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Theorem quart 26831

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 35340) 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 7376	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝐸) = (𝑋 − -(𝐴 / 4)))
11		4cn 12234	. . . . . . . 8 ⊢ 4 ∈ ℂ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ)
13		4ne0 12257	. . . . . . . 8 ⊢ 4 ≠ 0
14	13	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ≠ 0)
15	1, 12, 14	divcld 11921	. . . . . 6 ⊢ (𝜑 → (𝐴 / 4) ∈ ℂ)
16	8, 15	subnegd 11503	. . . . 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 26826	. . 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 26824	. . . 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 11483	. . . . 5 ⊢ (𝜑 → -(𝐴 / 4) ∈ ℂ)
24	9, 23	eqeltrd 2837	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ)
25	8, 24	subcld 11496	. . 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 26829	. . . 4 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))
34	33	simp1d 1143	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℂ)
35	29	oveq2d 7376	. . . . . 6 ⊢ (𝜑 → (2 · 𝑆) = (2 · ((√‘𝑀) / 2)))
36	33	simp2d 1144	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
37	36	sqrtcld 15367	. . . . . . 7 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
38		2cnd 12227	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
39		2ne0 12253	. . . . . . . 8 ⊢ 2 ≠ 0
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ≠ 0)
41	37, 38, 40	divcan2d 11923	. . . . . 6 ⊢ (𝜑 → (2 · ((√‘𝑀) / 2)) = (√‘𝑀))
42	35, 41	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (2 · 𝑆) = (√‘𝑀))
43	42	oveq1d 7375	. . . 4 ⊢ (𝜑 → ((2 · 𝑆)↑2) = ((√‘𝑀)↑2))
44	36	sqsqrtd 15369	. . . 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 26830	. . . 4 ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ))
50	49	simp2d 1144	. . 3 ⊢ (𝜑 → 𝐼 ∈ ℂ)
51	47	oveq1d 7375	. . . 4 ⊢ (𝜑 → (𝐼↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))↑2))
52	34	sqcld 14071	. . . . . . . 8 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ)
53	52	negcld 11483	. . . . . . 7 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ)
54	21	halfcld 12390	. . . . . . 7 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ)
55	53, 54	subcld 11496	. . . . . 6 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ)
56	22, 12, 14	divcld 11921	. . . . . . 7 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ)
57	49	simp1d 1143	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≠ 0)
58	56, 34, 57	divcld 11921	. . . . . 6 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ)
59	55, 58	addcld 11155	. . . . 5 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ)
60	59	sqsqrtd 15369	. . . 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 11131	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
64		3z 12528	. . . . . 6 ⊢ 3 ∈ ℤ
65		1exp 14018	. . . . . 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 11154	. . . . . . . . . 10 ⊢ (𝜑 → (1 · 𝑇) = 𝑇)
69	68	oveq2d 7376	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑃) + (1 · 𝑇)) = ((2 · 𝑃) + 𝑇))
70	68	oveq2d 7376	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / (1 · 𝑇)) = (𝑈 / 𝑇))
71	69, 70	oveq12d 7378	. . . . . . . 8 ⊢ (𝜑 → (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) = (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)))
72	71	oveq1d 7375	. . . . . . 7 ⊢ (𝜑 → ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3) = ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
73	72	negeqd 11378	. . . . . 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 7367	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑥↑3) = (1↑3))
76	75	eqeq1d 2739	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝑥↑3) = 1 ↔ (1↑3) = 1))
77		oveq1 7367	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 · 𝑇) = (1 · 𝑇))
78	77	oveq2d 7376	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((2 · 𝑃) + (𝑥 · 𝑇)) = ((2 · 𝑃) + (1 · 𝑇)))
79	77	oveq2d 7376	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑈 / (𝑥 · 𝑇)) = (𝑈 / (1 · 𝑇)))
80	78, 79	oveq12d 7378	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) = (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))))
81	80	oveq1d 7375	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3) = ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3))
