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 7247 |
. . . . 5
⊢ (𝜑 → (𝑋 − 𝐸) = (𝑋 − -(𝐴 / 4))) |
11 | | 4cn 11939 |
. . . . . . . 8
⊢ 4 ∈
ℂ |
12 | 11 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 4 ∈
ℂ) |
13 | | 4ne0 11962 |
. . . . . . . 8
⊢ 4 ≠
0 |
14 | 13 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 4 ≠ 0) |
15 | 1, 12, 14 | divcld 11632 |
. . . . . 6
⊢ (𝜑 → (𝐴 / 4) ∈ ℂ) |
16 | 8, 15 | subnegd 11220 |
. . . . 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 25763 |
. . 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 25761 |
. . . 4
⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) |
21 | 20 | simp1d 1144 |
. . 3
⊢ (𝜑 → 𝑃 ∈ ℂ) |
22 | 20 | simp2d 1145 |
. . 3
⊢ (𝜑 → 𝑄 ∈ ℂ) |
23 | 15 | negcld 11200 |
. . . . 5
⊢ (𝜑 → -(𝐴 / 4) ∈ ℂ) |
24 | 9, 23 | eqeltrd 2839 |
. . . 4
⊢ (𝜑 → 𝐸 ∈ ℂ) |
25 | 8, 24 | subcld 11213 |
. . 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 25766 |
. . . 4
⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ)) |
34 | 33 | simp1d 1144 |
. . 3
⊢ (𝜑 → 𝑆 ∈ ℂ) |
35 | 29 | oveq2d 7247 |
. . . . . 6
⊢ (𝜑 → (2 · 𝑆) = (2 ·
((√‘𝑀) /
2))) |
36 | 33 | simp2d 1145 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℂ) |
37 | 36 | sqrtcld 15025 |
. . . . . . 7
⊢ (𝜑 → (√‘𝑀) ∈
ℂ) |
38 | | 2cnd 11932 |
. . . . . . 7
⊢ (𝜑 → 2 ∈
ℂ) |
39 | | 2ne0 11958 |
. . . . . . . 8
⊢ 2 ≠
0 |
40 | 39 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 2 ≠ 0) |
41 | 37, 38, 40 | divcan2d 11634 |
. . . . . 6
⊢ (𝜑 → (2 ·
((√‘𝑀) / 2)) =
(√‘𝑀)) |
42 | 35, 41 | eqtrd 2778 |
. . . . 5
⊢ (𝜑 → (2 · 𝑆) = (√‘𝑀)) |
43 | 42 | oveq1d 7246 |
. . . 4
⊢ (𝜑 → ((2 · 𝑆)↑2) =
((√‘𝑀)↑2)) |
44 | 36 | sqsqrtd 15027 |
. . . 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 25767 |
. . . 4
⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ)) |
50 | 49 | simp2d 1145 |
. . 3
⊢ (𝜑 → 𝐼 ∈ ℂ) |
51 | 47 | oveq1d 7246 |
. . . 4
⊢ (𝜑 → (𝐼↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))↑2)) |
52 | 34 | sqcld 13738 |
. . . . . . . 8
⊢ (𝜑 → (𝑆↑2) ∈ ℂ) |
53 | 52 | negcld 11200 |
. . . . . . 7
⊢ (𝜑 → -(𝑆↑2) ∈ ℂ) |
54 | 21 | halfcld 12099 |
. . . . . . 7
⊢ (𝜑 → (𝑃 / 2) ∈ ℂ) |
55 | 53, 54 | subcld 11213 |
. . . . . 6
⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ) |
56 | 22, 12, 14 | divcld 11632 |
. . . . . . 7
⊢ (𝜑 → (𝑄 / 4) ∈ ℂ) |
57 | 49 | simp1d 1144 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ≠ 0) |
58 | 56, 34, 57 | divcld 11632 |
. . . . . 6
⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ) |
59 | 55, 58 | addcld 10876 |
. . . . 5
⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ) |
60 | 59 | sqsqrtd 15027 |
. . . 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 1146 |
. . 3
⊢ (𝜑 → 𝑅 ∈ ℂ) |
63 | | 1cnd 10852 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℂ) |
64 | | 3z 12234 |
. . . . . 6
⊢ 3 ∈
ℤ |
65 | | 1exp 13688 |
. . . . . 6
⊢ (3 ∈
ℤ → (1↑3) = 1) |
66 | 64, 65 | mp1i 13 |
. . . . 5
⊢ (𝜑 → (1↑3) =
1) |
67 | 33 | simp3d 1146 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇 ∈ ℂ) |
68 | 67 | mulid2d 10875 |
. . . . . . . . . 10
⊢ (𝜑 → (1 · 𝑇) = 𝑇) |
69 | 68 | oveq2d 7247 |
. . . . . . . . 9
⊢ (𝜑 → ((2 · 𝑃) + (1 · 𝑇)) = ((2 · 𝑃) + 𝑇)) |
70 | 68 | oveq2d 7247 |
