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Theorem quart 27099

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 35746) 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 7432	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝐸) = (𝑋 − -(𝐴 / 4)))
11		4cn 12353	. . . . . . . 8 ⊢ 4 ∈ ℂ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ)
13		4ne0 12379	. . . . . . . 8 ⊢ 4 ≠ 0
14	13	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ≠ 0)
15	1, 12, 14	divcld 12018	. . . . . 6 ⊢ (𝜑 → (𝐴 / 4) ∈ ℂ)
16	8, 15	subnegd 11603	. . . . 5 ⊢ (𝜑 → (𝑋 − -(𝐴 / 4)) = (𝑋 + (𝐴 / 4)))
17	10, 16	eqtrd 2797	. . . 4 ⊢ (𝜑 → (𝑋 − 𝐸) = (𝑋 + (𝐴 / 4)))
18	1, 2, 3, 4, 5, 6, 7, 8, 17	quart1 27094	. . 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 2764	. 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 27092	. . . 4 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
21	20	simp1d 1160	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℂ)
22	20	simp2d 1161	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℂ)
23	15	negcld 11583	. . . . 5 ⊢ (𝜑 → -(𝐴 / 4) ∈ ℂ)
24	9, 23	eqeltrd 2862	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ)
25	8, 24	subcld 11596	. . 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 27097	. . . 4 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))
34	33	simp1d 1160	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℂ)
35	29	oveq2d 7432	. . . . . 6 ⊢ (𝜑 → (2 · 𝑆) = (2 · ((√‘𝑀) / 2)))
36	33	simp2d 1161	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
37	36	sqrtcld 15529	. . . . . . 7 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
38		2cnd 12346	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
39		2ne0 12374	. . . . . . . 8 ⊢ 2 ≠ 0
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ≠ 0)
41	37, 38, 40	divcan2d 12020	. . . . . 6 ⊢ (𝜑 → (2 · ((√‘𝑀) / 2)) = (√‘𝑀))
42	35, 41	eqtrd 2797	. . . . 5 ⊢ (𝜑 → (2 · 𝑆) = (√‘𝑀))
43	42	oveq1d 7431	. . . 4 ⊢ (𝜑 → ((2 · 𝑆)↑2) = ((√‘𝑀)↑2))
44	36	sqsqrtd 15531	. . . 4 ⊢ (𝜑 → ((√‘𝑀)↑2) = 𝑀)
45	43, 44	eqtr2d 2798	. . 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 27098	. . . 4 ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ))
50	49	simp2d 1161	. . 3 ⊢ (𝜑 → 𝐼 ∈ ℂ)
51	47	oveq1d 7431	. . . 4 ⊢ (𝜑 → (𝐼↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))↑2))
52	34	sqcld 14210	. . . . . . . 8 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ)
53	52	negcld 11583	. . . . . . 7 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ)
54	21	halfcld 12516	. . . . . . 7 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ)
55	53, 54	subcld 11596	. . . . . 6 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ)
56	22, 12, 14	divcld 12018	. . . . . . 7 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ)
57	49	simp1d 1160	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≠ 0)
58	56, 34, 57	divcld 12018	. . . . . 6 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ)
59	55, 58	addcld 11255	. . . . 5 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ)
60	59	sqsqrtd 15531	. . . 4 ⊢ (𝜑 → ((√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))↑2) = ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))
61	51, 60	eqtrd 2797	. . 3 ⊢ (𝜑 → (𝐼↑2) = ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))
62	20	simp3d 1162	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℂ)
63		1cnd 11229	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
64		3z 12654	. . . . . 6 ⊢ 3 ∈ ℤ
65		1exp 14157	. . . . . 6 ⊢ (3 ∈ ℤ → (1↑3) = 1)
