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Theorem quart 26825

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 35347) 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 7383	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝐸) = (𝑋 − -(𝐴 / 4)))
11		4cn 12266	. . . . . . . 8 ⊢ 4 ∈ ℂ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ)
13		4ne0 12289	. . . . . . . 8 ⊢ 4 ≠ 0
14	13	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ≠ 0)
15	1, 12, 14	divcld 11931	. . . . . 6 ⊢ (𝜑 → (𝐴 / 4) ∈ ℂ)
16	8, 15	subnegd 11512	. . . . 5 ⊢ (𝜑 → (𝑋 − -(𝐴 / 4)) = (𝑋 + (𝐴 / 4)))
17	10, 16	eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝑋 − 𝐸) = (𝑋 + (𝐴 / 4)))
18	1, 2, 3, 4, 5, 6, 7, 8, 17	quart1 26820	. . 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 2739	. 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 26818	. . . 4 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
21	20	simp1d 1143	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℂ)
22	20	simp2d 1144	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℂ)
23	15	negcld 11492	. . . . 5 ⊢ (𝜑 → -(𝐴 / 4) ∈ ℂ)
24	9, 23	eqeltrd 2837	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ)
25	8, 24	subcld 11505	. . 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 26823	. . . 4 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))
34	33	simp1d 1143	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℂ)
35	29	oveq2d 7383	. . . . . 6 ⊢ (𝜑 → (2 · 𝑆) = (2 · ((√‘𝑀) / 2)))
36	33	simp2d 1144	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
37	36	sqrtcld 15402	. . . . . . 7 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
38		2cnd 12259	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
39		2ne0 12285	. . . . . . . 8 ⊢ 2 ≠ 0
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ≠ 0)
41	37, 38, 40	divcan2d 11933	. . . . . 6 ⊢ (𝜑 → (2 · ((√‘𝑀) / 2)) = (√‘𝑀))
42	35, 41	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (2 · 𝑆) = (√‘𝑀))
43	42	oveq1d 7382	. . . 4 ⊢ (𝜑 → ((2 · 𝑆)↑2) = ((√‘𝑀)↑2))
44	36	sqsqrtd 15404	. . . 4 ⊢ (𝜑 → ((√‘𝑀)↑2) = 𝑀)
45	43, 44	eqtr2d 2773	. . 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 26824	. . . 4 ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ))
50	49	simp2d 1144	. . 3 ⊢ (𝜑 → 𝐼 ∈ ℂ)
51	47	oveq1d 7382	. . . 4 ⊢ (𝜑 → (𝐼↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))↑2))
52	34	sqcld 14106	. . . . . . . 8 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ)
53	52	negcld 11492	. . . . . . 7 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ)
54	21	halfcld 12422	. . . . . . 7 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ)
55	53, 54	subcld 11505	. . . . . 6 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ)
56	22, 12, 14	divcld 11931	. . . . . . 7 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ)
57	49	simp1d 1143	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≠ 0)
58	56, 34, 57	divcld 11931	. . . . . 6 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ)
59	55, 58	addcld 11164	. . . . 5 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ)
60	59	sqsqrtd 15404	. . . 4 ⊢ (𝜑 → ((√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))↑2) = ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))
