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Theorem quart 25916

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 33027) if all the substitutions are performed. This is Metamath 100 proof #46. (Contributed by Mario Carneiro, 6-May-2015.)

**Hypotheses**
Ref	Expression
quart.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
quart.b	⊢ (𝜑 → 𝐵 ∈ ℂ)
quart.c	⊢ (𝜑 → 𝐶 ∈ ℂ)
quart.d	⊢ (𝜑 → 𝐷 ∈ ℂ)
quart.x	⊢ (𝜑 → 𝑋 ∈ ℂ)
quart.e	⊢ (𝜑 → 𝐸 = -(𝐴 / 4))
quart.p	⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2))))
quart.q	⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)))
quart.r	⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))
quart.u	⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅)))
quart.v	⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
quart.w	⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))
quart.s	⊢ (𝜑 → 𝑆 = ((√‘𝑀) / 2))
quart.m	⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
quart.t	⊢ (𝜑 → 𝑇 = (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)))
quart.t0	⊢ (𝜑 → 𝑇 ≠ 0)
quart.m0	⊢ (𝜑 → 𝑀 ≠ 0)
quart.i	⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))))
quart.j	⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))))

**Assertion**
Ref	Expression
quart	⊢ (𝜑 → ((((𝑋↑4) + (𝐴 · (𝑋↑3))) + ((𝐵 · (𝑋↑2)) + ((𝐶 · 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 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		quart.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		quart.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		quart.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ)
5		quart.p	. . . 4 ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2))))
6		quart.q	. . . 4 ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)))
7		quart.r	. . . 4 ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))
8		quart.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9		quart.e	. . . . . 6 ⊢ (𝜑 → 𝐸 = -(𝐴 / 4))
10	9	oveq2d 7271	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝐸) = (𝑋 − -(𝐴 / 4)))
11		4cn 11988	. . . . . . . 8 ⊢ 4 ∈ ℂ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ)
13		4ne0 12011	. . . . . . . 8 ⊢ 4 ≠ 0
14	13	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ≠ 0)
15	1, 12, 14	divcld 11681	. . . . . 6 ⊢ (𝜑 → (𝐴 / 4) ∈ ℂ)
16	8, 15	subnegd 11269	. . . . 5 ⊢ (𝜑 → (𝑋 − -(𝐴 / 4)) = (𝑋 + (𝐴 / 4)))
17	10, 16	eqtrd 2778	. . . 4 ⊢ (𝜑 → (𝑋 − 𝐸) = (𝑋 + (𝐴 / 4)))
18	1, 2, 3, 4, 5, 6, 7, 8, 17	quart1 25911	. . 3 ⊢ (𝜑 → (((𝑋↑4) + (𝐴 · (𝑋↑3))) + ((𝐵 · (𝑋↑2)) + ((𝐶 · 𝑋) + 𝐷))) = ((((𝑋 − 𝐸)↑4) + (𝑃 · ((𝑋 − 𝐸)↑2))) + ((𝑄 · (𝑋 − 𝐸)) + 𝑅)))
19	18	eqeq1d 2740	. 2 ⊢ (𝜑 → ((((𝑋↑4) + (𝐴 · (𝑋↑3))) + ((𝐵 · (𝑋↑2)) + ((𝐶 · 𝑋) + 𝐷))) = 0 ↔ ((((𝑋 − 𝐸)↑4) + (𝑃 · ((𝑋 − 𝐸)↑2))) + ((𝑄 · (𝑋 − 𝐸)) + 𝑅)) = 0))
20	1, 2, 3, 4, 5, 6, 7	quart1cl 25909	. . . 4 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
21	20	simp1d 1140	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℂ)
22	20	simp2d 1141	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℂ)
23	15	negcld 11249	. . . . 5 ⊢ (𝜑 → -(𝐴 / 4) ∈ ℂ)
24	9, 23	eqeltrd 2839	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ)
25	8, 24	subcld 11262	. . 3 ⊢ (𝜑 → (𝑋 − 𝐸) ∈ ℂ)
26		quart.u	. . . . 5 ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅)))
27		quart.v	. . . . 5 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