66	64, 65	mp1i 14	. . . . 5 ⊢ (𝜑 → (1↑3) = 1)
67	33	simp3d 1162	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
68	67	mullidd 11254	. . . . . . . . . 10 ⊢ (𝜑 → (1 · 𝑇) = 𝑇)
69	68	oveq2d 7432	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑃) + (1 · 𝑇)) = ((2 · 𝑃) + 𝑇))
70	68	oveq2d 7432	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / (1 · 𝑇)) = (𝑈 / 𝑇))
71	69, 70	oveq12d 7434	. . . . . . . 8 ⊢ (𝜑 → (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) = (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)))
72	71	oveq1d 7431	. . . . . . 7 ⊢ (𝜑 → ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3) = ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
73	72	negeqd 11478	. . . . . 6 ⊢ (𝜑 → -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3) = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
74	30, 73	eqtr4d 2800	. . . . 5 ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
75		oveq1 7423	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑥↑3) = (1↑3))
76	75	eqeq1d 2764	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝑥↑3) = 1 ↔ (1↑3) = 1))
77		oveq1 7423	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 · 𝑇) = (1 · 𝑇))
78	77	oveq2d 7432	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((2 · 𝑃) + (𝑥 · 𝑇)) = ((2 · 𝑃) + (1 · 𝑇)))
79	77	oveq2d 7432	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑈 / (𝑥 · 𝑇)) = (𝑈 / (1 · 𝑇)))
80	78, 79	oveq12d 7434	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) = (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))))
81	80	oveq1d 7431	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3) = ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
82	81	negeqd 11478	. . . . . . . 8 ⊢ (𝑥 = 1 → -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3) = -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
83	82	eqeq2d 2773	. . . . . . 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 644	. . . . . 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 3579	. . . . 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 850	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℂ ((𝑥↑3) = 1 ∧ 𝑀 = -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3)))
87		2cn 12343	. . . . . 6 ⊢ 2 ∈ ℂ
88		mulcl 11211	. . . . . 6 ⊢ ((2 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (2 · 𝑃) ∈ ℂ)
89	87, 21, 88	sylancr 599	. . . . 5 ⊢ (𝜑 → (2 · 𝑃) ∈ ℂ)
90	21	sqcld 14210	. . . . . 6 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ)
91		mulcl 11211	. . . . . . 7 ⊢ ((4 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (4 · 𝑅) ∈ ℂ)
92	11, 62, 91	sylancr 599	. . . . . 6 ⊢ (𝜑 → (4 · 𝑅) ∈ ℂ)
93	90, 92	subcld 11596	. . . . 5 ⊢ (𝜑 → ((𝑃↑2) − (4 · 𝑅)) ∈ ℂ)
94	22	sqcld 14210	. . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ)
95	94	negcld 11583	. . . . 5 ⊢ (𝜑 → -(𝑄↑2) ∈ ℂ)
96	31	oveq1d 7431	. . . . . 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 27096	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))
98	97	simp2d 1161	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ ℂ)
99	97	simp3d 1162	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℂ)
100	98, 99	addcld 11255	. . . . . . . 8 ⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ)
101	100	halfcld 12516	. . . . . . 7 ⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ)
102		3nn 12347	. . . . . . 7 ⊢ 3 ∈ ℕ
103		cxproot 26928	. . . . . . 7 ⊢ ((((𝑉 + 𝑊) / 2) ∈ ℂ ∧ 3 ∈ ℕ) → ((((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3))↑3) = ((𝑉 + 𝑊) / 2))