61	51, 60	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝐼↑2) = ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))
62	20	simp3d 1145	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℂ)
63		1cnd 11139	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
64		3z 12560	. . . . . 6 ⊢ 3 ∈ ℤ
65		1exp 14053	. . . . . 6 ⊢ (3 ∈ ℤ → (1↑3) = 1)
66	64, 65	mp1i 13	. . . . 5 ⊢ (𝜑 → (1↑3) = 1)
67	33	simp3d 1145	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
68	67	mullidd 11163	. . . . . . . . . 10 ⊢ (𝜑 → (1 · 𝑇) = 𝑇)
69	68	oveq2d 7383	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑃) + (1 · 𝑇)) = ((2 · 𝑃) + 𝑇))
70	68	oveq2d 7383	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / (1 · 𝑇)) = (𝑈 / 𝑇))
71	69, 70	oveq12d 7385	. . . . . . . 8 ⊢ (𝜑 → (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) = (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)))
72	71	oveq1d 7382	. . . . . . 7 ⊢ (𝜑 → ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3) = ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
73	72	negeqd 11387	. . . . . 6 ⊢ (𝜑 → -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3) = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
74	30, 73	eqtr4d 2775	. . . . 5 ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
75		oveq1 7374	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑥↑3) = (1↑3))
76	75	eqeq1d 2739	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝑥↑3) = 1 ↔ (1↑3) = 1))
77		oveq1 7374	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 · 𝑇) = (1 · 𝑇))
78	77	oveq2d 7383	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((2 · 𝑃) + (𝑥 · 𝑇)) = ((2 · 𝑃) + (1 · 𝑇)))
79	77	oveq2d 7383	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑈 / (𝑥 · 𝑇)) = (𝑈 / (1 · 𝑇)))
80	78, 79	oveq12d 7385	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) = (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))))
81	80	oveq1d 7382	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3) = ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
82	81	negeqd 11387	. . . . . . . 8 ⊢ (𝑥 = 1 → -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3) = -((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
83	82	eqeq2d 2748	. . . . . . 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 633	. . . . . 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 3565	. . . . 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 837	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℂ ((𝑥↑3) = 1 ∧ 𝑀 = -((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3)))
87		2cn 12256	. . . . . 6 ⊢ 2 ∈ ℂ
88		mulcl 11122	. . . . . 6 ⊢ ((2 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (2 · 𝑃) ∈ ℂ)
89	87, 21, 88	sylancr 588	. . . . 5 ⊢ (𝜑 → (2 · 𝑃) ∈ ℂ)
90	21	sqcld 14106	. . . . . 6 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ)
91		mulcl 11122	. . . . . . 7 ⊢ ((4 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (4 · 𝑅) ∈ ℂ)
92	11, 62, 91	sylancr 588	. . . . . 6 ⊢ (𝜑 → (4 · 𝑅) ∈ ℂ)
93	90, 92	subcld 11505	. . . . 5 ⊢ (𝜑 → ((𝑃↑2) − (4 · 𝑅)) ∈ ℂ)
94	22	sqcld 14106	. . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ)
95	94	negcld 11492	. . . . 5 ⊢ (𝜑 → -(𝑄↑2) ∈ ℂ)
