Step |
Hyp |
Ref |
Expression |
1 |
|
dquart.b |
|- ( ph -> B e. CC ) |
2 |
|
dquart.c |
|- ( ph -> C e. CC ) |
3 |
|
dquart.x |
|- ( ph -> X e. CC ) |
4 |
|
dquart.s |
|- ( ph -> S e. CC ) |
5 |
|
dquart.m |
|- ( ph -> M = ( ( 2 x. S ) ^ 2 ) ) |
6 |
|
dquart.m0 |
|- ( ph -> M =/= 0 ) |
7 |
|
dquart.i |
|- ( ph -> I e. CC ) |
8 |
|
dquart.i2 |
|- ( ph -> ( I ^ 2 ) = ( ( -u ( S ^ 2 ) - ( B / 2 ) ) + ( ( C / 4 ) / S ) ) ) |
9 |
|
dquart.d |
|- ( ph -> D e. CC ) |
10 |
|
dquart.3 |
|- ( ph -> ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) + -u ( C ^ 2 ) ) ) = 0 ) |
11 |
|
dquart.j |
|- ( ph -> J e. CC ) |
12 |
|
dquart.j2 |
|- ( ph -> ( J ^ 2 ) = ( ( -u ( S ^ 2 ) - ( B / 2 ) ) - ( ( C / 4 ) / S ) ) ) |
13 |
3
|
sqcld |
|- ( ph -> ( X ^ 2 ) e. CC ) |
14 |
|
2cn |
|- 2 e. CC |
15 |
|
mulcl |
|- ( ( 2 e. CC /\ S e. CC ) -> ( 2 x. S ) e. CC ) |
16 |
14 4 15
|
sylancr |
|- ( ph -> ( 2 x. S ) e. CC ) |
17 |
16
|
sqcld |
|- ( ph -> ( ( 2 x. S ) ^ 2 ) e. CC ) |
18 |
5 17
|
eqeltrd |
|- ( ph -> M e. CC ) |
19 |
18 1
|
addcld |
|- ( ph -> ( M + B ) e. CC ) |
20 |
19
|
halfcld |
|- ( ph -> ( ( M + B ) / 2 ) e. CC ) |
21 |
|
binom2 |
|- ( ( ( X ^ 2 ) e. CC /\ ( ( M + B ) / 2 ) e. CC ) -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) ^ 2 ) = ( ( ( ( X ^ 2 ) ^ 2 ) + ( 2 x. ( ( X ^ 2 ) x. ( ( M + B ) / 2 ) ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) ) |
22 |
13 20 21
|
syl2anc |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) ^ 2 ) = ( ( ( ( X ^ 2 ) ^ 2 ) + ( 2 x. ( ( X ^ 2 ) x. ( ( M + B ) / 2 ) ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) ) |
23 |
|
2t2e4 |
|- ( 2 x. 2 ) = 4 |
24 |
23
|
oveq2i |
|- ( X ^ ( 2 x. 2 ) ) = ( X ^ 4 ) |
25 |
|
2nn0 |
|- 2 e. NN0 |
26 |
25
|
a1i |
|- ( ph -> 2 e. NN0 ) |
27 |
3 26 26
|
expmuld |
|- ( ph -> ( X ^ ( 2 x. 2 ) ) = ( ( X ^ 2 ) ^ 2 ) ) |
28 |
24 27
|
syl5reqr |
|- ( ph -> ( ( X ^ 2 ) ^ 2 ) = ( X ^ 4 ) ) |
29 |
14
|
a1i |
|- ( ph -> 2 e. CC ) |
30 |
29 13 20
|
mul12d |
|- ( ph -> ( 2 x. ( ( X ^ 2 ) x. ( ( M + B ) / 2 ) ) ) = ( ( X ^ 2 ) x. ( 2 x. ( ( M + B ) / 2 ) ) ) ) |
31 |
|
2ne0 |
|- 2 =/= 0 |
32 |
31
|
a1i |
|- ( ph -> 2 =/= 0 ) |
33 |
19 29 32
|
divcan2d |
|- ( ph -> ( 2 x. ( ( M + B ) / 2 ) ) = ( M + B ) ) |
34 |
33
|
oveq2d |
|- ( ph -> ( ( X ^ 2 ) x. ( 2 x. ( ( M + B ) / 2 ) ) ) = ( ( X ^ 2 ) x. ( M + B ) ) ) |
35 |
13 19
|
mulcomd |
|- ( ph -> ( ( X ^ 2 ) x. ( M + B ) ) = ( ( M + B ) x. ( X ^ 2 ) ) ) |
36 |
34 35
|
eqtrd |
|- ( ph -> ( ( X ^ 2 ) x. ( 2 x. ( ( M + B ) / 2 ) ) ) = ( ( M + B ) x. ( X ^ 2 ) ) ) |
37 |
18 1 13
|
adddird |
|- ( ph -> ( ( M + B ) x. ( X ^ 2 ) ) = ( ( M x. ( X ^ 2 ) ) + ( B x. ( X ^ 2 ) ) ) ) |
38 |
30 36 37
|
3eqtrd |
|- ( ph -> ( 2 x. ( ( X ^ 2 ) x. ( ( M + B ) / 2 ) ) ) = ( ( M x. ( X ^ 2 ) ) + ( B x. ( X ^ 2 ) ) ) ) |
39 |
28 38
|
oveq12d |