104	101, 102, 103	sylancl 598	. . . . . 6 ⊢ (𝜑 → ((((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3))↑3) = ((𝑉 + 𝑊) / 2))
105	96, 104	eqtrd 2797	. . . . 5 ⊢ (𝜑 → (𝑇↑3) = ((𝑉 + 𝑊) / 2))
106	28	oveq1d 7431	. . . . . 6 ⊢ (𝜑 → (𝑊↑2) = ((√‘((𝑉↑2) − (4 · (𝑈↑3))))↑2))
107	98	sqcld 14210	. . . . . . . 8 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ)
108	97	simp1d 1160	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ)
109		3nn0 12549	. . . . . . . . . 10 ⊢ 3 ∈ ℕ₀
110		expcl 14145	. . . . . . . . . 10 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑈↑3) ∈ ℂ)
111	108, 109, 110	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ)
112		mulcl 11211	. . . . . . . . 9 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ)
113	11, 111, 112	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ)
114	107, 113	subcld 11596	. . . . . . 7 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ)
115	114	sqsqrtd 15531	. . . . . 6 ⊢ (𝜑 → ((√‘((𝑉↑2) − (4 · (𝑈↑3))))↑2) = ((𝑉↑2) − (4 · (𝑈↑3))))
116	106, 115	eqtrd 2797	. . . . 5 ⊢ (𝜑 → (𝑊↑2) = ((𝑉↑2) − (4 · (𝑈↑3))))
117	21, 22, 62, 26, 27	quartlem1 27095	. . . . . 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 500	. . . . 5 ⊢ (𝜑 → 𝑈 = (((2 · 𝑃)↑2) − (3 · ((𝑃↑2) − (4 · 𝑅)))))
119	117	simprd 501	. . . . 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 27085	. . . 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 260	. . 3 ⊢ (𝜑 → (((𝑀↑3) + ((2 · 𝑃) · (𝑀↑2))) + ((((𝑃↑2) − (4 · 𝑅)) · 𝑀) + -(𝑄↑2))) = 0)
122	49	simp3d 1162	. . 3 ⊢ (𝜑 → 𝐽 ∈ ℂ)
123	48	oveq1d 7431	. . . 4 ⊢ (𝜑 → (𝐽↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))↑2))
124	55, 58	subcld 11596	. . . . 5 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ)
125	124	sqsqrtd 15531	. . . 4 ⊢ (𝜑 → ((√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))↑2) = ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))
126	123, 125	eqtrd 2797	. . 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 27091	. 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 11583	. . . . . . . 8 ⊢ (𝜑 → -𝑆 ∈ ℂ)
129	128, 50	addcld 11255	. . . . . . 7 ⊢ (𝜑 → (-𝑆 + 𝐼) ∈ ℂ)
130	8, 24, 129	subaddd 11614	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ (𝐸 + (-𝑆 + 𝐼)) = 𝑋))
131	24, 34	negsubd 11602	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 + -𝑆) = (𝐸 − 𝑆))
132	131	oveq1d 7431	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) + 𝐼) = ((𝐸 − 𝑆) + 𝐼))
133	24, 128, 50	addassd 11258	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) + 𝐼) = (𝐸 + (-𝑆 + 𝐼)))
134	132, 133	eqtr3d 2799	. . . . . . 7 ⊢ (𝜑 → ((𝐸 − 𝑆) + 𝐼) = (𝐸 + (-𝑆 + 𝐼)))
135	134	eqeq1d 2764	. . . . . 6 ⊢ (𝜑 → (((𝐸 − 𝑆) + 𝐼) = 𝑋 ↔ (𝐸 + (-𝑆 + 𝐼)) = 𝑋))
136	130, 135	bitr4d 285	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ ((𝐸 − 𝑆) + 𝐼) = 𝑋))
137		eqcom 2769	. . . . 5 ⊢ (((𝐸 − 𝑆) + 𝐼) = 𝑋 ↔ 𝑋 = ((𝐸 − 𝑆) + 𝐼))
138	136, 137	bitrdi 290	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ 𝑋 = ((𝐸 − 𝑆) + 𝐼)))
139	128, 50	subcld 11596	. . . . . . 7 ⊢ (𝜑 → (-𝑆 − 𝐼) ∈ ℂ)
140	8, 24, 139	subaddd 11614	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ (𝐸 + (-𝑆 − 𝐼)) = 𝑋))
141	131	oveq1d 7431	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) − 𝐼) = ((𝐸 − 𝑆) − 𝐼))
142	24, 128, 50	addsubassd 11616	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) − 𝐼) = (𝐸 + (-𝑆 − 𝐼)))
143	141, 142	eqtr3d 2799	. . . . . . 7 ⊢ (𝜑 → ((𝐸 − 𝑆) − 𝐼) = (𝐸 + (-𝑆 − 𝐼)))
144	143	eqeq1d 2764	. . . . . 6 ⊢ (𝜑 → (((𝐸 − 𝑆) − 𝐼) = 𝑋 ↔ (𝐸 + (-𝑆 − 𝐼)) = 𝑋))