. . . . . . . . 9
⊢ (𝜑 → (𝑈 / (1 · 𝑇)) = (𝑈 / 𝑇)) |
71 | 69, 70 | oveq12d 7249 |
. . . . . . . 8
⊢ (𝜑 → (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) = (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇))) |
72 | 71 | oveq1d 7246 |
. . . . . . 7
⊢ (𝜑 → ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3) = ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3)) |
73 | 72 | negeqd 11096 |
. . . . . 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 7238 |
. . . . . . . 8
⊢ (𝑥 = 1 → (𝑥↑3) = (1↑3)) |
76 | 75 | eqeq1d 2740 |
. . . . . . 7
⊢ (𝑥 = 1 → ((𝑥↑3) = 1 ↔ (1↑3) =
1)) |
77 | | oveq1 7238 |
. . . . . . . . . . . 12
⊢ (𝑥 = 1 → (𝑥 · 𝑇) = (1 · 𝑇)) |
78 | 77 | oveq2d 7247 |
. . . . . . . . . . 11
⊢ (𝑥 = 1 → ((2 · 𝑃) + (𝑥 · 𝑇)) = ((2 · 𝑃) + (1 · 𝑇))) |
79 | 77 | oveq2d 7247 |
. . . . . . . . . . 11
⊢ (𝑥 = 1 → (𝑈 / (𝑥 · 𝑇)) = (𝑈 / (1 · 𝑇))) |
80 | 78, 79 | oveq12d 7249 |
. . . . . . . . . 10
⊢ (𝑥 = 1 → (((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) = (((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇)))) |
81 | 80 | oveq1d 7246 |
. . . . . . . . 9
⊢ (𝑥 = 1 → ((((2 · 𝑃) + (𝑥 · 𝑇)) + (𝑈 / (𝑥 · 𝑇))) / 3) = ((((2 · 𝑃) + (1 · 𝑇)) + (𝑈 / (1 · 𝑇))) / 3)) |
82 | 81 | negeqd 11096 |
. . . . . . . 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 634 |
. . . . . 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 3549 |
. . . . 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 11929 |
. . . . . 6
⊢ 2 ∈
ℂ |
88 | | mulcl 10837 |
. . . . . 6
⊢ ((2
∈ ℂ ∧ 𝑃
∈ ℂ) → (2 · 𝑃) ∈ ℂ) |
89 | 87, 21, 88 | sylancr 590 |
. . . . 5
⊢ (𝜑 → (2 · 𝑃) ∈
ℂ) |
90 | 21 | sqcld 13738 |
. . . . . 6
⊢ (𝜑 → (𝑃↑2) ∈ ℂ) |
91 | | mulcl 10837 |
. . . . . . 7
⊢ ((4
∈ ℂ ∧ 𝑅
∈ ℂ) → (4 · 𝑅) ∈ ℂ) |
92 | 11, 62, 91 | sylancr 590 |
. . . . . 6
⊢ (𝜑 → (4 · 𝑅) ∈
ℂ) |
93 | 90, 92 | subcld 11213 |
. . . . 5
⊢ (𝜑 → ((𝑃↑2) − (4 · 𝑅)) ∈
ℂ) |
94 | 22 | sqcld 13738 |
. . . . . 6
⊢ (𝜑 → (𝑄↑2) ∈ ℂ) |
95 | 94 | negcld 11200 |
. . . . 5
⊢ (𝜑 → -(𝑄↑2) ∈ ℂ) |
96 | 31 | oveq1d 7246 |
. . . . . 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 25765 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ)) |
98 | 97 | simp2d 1145 |
. . . . . . . . 9
⊢ (𝜑 → 𝑉 ∈ ℂ) |
99 | 97 | simp3d 1146 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ ℂ) |
100 | 98, 99 | addcld 10876 |
. . . . . . . 8
⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ) |
101 | 100 | halfcld 12099 |
. . . . . . 7
⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ) |
102 | | 3nn 11933 |
. . . . . . 7
⊢ 3 ∈
ℕ |
103 | | cxproot 25602 |
. . . . . . 7
⊢ ((((𝑉 + 𝑊) / 2) ∈ ℂ ∧ 3 ∈
ℕ) → ((((𝑉 +
𝑊) /
2)↑𝑐(1 / 3))↑3) = ((𝑉 + 𝑊) / 2)) |
104 | 101, 102,
103 | sylancl 589 |
. . . . . 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 7246 |
. . . . . 6
⊢ (𝜑 → (𝑊↑2) = ((√‘((𝑉↑2) − (4 ·
(𝑈↑3))))↑2)) |
107 | 98 | sqcld 13738 |
. . . . . . . 8
⊢ (𝜑 → (𝑉↑2) ∈ ℂ) |
108 | 97 | simp1d 1144 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑈 ∈ ℂ) |
109 | | 3nn0 12132 |
. . . . . . . . . 10
⊢ 3 ∈
ℕ0 |
110 | | expcl 13677 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ ℂ ∧ 3 ∈
ℕ0) → (𝑈↑3) ∈ ℂ) |
111 | 108, 109,
110 | sylancl 589 |
. . . . . . . . 9
⊢ (𝜑 → (𝑈↑3) ∈ ℂ) |