28		quart.w	. . . . 5 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))
29		quart.s	. . . . 5 ⊢ (𝜑 → 𝑆 = ((√‘𝑀) / 2))
30		quart.m	. . . . 5 ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
31		quart.t	. . . . 5 ⊢ (𝜑 → 𝑇 = (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)))
32		quart.t0	. . . . 5 ⊢ (𝜑 → 𝑇 ≠ 0)
33	1, 2, 3, 4, 1, 9, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32	quartlem3 25914	. . . 4 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))
34	33	simp1d 1140	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℂ)
35	29	oveq2d 7271	. . . . . 6 ⊢ (𝜑 → (2 · 𝑆) = (2 · ((√‘𝑀) / 2)))
36	33	simp2d 1141	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
37	36	sqrtcld 15077	. . . . . . 7 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
38		2cnd 11981	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
39		2ne0 12007	. . . . . . . 8 ⊢ 2 ≠ 0
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ≠ 0)
41	37, 38, 40	divcan2d 11683	. . . . . 6 ⊢ (𝜑 → (2 · ((√‘𝑀) / 2)) = (√‘𝑀))
42	35, 41	eqtrd 2778	. . . . 5 ⊢ (𝜑 → (2 · 𝑆) = (√‘𝑀))
43	42	oveq1d 7270	. . . 4 ⊢ (𝜑 → ((2 · 𝑆)↑2) = ((√‘𝑀)↑2))
44	36	sqsqrtd 15079	. . . 4 ⊢ (𝜑 → ((√‘𝑀)↑2) = 𝑀)
45	43, 44	eqtr2d 2779	. . 3 ⊢ (𝜑 → 𝑀 = ((2 · 𝑆)↑2))
46		quart.m0	. . 3 ⊢ (𝜑 → 𝑀 ≠ 0)
47		quart.i	. . . . 5 ⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))))
48		quart.j	. . . . 5 ⊢ (𝜑 → 𝐽 = (√‘((-(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 25915	. . . 4 ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ))
50	49	simp2d 1141	. . 3 ⊢ (𝜑 → 𝐼 ∈ ℂ)
51	47	oveq1d 7270	. . . 4 ⊢ (𝜑 → (𝐼↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))↑2))
52	34	sqcld 13790	. . . . . . . 8 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ)
53	52	negcld 11249	. . . . . . 7 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ)
54	21	halfcld 12148	. . . . . . 7 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ)
55	53, 54	subcld 11262	. . . . . 6 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ)
56	22, 12, 14	divcld 11681	. . . . . . 7 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ)
57	49	simp1d 1140	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≠ 0)
58	56, 34, 57	divcld 11681	. . . . . 6 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ)
59	55, 58	addcld 10925	. . . . 5 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ)
60	59	sqsqrtd 15079	. . . 4 ⊢ (𝜑 → ((√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))↑2) = ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))
61	51, 60	eqtrd 2778	. . 3 ⊢ (𝜑 → (𝐼↑2) = ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))
62	20	simp3d 1142	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℂ)
63		1cnd 10901	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
64		3z 12283	. . . . . 6 ⊢ 3 ∈ ℤ
65		1exp 13740	. . . . . 6 ⊢ (3 ∈ ℤ → (1↑3) = 1)
66	64, 65	mp1i 13	. . . . 5 ⊢ (𝜑 → (1↑3) = 1)
67	33	simp3d 1142	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
68	67	mulid2d 10924	. . . . . . . . . 10 ⊢ (𝜑 → (1 · 𝑇) = 𝑇)
69	68	oveq2d 7271	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑃) + (1 · 𝑇)) = ((2 · 𝑃) + 𝑇))
70	68	oveq2d 7271	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / (1 · 𝑇)) = (𝑈 / 𝑇))