|- ( ph -> ( ( ( X ^ 2 ) ^ 2 ) + ( 2 x. ( ( X ^ 2 ) x. ( ( M + B ) / 2 ) ) ) ) = ( ( X ^ 4 ) + ( ( M x. ( X ^ 2 ) ) + ( B x. ( X ^ 2 ) ) ) ) ) |
40 |
|
4nn0 |
|- 4 e. NN0 |
41 |
|
expcl |
|- ( ( X e. CC /\ 4 e. NN0 ) -> ( X ^ 4 ) e. CC ) |
42 |
3 40 41
|
sylancl |
|- ( ph -> ( X ^ 4 ) e. CC ) |
43 |
18 13
|
mulcld |
|- ( ph -> ( M x. ( X ^ 2 ) ) e. CC ) |
44 |
1 13
|
mulcld |
|- ( ph -> ( B x. ( X ^ 2 ) ) e. CC ) |
45 |
42 43 44
|
add12d |
|- ( ph -> ( ( X ^ 4 ) + ( ( M x. ( X ^ 2 ) ) + ( B x. ( X ^ 2 ) ) ) ) = ( ( M x. ( X ^ 2 ) ) + ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) ) ) |
46 |
39 45
|
eqtrd |
|- ( ph -> ( ( ( X ^ 2 ) ^ 2 ) + ( 2 x. ( ( X ^ 2 ) x. ( ( M + B ) / 2 ) ) ) ) = ( ( M x. ( X ^ 2 ) ) + ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) ) ) |
47 |
46
|
oveq1d |
|- ( ph -> ( ( ( ( X ^ 2 ) ^ 2 ) + ( 2 x. ( ( X ^ 2 ) x. ( ( M + B ) / 2 ) ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) = ( ( ( M x. ( X ^ 2 ) ) + ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) ) |
48 |
42 44
|
addcld |
|- ( ph -> ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) e. CC ) |
49 |
20
|
sqcld |
|- ( ph -> ( ( ( M + B ) / 2 ) ^ 2 ) e. CC ) |
50 |
43 48 49
|
addassd |
|- ( ph -> ( ( ( M x. ( X ^ 2 ) ) + ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) = ( ( M x. ( X ^ 2 ) ) + ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) ) ) |
51 |
22 47 50
|
3eqtrd |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) ^ 2 ) = ( ( M x. ( X ^ 2 ) ) + ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) ) ) |
52 |
18
|
halfcld |
|- ( ph -> ( M / 2 ) e. CC ) |
53 |
52 3
|
mulcld |
|- ( ph -> ( ( M / 2 ) x. X ) e. CC ) |
54 |
|
4cn |
|- 4 e. CC |
55 |
54
|
a1i |
|- ( ph -> 4 e. CC ) |
56 |
|
4ne0 |
|- 4 =/= 0 |
57 |
56
|
a1i |
|- ( ph -> 4 =/= 0 ) |
58 |
2 55 57
|
divcld |
|- ( ph -> ( C / 4 ) e. CC ) |
59 |
53 58
|
subcld |
|- ( ph -> ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) e. CC ) |
60 |
5 6
|
eqnetrrd |
|- ( ph -> ( ( 2 x. S ) ^ 2 ) =/= 0 ) |
61 |
|
sqne0 |
|- ( ( 2 x. S ) e. CC -> ( ( ( 2 x. S ) ^ 2 ) =/= 0 <-> ( 2 x. S ) =/= 0 ) ) |
62 |
16 61
|
syl |
|- ( ph -> ( ( ( 2 x. S ) ^ 2 ) =/= 0 <-> ( 2 x. S ) =/= 0 ) ) |
63 |
60 62
|
mpbid |
|- ( ph -> ( 2 x. S ) =/= 0 ) |
64 |
|
mulne0b |
|- ( ( 2 e. CC /\ S e. CC ) -> ( ( 2 =/= 0 /\ S =/= 0 ) <-> ( 2 x. S ) =/= 0 ) ) |
65 |
14 4 64
|
sylancr |
|- ( ph -> ( ( 2 =/= 0 /\ S =/= 0 ) <-> ( 2 x. S ) =/= 0 ) ) |
66 |
63 65
|
mpbird |
|- ( ph -> ( 2 =/= 0 /\ S =/= 0 ) ) |
67 |
66
|
simprd |
|- ( ph -> S =/= 0 ) |
68 |
59 4 29 67 32
|
divcan5d |
|- ( ph -> ( ( 2 x. ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) ) / ( 2 x. S ) ) = ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) |
69 |
29 53 58
|
subdid |
|- ( ph -> ( 2 x. ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) ) = ( ( 2 x. ( ( M / 2 ) x. X ) ) - ( 2 x. ( C / 4 ) ) ) ) |
70 |
29 52 3
|
mulassd |
