Metamath Proof Explorer


Theorem dcubic1lem

Description: Lemma for dcubic1 and dcubic2 : simplify the cubic equation under the substitution X = U - M / U . (Contributed by Mario Carneiro, 26-Apr-2015)

Ref Expression
Hypotheses dcubic.c φ P
dcubic.d φ Q
dcubic.x φ X
dcubic.t φ T
dcubic.3 φ T 3 = G N
dcubic.g φ G
dcubic.2 φ G 2 = N 2 + M 3
dcubic.m φ M = P 3
dcubic.n φ N = Q 2
dcubic.0 φ T 0
dcubic2.u φ U
dcubic2.z φ U 0
dcubic2.2 φ X = U M U
Assertion dcubic1lem φ X 3 + P X + Q = 0 U 3 2 + Q U 3 - M 3 = 0

Proof

Step Hyp Ref Expression
1 dcubic.c φ P
2 dcubic.d φ Q
3 dcubic.x φ X
4 dcubic.t φ T
5 dcubic.3 φ T 3 = G N
6 dcubic.g φ G
7 dcubic.2 φ G 2 = N 2 + M 3
8 dcubic.m φ M = P 3
9 dcubic.n φ N = Q 2
10 dcubic.0 φ T 0
11 dcubic2.u φ U
12 dcubic2.z φ U 0
13 dcubic2.2 φ X = U M U
14 3nn0 3 0
15 expcl U 3 0 U 3
16 11 14 15 sylancl φ U 3
17 16 sqvald φ U 3 2 = U 3 U 3
18 17 oveq1d φ U 3 2 U 3 = U 3 U 3 U 3
19 3z 3
20 19 a1i φ 3
21 11 12 20 expne0d φ U 3 0
22 16 16 21 divcan4d φ U 3 U 3 U 3 = U 3
23 18 22 eqtr2d φ U 3 = U 3 2 U 3
24 3cn 3
25 24 a1i φ 3
26 3ne0 3 0
27 26 a1i φ 3 0
28 1 25 27 divcld φ P 3
29 8 28 eqeltrd φ M
30 expcl M 3 0 M 3
31 29 14 30 sylancl φ M 3
32 31 16 21 divcld φ M 3 U 3
33 2 32 negsubd φ Q + M 3 U 3 = Q M 3 U 3
34 2 16 21 divcan4d φ Q U 3 U 3 = Q
35 34 oveq1d φ Q U 3 U 3 M 3 U 3 = Q M 3 U 3
36 33 35 eqtr4d φ Q + M 3 U 3 = Q U 3 U 3 M 3 U 3
37 1 3 mulcld φ P X
38 37 negcld φ P X
39 32 negcld φ M 3 U 3
40 38 39 37 2 add42d φ P X + M 3 U 3 + P X + Q = P X + P X + Q + M 3 U 3
41 1 3 mulneg2d φ P X = P X
42 13 negeqd φ X = U M U
43 29 11 12 divcld φ M U
44 11 43 negsubdid φ U M U = - U + M U
45 42 44 eqtrd φ X = - U + M U
46 45 oveq2d φ P X = P - U + M U
47 41 46 eqtr3d φ P X = P - U + M U
48 11 negcld φ U
49 1 48 43 adddid φ P - U + M U = P U + P M U
50 1 11 mulneg2d φ P U = P U
51 50 oveq1d φ P U + P M U = - P U + P M U
52 47 49 51 3eqtrd φ P X = - P U + P M U
53 52 oveq1d φ - P X + M 3 U 3 = P U + P M U + M 3 U 3
54 1 11 mulcld φ P U
55 54 negcld φ P U
56 1 43 mulcld φ P M U
57 55 56 39 addassd φ P U + P M U + M 3 U 3 = P U + P M U + M 3 U 3
58 53 57 eqtrd φ - P X + M 3 U 3 = P U + P M U + M 3 U 3
59 58 oveq1d φ P X + M 3 U 3 + P X + Q = P U + P M U + M 3 U 3 + P X + Q
60 38 37 addcomd φ - P X + P X = P X + P X
61 37 negidd φ P X + P X = 0
62 60 61 eqtrd φ - P X + P X = 0
63 62 oveq1d φ P X + P X + Q + M 3 U 3 = 0 + Q + M 3 U 3
64 2 39 addcld φ Q + M 3 U 3
65 64 addid2d φ 0 + Q + M 3 U 3 = Q + M 3 U 3
66 63 65 eqtrd φ P X + P X + Q + M 3 U 3 = Q + M 3 U 3
67 40 59 66 3eqtr3d φ P U + P M U + M 3 U 3 + P X + Q = Q + M 3 U 3
68 2 16 mulcld φ Q U 3
69 68 31 16 21 divsubdird φ Q U 3 M 3 U 3 = Q U 3 U 3 M 3 U 3
70 36 67 69 3eqtr4d φ P U + P M U + M 3 U 3 + P X + Q = Q U 3 M 3 U 3
71 23 70 oveq12d φ U 3 + P U + P M U + M 3 U 3 + P X + Q = U 3 2 U 3 + Q U 3 M 3 U 3