82	81	negeqd 11378	. . . . . . . 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 3577	. . . . 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 12224	. . . . . 6 ⊢ 2 ∈ ℂ
88		mulcl 11114	. . . . . 6 ⊢ ((2 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (2 · 𝑃) ∈ ℂ)
89	87, 21, 88	sylancr 588	. . . . 5 ⊢ (𝜑 → (2 · 𝑃) ∈ ℂ)
90	21	sqcld 14071	. . . . . 6 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ)
91		mulcl 11114	. . . . . . 7 ⊢ ((4 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (4 · 𝑅) ∈ ℂ)
92	11, 62, 91	sylancr 588	. . . . . 6 ⊢ (𝜑 → (4 · 𝑅) ∈ ℂ)
93	90, 92	subcld 11496	. . . . 5 ⊢ (𝜑 → ((𝑃↑2) − (4 · 𝑅)) ∈ ℂ)
94	22	sqcld 14071	. . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ)
95	94	negcld 11483	. . . . 5 ⊢ (𝜑 → -(𝑄↑2) ∈ ℂ)
96	31	oveq1d 7375	. . . . . 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 26828	. . . . . . . . . 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 11155	. . . . . . . 8 ⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ)
101	100	halfcld 12390	. . . . . . 7 ⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ)
102		3nn 12228	. . . . . . 7 ⊢ 3 ∈ ℕ
103		cxproot 26659	. . . . . . 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 7375	. . . . . 6 ⊢ (𝜑 → (𝑊↑2) = ((√‘((𝑉↑2) − (4 · (𝑈↑3))))↑2))
107	98	sqcld 14071	. . . . . . . 8 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ)
108	97	simp1d 1143	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ)
109		3nn0 12423	. . . . . . . . . 10 ⊢ 3 ∈ ℕ₀
110		expcl 14006	. . . . . . . . . 10 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑈↑3) ∈ ℂ)
111	108, 109, 110	sylancl 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ)
112		mulcl 11114	. . . . . . . . 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 11496	. . . . . . 7 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ)
115	114	sqsqrtd 15369	. . . . . 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 26827	. . . . . 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 26817	. . . 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 7375	. . . 4 ⊢ (𝜑 → (𝐽↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))↑2))
124	55, 58	subcld 11496	. . . . 5 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ)
125	124	sqsqrtd 15369	. . . 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 26823	. 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 11483	. . . . . . . 8 ⊢ (𝜑 → -𝑆 ∈ ℂ)
129	128, 50	addcld 11155	. . . . . . 7 ⊢ (𝜑 → (-𝑆 + 𝐼) ∈ ℂ)
130	8, 24, 129	subaddd 11514	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ (𝐸 + (-𝑆 + 𝐼)) = 𝑋))
131	24, 34	negsubd 11502	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 + -𝑆) = (𝐸 − 𝑆))
132	131	oveq1d 7375	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) + 𝐼) = ((𝐸 − 𝑆) + 𝐼))
133	24, 128, 50	addassd 11158	. . . . . . . 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 11496	. . . . . . 7 ⊢ (𝜑 → (-𝑆 − 𝐼) ∈ ℂ)
140	8, 24, 139	subaddd 11514	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ (𝐸 + (-𝑆 − 𝐼)) = 𝑋))
141	131	oveq1d 7375	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + -𝑆) − 𝐼) = ((𝐸 − 𝑆) − 𝐼))
142	24, 128, 50	addsubassd 11516	. . . . . . . 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 11155	. . . . . . 7 ⊢ (𝜑 → (𝑆 + 𝐽) ∈ ℂ)
150	8, 24, 149	subaddd 11514	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ (𝐸 + (𝑆 + 𝐽)) = 𝑋))
151	24, 34, 122	addassd 11158	. . . . . . 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 11496	. . . . . . 7 ⊢ (𝜑 → (𝑆 − 𝐽) ∈ ℂ)
157	8, 24, 156	subaddd 11514	. . . . . 6 ⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ (𝐸 + (𝑆 − 𝐽)) = 𝑋))
158	24, 34, 122	addsubassd 11516	. . . . . . 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 3061 ‘cfv 6493 (class class class)co 7360 ℂcc 11028 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 − cmin 11368 -cneg 11369 / cdiv 11798 ℕcn 12149 2c2 12204 3c3 12205 4c4 12206 5c5 12207 6c6 12208 7c7 12209 8c8 12210 9c9 12211 ℕ₀cn0 12405 ℤcz 12492 cdc 12611 ↑cexp 13988 √csqrt 15160 ↑_𝑐ccxp 26524

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ioc 13270 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-shft 14994 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-ef 15994 df-sin 15996 df-cos 15997 df-pi 15999 df-dvds 16184 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-limc 25827 df-dv 25828 df-log 26525 df-cxp 26526

This theorem is referenced by: quartfull 35340

W3C validator