112 | | mulcl 10837 |
. . . . . . . . 9
⊢ ((4
∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 ·
(𝑈↑3)) ∈
ℂ) |
113 | 11, 111, 112 | sylancr 590 |
. . . . . . . 8
⊢ (𝜑 → (4 · (𝑈↑3)) ∈
ℂ) |
114 | 107, 113 | subcld 11213 |
. . . . . . 7
⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈
ℂ) |
115 | 114 | sqsqrtd 15027 |
. . . . . 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 25764 |
. . . . . 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 498 |
. . . . 5
⊢ (𝜑 → 𝑈 = (((2 · 𝑃)↑2) − (3 · ((𝑃↑2) − (4 ·
𝑅))))) |
119 | 117 | simprd 499 |
. . . . 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 25754 |
. . . 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 1146 |
. . 3
⊢ (𝜑 → 𝐽 ∈ ℂ) |
123 | 48 | oveq1d 7246 |
. . . 4
⊢ (𝜑 → (𝐽↑2) = ((√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))↑2)) |
124 | 55, 58 | subcld 11213 |
. . . . 5
⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ) |
125 | 124 | sqsqrtd 15027 |
. . . 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 25760 |
. 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 11200 |
. . . . . . . 8
⊢ (𝜑 → -𝑆 ∈ ℂ) |
129 | 128, 50 | addcld 10876 |
. . . . . . 7
⊢ (𝜑 → (-𝑆 + 𝐼) ∈ ℂ) |
130 | 8, 24, 129 | subaddd 11231 |
. . . . . 6
⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ (𝐸 + (-𝑆 + 𝐼)) = 𝑋)) |
131 | 24, 34 | negsubd 11219 |
. . . . . . . . 9
⊢ (𝜑 → (𝐸 + -𝑆) = (𝐸 − 𝑆)) |
132 | 131 | oveq1d 7246 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸 + -𝑆) + 𝐼) = ((𝐸 − 𝑆) + 𝐼)) |
133 | 24, 128, 50 | addassd 10879 |
. . . . . . . 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 285 |
. . . . 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 290 |
. . . 4
⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 + 𝐼) ↔ 𝑋 = ((𝐸 − 𝑆) + 𝐼))) |
139 | 128, 50 | subcld 11213 |
. . . . . . 7
⊢ (𝜑 → (-𝑆 − 𝐼) ∈ ℂ) |
140 | 8, 24, 139 | subaddd 11231 |
. . . . . 6
⊢ (𝜑 → ((𝑋 − 𝐸) = (-𝑆 − 𝐼) ↔ (𝐸 + (-𝑆 − 𝐼)) = 𝑋)) |
141 | 131 | oveq1d 7246 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸 + -𝑆) − 𝐼) = ((𝐸 − 𝑆) − 𝐼)) |
142 | 24, 128, 50 | addsubassd 11233 |
. . . . . . . 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 285 |
. . . . 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 290 |
. . . 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 10876 |
. . . . . . 7
⊢ (𝜑 → (𝑆 + 𝐽) ∈ ℂ) |
150 | 8, 24, 149 | subaddd 11231 |
. . . . . 6
⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ (𝐸 + (𝑆 + 𝐽)) = 𝑋)) |
151 | 24, 34, 122 | addassd 10879 |
. . . . . . 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 285 |
. . . . 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 290 |
. . . 4
⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 + 𝐽) ↔ 𝑋 = ((𝐸 + 𝑆) + 𝐽))) |
156 | 34, 122 | subcld 11213 |
. . . . . . 7
⊢ (𝜑 → (𝑆 − 𝐽) ∈ ℂ) |
157 | 8, 24, 156 | subaddd 11231 |
. . . . . 6
⊢ (𝜑 → ((𝑋 − 𝐸) = (𝑆 − 𝐽) ↔ (𝐸 + (𝑆 − 𝐽)) = 𝑋)) |
158 | 24, 34, 122 | addsubassd 11233 |
. . . . . . 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 285 |
. . . . 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 290 |
. . . 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 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))))) |