71	69, 70	oveq12d 7273	. . . . . . . 8 ⊢ (𝜑 → (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) = (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)))
72	71	oveq1d 7270	. . . . . . 7 ⊢ (𝜑 → ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3) = ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
73	72	negeqd 11145	. . . . . 6 ⊢ (𝜑 → -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3) = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
74	30, 73	eqtr4d 2781	. . . . 5 ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
75		oveq1 7262	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑥↑3) = (1↑3))
76	75	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝑥↑3) = 1 ↔ (1↑3) = 1))
77		oveq1 7262	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 · 𝑇) = (1 · 𝑇))
78	77	oveq2d 7271	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((2 · 𝑃) + (𝑥 · 𝑇)) = ((2 · 𝑃) + (1 · 𝑇)))
79	77	oveq2d 7271	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑈 / (𝑥 · 𝑇)) = (𝑈 / (1 · 𝑇)))
80	78, 79	oveq12d 7273	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) = (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))))
81	80	oveq1d 7270	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3) = ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
82	81	negeqd 11145	. . . . . . . 8 ⊢ (𝑥 = 1 → -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3) = -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
83	82	eqeq2d 2749	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑀 = -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3) ↔ 𝑀 = -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3)))
84	76, 83	anbi12d 630	. . . . . 6 ⊢ (𝑥 = 1 → (((𝑥↑3) = 1 ∧ 𝑀 = -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3)) ↔ ((1↑3) = 1 ∧ 𝑀 = -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))))
85	84	rspcev 3552	. . . . 5 ⊢ ((1 ∈ ℂ ∧ ((1↑3) = 1 ∧ 𝑀 = -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))) → ∃𝑥 ∈ ℂ ((𝑥↑3) = 1 ∧ 𝑀 = -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3)))
86	63, 66, 74, 85	syl12anc 833	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℂ ((𝑥↑3) = 1 ∧ 𝑀 = -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3)))
87		2cn 11978	. . . . . 6 ⊢ 2 ∈ ℂ
88		mulcl 10886	. . . . . 6 ⊢ ((2 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (2 · 𝑃) ∈ ℂ)
89	87, 21, 88	sylancr 586	. . . . 5 ⊢ (𝜑 → (2 · 𝑃) ∈ ℂ)
90	21	sqcld 13790	. . . . . 6 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ)
91		mulcl 10886	. . . . . . 7 ⊢ ((4 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (4 · 𝑅) ∈ ℂ)
92	11, 62, 91	sylancr 586	. . . . . 6 ⊢ (𝜑 → (4 · 𝑅) ∈ ℂ)
93	90, 92	subcld 11262	. . . . 5 ⊢ (𝜑 → ((𝑃↑2) − (4 · 𝑅)) ∈ ℂ)
94	22	sqcld 13790	. . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ)
95	94	negcld 11249	. . . . 5 ⊢ (𝜑 → -(𝑄↑2) ∈ ℂ)
96	31	oveq1d 7270	. . . . . 6 ⊢ (𝜑 → (𝑇↑3) = ((((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3))↑3))
97	1, 2, 3, 4, 1, 9, 5, 6, 7, 26, 27, 28	quartlem2 25913	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))
98	97	simp2d 1141	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ ℂ)
99	97	simp3d 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℂ)
100	98, 99	addcld 10925	. . . . . . . 8 ⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ)
101	100	halfcld 12148	. . . . . . 7 ⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ)