96	31	oveq1d 7382	. . . . . 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 26822	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))
98	97	simp2d 1144	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ ℂ)
99	97	simp3d 1145	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℂ)
100	98, 99	addcld 11164	. . . . . . . 8 ⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ)
101	100	halfcld 12422	. . . . . . 7 ⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ)
102		3nn 12260	. . . . . . 7 ⊢ 3 ∈ ℕ
103		cxproot 26654	. . . . . . 7 ⊢ ((((𝑉 + 𝑊) / 2) ∈ ℂ ∧ 3 ∈ ℕ) → ((((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3))↑3) = ((𝑉 + 𝑊) / 2))
104	101, 102, 103	sylancl 587	. . . . . 6 ⊢ (𝜑 → ((((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3))↑3) = ((𝑉 + 𝑊) / 2))
105	96, 104	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (𝑇↑3) = ((𝑉 + 𝑊) / 2))
106	28	oveq1d 7382	. . . . . 6 ⊢ (𝜑 → (𝑊↑2) = ((√‘((𝑉↑2) − (4 · (𝑈↑3))))↑2))
107	98	sqcld 14106	. . . . . . . 8 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ)
108	97	simp1d 1143	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ)
109		3nn0 12455	. . . . . . . . . 10 ⊢ 3 ∈ ℕ₀
110		expcl 14041	. . . . . . . . . 10 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑈↑3) ∈ ℂ)
111	108, 109, 110	sylancl 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ)
112		mulcl 11122	. . . . . . . . 9 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ)
113	11, 111, 112	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ)
114	107, 113	subcld 11505	. . . . . . 7 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ)
115	114	sqsqrtd 15404	. . . . . 6 ⊢ (𝜑 → ((√‘((𝑉↑2) − (4 · (𝑈↑3))))↑2) = ((𝑉↑2) − (4 · (𝑈↑3))))
116	106, 115	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (𝑊↑2) = ((𝑉↑2) − (4 · (𝑈↑3))))
117	21, 22, 62, 26, 27	quartlem1 26821	. . . . . 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 26811	. . . 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 257	. . 3 ⊢ (𝜑 → (((𝑀↑3) + ((2 · 𝑃) · (𝑀↑2))) + ((((𝑃↑2) − (4 · 𝑅)) · 𝑀) + -(𝑄↑2))) = 0)
122	49	simp3d 1145	. . 3 ⊢ (𝜑 → 𝐽 ∈ ℂ)
123	48	oveq1d 7382	. . . 4 ⊢ (𝜑 → (𝐽↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))↑2))
124	55, 58	subcld 11505	. . . . 5 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ)
125	124	sqsqrtd 15404	. . . 4 ⊢ (𝜑 → ((√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))↑2) = ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))
126	123, 125	eqtrd 2772	. . 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 26817	. 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 11492	. . . . . . . 8 ⊢ (𝜑 → -𝑆 ∈ ℂ)
129	128, 50	addcld 11164	. . . . . . 7 ⊢ (𝜑 → (-𝑆 + 𝐼) ∈ ℂ)
130	8, 24, 129	subaddd 11523	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ (𝐸 + (-𝑆 + 𝐼)) = 𝑋))
131	24, 34	negsubd 11511	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 + -𝑆) = (𝐸 − 𝑆))
132	131	oveq1d 7382	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) + 𝐼) = ((𝐸 − 𝑆) + 𝐼))
133	24, 128, 50	addassd 11167	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) + 𝐼) = (𝐸 + (-𝑆 + 𝐼)))
134	132, 133	eqtr3d 2774	. . . . . . 7 ⊢ (𝜑 → ((𝐸 − 𝑆) + 𝐼) = (𝐸 + (-𝑆 + 𝐼)))
135	134	eqeq1d 2739	. . . . . 6 ⊢ (𝜑 → (((𝐸 − 𝑆) + 𝐼) = 𝑋 ↔ (𝐸 + (-𝑆 + 𝐼)) = 𝑋))