145	140, 144	bitr4d 285	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ ((𝐸 − 𝑆) − 𝐼) = 𝑋))
146		eqcom 2769	. . . . 5 ⊢ (((𝐸 − 𝑆) − 𝐼) = 𝑋 ↔ 𝑋 = ((𝐸 − 𝑆) − 𝐼))
147	145, 146	bitrdi 290	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ 𝑋 = ((𝐸 − 𝑆) − 𝐼)))
148	138, 147	orbi12d 932	. . 3 ⊢ (𝜑 → (((𝑋 − 𝐸) = (-𝑆 + 𝐼) ∨ (𝑋 − 𝐸) = (-𝑆 − 𝐼)) ↔ (𝑋 = ((𝐸 − 𝑆) + 𝐼) ∨ 𝑋 = ((𝐸 − 𝑆) − 𝐼))))
149	34, 122	addcld 11255	. . . . . . 7 ⊢ (𝜑 → (𝑆 + 𝐽) ∈ ℂ)
150	8, 24, 149	subaddd 11614	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ (𝐸 + (𝑆 + 𝐽)) = 𝑋))
151	24, 34, 122	addassd 11258	. . . . . . 7 ⊢ (𝜑 → ((𝐸 + 𝑆) + 𝐽) = (𝐸 + (𝑆 + 𝐽)))
152	151	eqeq1d 2764	. . . . . 6 ⊢ (𝜑 → (((𝐸 + 𝑆) + 𝐽) = 𝑋 ↔ (𝐸 + (𝑆 + 𝐽)) = 𝑋))
153	150, 152	bitr4d 285	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ ((𝐸 + 𝑆) + 𝐽) = 𝑋))
154		eqcom 2769	. . . . 5 ⊢ (((𝐸 + 𝑆) + 𝐽) = 𝑋 ↔ 𝑋 = ((𝐸 + 𝑆) + 𝐽))
155	153, 154	bitrdi 290	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ 𝑋 = ((𝐸 + 𝑆) + 𝐽)))
156	34, 122	subcld 11596	. . . . . . 7 ⊢ (𝜑 → (𝑆 − 𝐽) ∈ ℂ)
157	8, 24, 156	subaddd 11614	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ (𝐸 + (𝑆 − 𝐽)) = 𝑋))
158	24, 34, 122	addsubassd 11616	. . . . . . 7 ⊢ (𝜑 → ((𝐸 + 𝑆) − 𝐽) = (𝐸 + (𝑆 − 𝐽)))
159	158	eqeq1d 2764	. . . . . 6 ⊢ (𝜑 → (((𝐸 + 𝑆) − 𝐽) = 𝑋 ↔ (𝐸 + (𝑆 − 𝐽)) = 𝑋))
160	157, 159	bitr4d 285	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ ((𝐸 + 𝑆) − 𝐽) = 𝑋))
161		eqcom 2769	. . . . 5 ⊢ (((𝐸 + 𝑆) − 𝐽) = 𝑋 ↔ 𝑋 = ((𝐸 + 𝑆) − 𝐽))
162	160, 161	bitrdi 290	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ 𝑋 = ((𝐸 + 𝑆) − 𝐽)))
163	155, 162	orbi12d 932	. . 3 ⊢ (𝜑 → (((𝑋 − 𝐸) = (𝑆 + 𝐽) ∨ (𝑋 − 𝐸) = (𝑆 − 𝐽)) ↔ (𝑋 = ((𝐸 + 𝑆) + 𝐽) ∨ 𝑋 = ((𝐸 + 𝑆) − 𝐽))))
164	148, 163	orbi12d 932	. 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 308	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∃wrex 3088 ‘cfv 6537 (class class class)co 7416 ℂcc 11125 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 − cmin 11468 -cneg 11469 / cdiv 11898 ℕcn 12260 2c2 12322 3c3 12323 4c4 12324 5c5 12325 6c6 12326 7c7 12327 8c8 12328 9c9 12329 ℕ₀cn0 12531 ℤcz 12618 cdc 12739 ↑cexp 14127 √csqrt 15322 ↑_𝑐ccxp 26793

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13045 df-xneg 13165 df-xadd 13166 df-xmul 13167 df-ioo 13404 df-ioc 13405 df-ico 13406 df-icc 13407 df-fz 13564 df-fzo 13712 df-fl 13855 df-mod 13933 df-seq 14068 df-exp 14128 df-fac 14340 df-bc 14369 df-hash 14397 df-shft 15142 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-limsup 15560 df-clim 15577 df-rlim 15578 df-sum 15776 df-ef 16157 df-sin 16159 df-cos 16160 df-pi 16162 df-dvds 16347 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-rest 17511 df-topn 17512 df-0g 17530 df-gsum 17531 df-topgen 17532 df-pt 17533 df-prds 17536 df-xrs 17592 df-qtop 17597 df-imas 17598 df-xps 17600 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18825 df-mnd 18841 df-submnd 18896 df-mulg 19195 df-cntz 19448 df-cmn 19913 df-psmet 21581 df-xmet 21582 df-met 21583 df-bl 21584 df-mopn 21585 df-fbas 21586 df-fg 21587 df-cnfld 21590 df-top 23123 df-topon 23140 df-topsp 23162 df-bases 23175 df-cld 23248 df-ntr 23249 df-cls 23250 df-nei 23327 df-lp 23365 df-perf 23366 df-cn 23456 df-cnp 23457 df-haus 23544 df-tx 23792 df-hmeo 23985 df-fil 24076 df-fm 24168 df-flim 24169 df-flf 24170 df-xms 24550 df-ms 24551 df-tms 24552 df-cncf 25110 df-limc 26098 df-dv 26099 df-log 26794 df-cxp 26795

This theorem is used by: quartfull 35746

W3C validator