72 11 43 negsubd φ U + M U = U M U
73 13 72 eqtr4d φ X = U + M U
74 73 oveq1d φ X 3 = U + M U 3
75 43 negcld φ M U
76 binom3 U M U U + M U 3 = U 3 + 3 U 2 M U + 3 U M U 2 + M U 3
77 11 75 76 syl2anc φ U + M U 3 = U 3 + 3 U 2 M U + 3 U M U 2 + M U 3
78 11 sqcld φ U 2
79 78 43 mulneg2d φ U 2 M U = U 2 M U
80 78 29 11 12 div12d φ U 2 M U = M U 2 U
81 11 sqvald φ U 2 = U U
82 81 oveq1d φ U 2 U = U U U
83 11 11 12 divcan4d φ U U U = U
84 82 83 eqtrd φ U 2 U = U
85 84 oveq2d φ M U 2 U = M U
86 80 85 eqtrd φ U 2 M U = M U
87 86 negeqd φ U 2 M U = M U
88 79 87 eqtrd φ U 2 M U = M U
89 88 oveq2d φ 3 U 2 M U = 3 M U
90 29 11 mulcld φ M U
91 25 90 mulneg2d φ 3 M U = 3 M U
92 25 29 11 mulassd φ 3 M U = 3 M U
93 8 oveq2d φ 3 M = 3 P 3
94 1 25 27 divcan2d φ 3 P 3 = P
95 93 94 eqtrd φ 3 M = P
96 95 oveq1d φ 3 M U = P U
97 92 96 eqtr3d φ 3 M U = P U
98 97 negeqd φ 3 M U = P U
99 89 91 98 3eqtrd φ 3 U 2 M U = P U
100 99 oveq2d φ U 3 + 3 U 2 M U = U 3 + P U
101 sqneg M U M U 2 = M U 2
102 43 101 syl φ M U 2 = M U 2
103 43 sqvald φ M U 2 = M U M U
104 102 103 eqtrd φ M U 2 = M U M U
105 104 oveq2d φ U M U 2 = U M U M U
106 11 43 43 mulassd φ U M U M U = U M U M U
107 29 11 12 divcan2d φ U M U = M
108 107 oveq1d φ U M U M U = M M U
109 105 106 108 3eqtr2d φ U M U 2 = M M U
110 109 oveq2d φ 3 U M U 2 = 3 M M U
111 25 29 43 mulassd φ 3 M M U = 3 M M U
112 95 oveq1d φ 3 M M U = P M U
113 110 111 112 3eqtr2d φ 3 U M U 2 = P M U
114 3nn 3
115 114 a1i φ 3
116 n2dvds3 ¬ 2 3
117 116 a1i φ ¬ 2 3
118 oexpneg M U 3 ¬ 2 3 M U 3 = M U 3
119 43 115 117 118 syl3anc φ M U 3 = M U 3
120 14 a1i φ 3 0
121 29 11 12 120 expdivd φ M U 3 = M 3 U 3
122 121 negeqd φ M U 3 = M 3 U 3
123 119 122 eqtrd φ M U 3 = M 3 U 3
124 113 123 oveq12d φ 3 U M U 2 + M U 3 = P M U + M 3 U 3
125 100 124 oveq12d φ U 3 + 3 U 2 M U + 3 U M U 2 + M U 3 = U 3 + P U + P M U + M 3 U 3
126 74 77 125 3eqtrd φ X 3 = U 3 + P U + P M U + M 3 U 3
127 56 39 addcld φ P M U + M 3 U 3
128 16 55 127 addassd φ U 3 + P U + P M U + M 3 U 3 = U 3 + P U + P M U + M 3 U 3
129 126 128 eqtrd φ X 3 = U 3 + P U + P M U + M 3 U 3
130 129 oveq1d φ X 3 + P X + Q = U 3 + P U + P M U + M 3 U 3 + P X + Q
131 55 127 addcld φ P U + P M U + M 3 U 3
132 37 2 addcld φ P X + Q
133 16 131 132 addassd φ U 3 + P U + P M U + M 3 U 3 + P X + Q = U 3 + P U + P M U + M 3 U 3 + P X + Q
134 130 133 eqtrd φ X 3 + P X + Q = U 3 + P U + P M U + M 3 U 3 + P X + Q
135 16 sqcld φ U 3 2
136 68 31 subcld φ Q U 3 M 3
137 135 136 16 21 divdird φ U 3 2 + Q U 3 - M 3 U 3 = U 3 2 U 3 + Q U 3 M 3 U 3
138 71 134 137 3eqtr4d φ X 3 + P X + Q = U 3 2 + Q U 3 - M 3 U 3
139 138 eqeq1d φ X 3 + P X + Q = 0 U 3 2 + Q U 3 - M 3 U 3 = 0
140 135 136 addcld φ U 3 2 + Q U 3 - M 3
141 140 16 21 diveq0ad φ U 3 2 + Q U 3 - M 3 U 3 = 0 U 3 2 + Q U 3 - M 3 = 0
142 139 141 bitrd φ X 3 + P X + Q = 0 U 3 2 + Q U 3 - M 3 = 0