136	130, 135	bitr4d 282	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ ((𝐸 − 𝑆) + 𝐼) = 𝑋))
137		eqcom 2744	. . . . 5 ⊢ (((𝐸 − 𝑆) + 𝐼) = 𝑋 ↔ 𝑋 = ((𝐸 − 𝑆) + 𝐼))
138	136, 137	bitrdi 287	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ 𝑋 = ((𝐸 − 𝑆) + 𝐼)))
139	128, 50	subcld 11505	. . . . . . 7 ⊢ (𝜑 → (-𝑆 − 𝐼) ∈ ℂ)
140	8, 24, 139	subaddd 11523	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ (𝐸 + (-𝑆 − 𝐼)) = 𝑋))
141	131	oveq1d 7382	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) − 𝐼) = ((𝐸 − 𝑆) − 𝐼))
142	24, 128, 50	addsubassd 11525	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) − 𝐼) = (𝐸 + (-𝑆 − 𝐼)))
143	141, 142	eqtr3d 2774	. . . . . . 7 ⊢ (𝜑 → ((𝐸 − 𝑆) − 𝐼) = (𝐸 + (-𝑆 − 𝐼)))
144	143	eqeq1d 2739	. . . . . 6 ⊢ (𝜑 → (((𝐸 − 𝑆) − 𝐼) = 𝑋 ↔ (𝐸 + (-𝑆 − 𝐼)) = 𝑋))
145	140, 144	bitr4d 282	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ ((𝐸 − 𝑆) − 𝐼) = 𝑋))
146		eqcom 2744	. . . . 5 ⊢ (((𝐸 − 𝑆) − 𝐼) = 𝑋 ↔ 𝑋 = ((𝐸 − 𝑆) − 𝐼))
147	145, 146	bitrdi 287	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ 𝑋 = ((𝐸 − 𝑆) − 𝐼)))
148	138, 147	orbi12d 919	. . 3 ⊢ (𝜑 → (((𝑋 − 𝐸) = (-𝑆 + 𝐼) ∨ (𝑋 − 𝐸) = (-𝑆 − 𝐼)) ↔ (𝑋 = ((𝐸 − 𝑆) + 𝐼) ∨ 𝑋 = ((𝐸 − 𝑆) − 𝐼))))
149	34, 122	addcld 11164	. . . . . . 7 ⊢ (𝜑 → (𝑆 + 𝐽) ∈ ℂ)
150	8, 24, 149	subaddd 11523	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ (𝐸 + (𝑆 + 𝐽)) = 𝑋))
151	24, 34, 122	addassd 11167	. . . . . . 7 ⊢ (𝜑 → ((𝐸 + 𝑆) + 𝐽) = (𝐸 + (𝑆 + 𝐽)))
152	151	eqeq1d 2739	. . . . . 6 ⊢ (𝜑 → (((𝐸 + 𝑆) + 𝐽) = 𝑋 ↔ (𝐸 + (𝑆 + 𝐽)) = 𝑋))
153	150, 152	bitr4d 282	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ ((𝐸 + 𝑆) + 𝐽) = 𝑋))
154		eqcom 2744	. . . . 5 ⊢ (((𝐸 + 𝑆) + 𝐽) = 𝑋 ↔ 𝑋 = ((𝐸 + 𝑆) + 𝐽))
155	153, 154	bitrdi 287	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ 𝑋 = ((𝐸 + 𝑆) + 𝐽)))
156	34, 122	subcld 11505	. . . . . . 7 ⊢ (𝜑 → (𝑆 − 𝐽) ∈ ℂ)
157	8, 24, 156	subaddd 11523	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ (𝐸 + (𝑆 − 𝐽)) = 𝑋))
158	24, 34, 122	addsubassd 11525	. . . . . . 7 ⊢ (𝜑 → ((𝐸 + 𝑆) − 𝐽) = (𝐸 + (𝑆 − 𝐽)))
159	158	eqeq1d 2739	. . . . . 6 ⊢ (𝜑 → (((𝐸 + 𝑆) − 𝐽) = 𝑋 ↔ (𝐸 + (𝑆 − 𝐽)) = 𝑋))
160	157, 159	bitr4d 282	. . . . 5 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ ((𝐸 + 𝑆) − 𝐽) = 𝑋))
161		eqcom 2744	. . . . 5 ⊢ (((𝐸 + 𝑆) − 𝐽) = 𝑋 ↔ 𝑋 = ((𝐸 + 𝑆) − 𝐽))
162	160, 161	bitrdi 287	. . . 4 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ 𝑋 = ((𝐸 + 𝑆) − 𝐽)))
163	155, 162	orbi12d 919	. . 3 ⊢ (𝜑 → (((𝑋 − 𝐸) = (𝑆 + 𝐽) ∨ (𝑋 − 𝐸) = (𝑆 − 𝐽)) ↔ (𝑋 = ((𝐸 + 𝑆) + 𝐽) ∨ 𝑋 = ((𝐸 + 𝑆) − 𝐽))))
164	148, 163	orbi12d 919	. 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 305	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 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 ‘cfv 6499 (class class class)co 7367 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 − cmin 11377 -cneg 11378 / cdiv 11807 ℕcn 12174 2c2 12236 3c3 12237 4c4 12238 5c5 12239 6c6 12240 7c7 12241 8c8 12242 9c9 12243 ℕ₀cn0 12437 ℤcz 12524 cdc 12644 ↑cexp 14023 √csqrt 15195 ↑_𝑐ccxp 26519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-dvds 16222 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-cxp 26521

This theorem is referenced by: quartfull 35347

W3C validator