102		3nn 11982	. . . . . . 7 ⊢ 3 ∈ ℕ
103		cxproot 25750	. . . . . . 7 ⊢ ((((𝑉 + 𝑊) / 2) ∈ ℂ ∧ 3 ∈ ℕ) → ((((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3))↑3) = ((𝑉 + 𝑊) / 2))
104	101, 102, 103	sylancl 585	. . . . . 6 ⊢ (𝜑 → ((((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3))↑3) = ((𝑉 + 𝑊) / 2))
105	96, 104	eqtrd 2778	. . . . 5 ⊢ (𝜑 → (𝑇↑3) = ((𝑉 + 𝑊) / 2))
106	28	oveq1d 7270	. . . . . 6 ⊢ (𝜑 → (𝑊↑2) = ((√‘((𝑉↑2) − (4 · (𝑈↑3))))↑2))
107	98	sqcld 13790	. . . . . . . 8 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ)
108	97	simp1d 1140	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ)
109		3nn0 12181	. . . . . . . . . 10 ⊢ 3 ∈ ℕ₀
110		expcl 13728	. . . . . . . . . 10 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑈↑3) ∈ ℂ)
111	108, 109, 110	sylancl 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ)
112		mulcl 10886	. . . . . . . . 9 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ)
113	11, 111, 112	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ)
114	107, 113	subcld 11262	. . . . . . 7 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ)
115	114	sqsqrtd 15079	. . . . . 6 ⊢ (𝜑 → ((√‘((𝑉↑2) − (4 · (𝑈↑3))))↑2) = ((𝑉↑2) − (4 · (𝑈↑3))))
116	106, 115	eqtrd 2778	. . . . 5 ⊢ (𝜑 → (𝑊↑2) = ((𝑉↑2) − (4 · (𝑈↑3))))
117	21, 22, 62, 26, 27	quartlem1 25912	. . . . . 6 ⊢ (𝜑 → (𝑈 = (((2 · 𝑃)↑2) − (3 · ((𝑃↑2) − (4 · 𝑅)))) ∧ 𝑉 = (((2 · ((2 · 𝑃)↑3)) − (9 · ((2 · 𝑃) · ((𝑃↑2) − (4 · 𝑅))))) + (;27 · -(𝑄↑2)))))
118	117	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑈 = (((2 · 𝑃)↑2) − (3 · ((𝑃↑2) − (4 · 𝑅)))))
119	117	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑉 = (((2 · ((2 · 𝑃)↑3)) − (9 · ((2 · 𝑃) · ((𝑃↑2) − (4 · 𝑅))))) + (;27 · -(𝑄↑2))))
120	89, 93, 95, 36, 67, 105, 99, 116, 118, 119, 32	mcubic 25902	. . . 4 ⊢ (𝜑 → ((((𝑀↑3) + ((2 · 𝑃) · (𝑀↑2))) + ((((𝑃↑2) − (4 · 𝑅)) · 𝑀) + -(𝑄↑2))) = 0 ↔ ∃𝑥 ∈ ℂ ((𝑥↑3) = 1 ∧ 𝑀 = -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3))))
121	86, 120	mpbird 256	. . 3 ⊢ (𝜑 → (((𝑀↑3) + ((2 · 𝑃) · (𝑀↑2))) + ((((𝑃↑2) − (4 · 𝑅)) · 𝑀) + -(𝑄↑2))) = 0)
122	49	simp3d 1142	. . 3 ⊢ (𝜑 → 𝐽 ∈ ℂ)
123	48	oveq1d 7270	. . . 4 ⊢ (𝜑 → (𝐽↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))↑2))
124	55, 58	subcld 11262	. . . . 5 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ)
125	124	sqsqrtd 15079	. . . 4 ⊢ (𝜑 → ((√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))↑2) = ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))
126	123, 125	eqtrd 2778	. . 3 ⊢ (𝜑 → (𝐽↑2) = ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))
127	21, 22, 25, 34, 45, 46, 50, 61, 62, 121, 122, 126	dquart 25908	. 2 ⊢ (𝜑 → (((((𝑋 − 𝐸)↑4) + (𝑃 · ((𝑋 − 𝐸)↑2))) + ((𝑄 · (𝑋 − 𝐸)) + 𝑅)) = 0 ↔ (((𝑋 − 𝐸) = (-𝑆 + 𝐼) ∨ (𝑋 − 𝐸) = (-𝑆 − 𝐼)) ∨ ((𝑋 − 𝐸) = (𝑆 + 𝐽) ∨ (𝑋 − 𝐸) = (𝑆 − 𝐽)))))
128	34	negcld 11249	. . . . . . . 8 ⊢ (𝜑 → -𝑆 ∈ ℂ)
129	128, 50	addcld 10925	. . . . . . 7 ⊢ (𝜑 → (-𝑆 + 𝐼) ∈ ℂ)
130	8, 24, 129	subaddd 11280	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ (𝐸 + (-𝑆 + 𝐼)) = 𝑋))
131	24, 34	negsubd 11268	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 + -𝑆) = (𝐸 − 𝑆))
132	131	oveq1d 7270	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) + 𝐼) = ((𝐸 − 𝑆) + 𝐼))
133	24, 128, 50	addassd 10928	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) + 𝐼) = (𝐸 + (-𝑆 + 𝐼)))
