/ ( ( X - E ) = ( S + J ) \/ ( X - E ) = ( S - J ) ) ) <-> ( ( X = ( ( E - S ) + I ) \/ X = ( ( E - S ) - I ) ) \/ ( X = ( ( E + S ) + J ) \/ X = ( ( E + S ) - J ) ) ) ) )
165	19, 127, 164	3bitrd 294	1 |- ( ph -> ( ( ( ( X ^ 4 ) + ( A x. ( X ^ 3 ) ) ) + ( ( B x. ( X ^ 2 ) ) + ( ( C x. X ) + D ) ) ) = 0 <-> ( ( X = ( ( E - S ) + I ) \/ X = ( ( E - S ) - I ) ) \/ ( X = ( ( E + S ) + J ) \/ X = ( ( E + S ) - J ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   5c5 11073   6c6 11074   7c7 11075   8c8 11076   9c9 11077   NN0cn0 11292   ZZcz 11377  ;cdc 11493   ^cexp 12860   sqrcsqrt 13973   ^c ccxp 24302

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-int 4476  df-iun 4522  df-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  quartfull  31147