134	132, 133	eqtr3d 2780	. . . . . . 7 ⊢ (𝜑 → ((𝐸 − 𝑆) + 𝐼) = (𝐸 + (-𝑆 + 𝐼)))
135	134	eqeq1d 2740	. . . . . 6 ⊢ (𝜑 → (((𝐸 − 𝑆) + 𝐼) = 𝑋 ↔ (𝐸 + (-𝑆 + 𝐼)) = 𝑋))
136	130, 135	bitr4d 281	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ ((𝐸 − 𝑆) + 𝐼) = 𝑋))
137		eqcom 2745	. . . . 5 ⊢ (((𝐸 − 𝑆) + 𝐼) = 𝑋 ↔ 𝑋 = ((𝐸 − 𝑆) + 𝐼))
138	136, 137	bitrdi 286	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ 𝑋 = ((𝐸 − 𝑆) + 𝐼)))
139	128, 50	subcld 11262	. . . . . . 7 ⊢ (𝜑 → (-𝑆 − 𝐼) ∈ ℂ)
140	8, 24, 139	subaddd 11280	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ (𝐸 + (-𝑆 − 𝐼)) = 𝑋))
141	131	oveq1d 7270	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) − 𝐼) = ((𝐸 − 𝑆) − 𝐼))
142	24, 128, 50	addsubassd 11282	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) − 𝐼) = (𝐸 + (-𝑆 − 𝐼)))
143	141, 142	eqtr3d 2780	. . . . . . 7 ⊢ (𝜑 → ((𝐸 − 𝑆) − 𝐼) = (𝐸 + (-𝑆 − 𝐼)))
144	143	eqeq1d 2740	. . . . . 6 ⊢ (𝜑 → (((𝐸 − 𝑆) − 𝐼) = 𝑋 ↔ (𝐸 + (-𝑆 − 𝐼)) = 𝑋))
145	140, 144	bitr4d 281	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ ((𝐸 − 𝑆) − 𝐼) = 𝑋))
146		eqcom 2745	. . . . 5 ⊢ (((𝐸 − 𝑆) − 𝐼) = 𝑋 ↔ 𝑋 = ((𝐸 − 𝑆) − 𝐼))
147	145, 146	bitrdi 286	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ 𝑋 = ((𝐸 − 𝑆) − 𝐼)))
148	138, 147	orbi12d 915	. . 3 ⊢ (𝜑 → (((𝑋 − 𝐸) = (-𝑆 + 𝐼) ∨ (𝑋 − 𝐸) = (-𝑆 − 𝐼)) ↔ (𝑋 = ((𝐸 − 𝑆) + 𝐼) ∨ 𝑋 = ((𝐸 − 𝑆) − 𝐼))))
149	34, 122	addcld 10925	. . . . . . 7 ⊢ (𝜑 → (𝑆 + 𝐽) ∈ ℂ)
150	8, 24, 149	subaddd 11280	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ (𝐸 + (𝑆 + 𝐽)) = 𝑋))
151	24, 34, 122	addassd 10928	. . . . . . 7 ⊢ (𝜑 → ((𝐸 + 𝑆) + 𝐽) = (𝐸 + (𝑆 + 𝐽)))
152	151	eqeq1d 2740	. . . . . 6 ⊢ (𝜑 → (((𝐸 + 𝑆) + 𝐽) = 𝑋 ↔ (𝐸 + (𝑆 + 𝐽)) = 𝑋))
153	150, 152	bitr4d 281	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ ((𝐸 + 𝑆) + 𝐽) = 𝑋))
154		eqcom 2745	. . . . 5 ⊢ (((𝐸 + 𝑆) + 𝐽) = 𝑋 ↔ 𝑋 = ((𝐸 + 𝑆) + 𝐽))
155	153, 154	bitrdi 286	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ 𝑋 = ((𝐸 + 𝑆) + 𝐽)))
156	34, 122	subcld 11262	. . . . . . 7 ⊢ (𝜑 → (𝑆 − 𝐽) ∈ ℂ)
157	8, 24, 156	subaddd 11280	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ (𝐸 + (𝑆 − 𝐽)) = 𝑋))
158	24, 34, 122	addsubassd 11282	. . . . . . 7 ⊢ (𝜑 → ((𝐸 + 𝑆) − 𝐽) = (𝐸 + (𝑆 − 𝐽)))
159	158	eqeq1d 2740	. . . . . 6 ⊢ (𝜑 → (((𝐸 + 𝑆) − 𝐽) = 𝑋 ↔ (𝐸 + (𝑆 − 𝐽)) = 𝑋))
160	157, 159	bitr4d 281	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ ((𝐸 + 𝑆) − 𝐽) = 𝑋))
161		eqcom 2745	. . . . 5 ⊢ (((𝐸 + 𝑆) − 𝐽) = 𝑋 ↔ 𝑋 = ((𝐸 + 𝑆) − 𝐽))
162	160, 161	bitrdi 286	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ 𝑋 = ((𝐸 + 𝑆) − 𝐽)))
163	155, 162	orbi12d 915	. . 3 ⊢ (𝜑 → (((𝑋 − 𝐸) = (𝑆 + 𝐽) ∨ (𝑋 − 𝐸) = (𝑆 − 𝐽)) ↔ (𝑋 = ((𝐸 + 𝑆) + 𝐽) ∨ 𝑋 = ((𝐸 + 𝑆) − 𝐽))))
164	148, 163	orbi12d 915	. 2 ⊢ (𝜑 → ((((𝑋 − 𝐸) = (-𝑆 + 𝐼) ∨ (𝑋 − 𝐸) = (-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 304	1 ⊢ (𝜑 → ((((𝑋↑4) + (𝐴 · (𝑋↑3))) + ((𝐵 · (𝑋↑2)) + ((𝐶 · 𝑋) + 𝐷))) = 0 ↔ ((𝑋 = ((𝐸 − 𝑆) + 𝐼) ∨ 𝑋 = ((𝐸 − 𝑆) − 𝐼)) ∨ (𝑋 = ((𝐸 + 𝑆) + 𝐽) ∨ 𝑋 = ((𝐸 + 𝑆) − 𝐽)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 3c3 11959 4c4 11960 5c5 11961 6c6 11962 7c7 11963 8c8 11964 9c9 11965 ℕ₀cn0 12163 ℤcz 12249 cdc 12366 ↑cexp 13710 √csqrt 14872 ↑_𝑐ccxp 25616

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-dvds 15892 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-cxp 25618

This theorem is referenced by: quartfull 33027

W3C validator