|- ( ph -> ( ( 2 x. ( M / 2 ) ) x. X ) = ( 2 x. ( ( M / 2 ) x. X ) ) ) |
71 |
18 29 32
|
divcan2d |
|- ( ph -> ( 2 x. ( M / 2 ) ) = M ) |
72 |
71
|
oveq1d |
|- ( ph -> ( ( 2 x. ( M / 2 ) ) x. X ) = ( M x. X ) ) |
73 |
70 72
|
eqtr3d |
|- ( ph -> ( 2 x. ( ( M / 2 ) x. X ) ) = ( M x. X ) ) |
74 |
29 2 55 57
|
divassd |
|- ( ph -> ( ( 2 x. C ) / 4 ) = ( 2 x. ( C / 4 ) ) ) |
75 |
23
|
oveq2i |
|- ( ( 2 x. C ) / ( 2 x. 2 ) ) = ( ( 2 x. C ) / 4 ) |
76 |
2 29 29 32 32
|
divcan5d |
|- ( ph -> ( ( 2 x. C ) / ( 2 x. 2 ) ) = ( C / 2 ) ) |
77 |
75 76
|
syl5eqr |
|- ( ph -> ( ( 2 x. C ) / 4 ) = ( C / 2 ) ) |
78 |
74 77
|
eqtr3d |
|- ( ph -> ( 2 x. ( C / 4 ) ) = ( C / 2 ) ) |
79 |
73 78
|
oveq12d |
|- ( ph -> ( ( 2 x. ( ( M / 2 ) x. X ) ) - ( 2 x. ( C / 4 ) ) ) = ( ( M x. X ) - ( C / 2 ) ) ) |
80 |
69 79
|
eqtrd |
|- ( ph -> ( 2 x. ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) ) = ( ( M x. X ) - ( C / 2 ) ) ) |
81 |
80
|
oveq1d |
|- ( ph -> ( ( 2 x. ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) ) / ( 2 x. S ) ) = ( ( ( M x. X ) - ( C / 2 ) ) / ( 2 x. S ) ) ) |
82 |
68 81
|
eqtr3d |
|- ( ph -> ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) = ( ( ( M x. X ) - ( C / 2 ) ) / ( 2 x. S ) ) ) |
83 |
82
|
oveq1d |
|- ( ph -> ( ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ^ 2 ) = ( ( ( ( M x. X ) - ( C / 2 ) ) / ( 2 x. S ) ) ^ 2 ) ) |
84 |
18 3
|
mulcld |
|- ( ph -> ( M x. X ) e. CC ) |
85 |
2
|
halfcld |
|- ( ph -> ( C / 2 ) e. CC ) |
86 |
84 85
|
subcld |
|- ( ph -> ( ( M x. X ) - ( C / 2 ) ) e. CC ) |
87 |
86 16 63
|
sqdivd |
|- ( ph -> ( ( ( ( M x. X ) - ( C / 2 ) ) / ( 2 x. S ) ) ^ 2 ) = ( ( ( ( M x. X ) - ( C / 2 ) ) ^ 2 ) / ( ( 2 x. S ) ^ 2 ) ) ) |
88 |
18
|
sqcld |
|- ( ph -> ( M ^ 2 ) e. CC ) |
89 |
88 13
|
mulcld |
|- ( ph -> ( ( M ^ 2 ) x. ( X ^ 2 ) ) e. CC ) |
90 |
84 2
|
mulcld |
|- ( ph -> ( ( M x. X ) x. C ) e. CC ) |
91 |
89 90
|
subcld |
|- ( ph -> ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) e. CC ) |
92 |
2
|
sqcld |
|- ( ph -> ( C ^ 2 ) e. CC ) |
93 |
92 55 57
|
divcld |
|- ( ph -> ( ( C ^ 2 ) / 4 ) e. CC ) |
94 |
91 93 18 6
|
divdird |
|- ( ph -> ( ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) + ( ( C ^ 2 ) / 4 ) ) / M ) = ( ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) / M ) + ( ( ( C ^ 2 ) / 4 ) / M ) ) ) |
95 |
|
binom2sub |
|- ( ( ( M x. X ) e. CC /\ ( C / 2 ) e. CC ) -> ( ( ( M x. X ) - ( C / 2 ) ) ^ 2 ) = ( ( ( ( M x. X ) ^ 2 ) - ( 2 x. ( ( M x. X ) x. ( C / 2 ) ) ) ) + ( ( C / 2 ) ^ 2 ) ) ) |
96 |
84 85 95
|
syl2anc |
|- ( ph -> ( ( ( M x. X ) - ( C / 2 ) ) ^ 2 ) = ( ( ( ( M x. X ) ^ 2 ) - ( 2 x. ( ( M x. X ) x. ( C / 2 ) ) ) ) + ( ( C / 2 ) ^ 2 ) ) ) |
97 |
18 3
|
sqmuld |
|- ( ph -> ( ( M x. X ) ^ 2 ) = ( ( M ^ 2 ) x. ( X ^ 2 ) ) ) |
98 |
29 84 85
|
mul12d |
|- ( ph -> ( 2 x. ( ( M x. X ) x. ( C / 2 ) ) ) = ( ( M x. X ) x. ( 2 x. ( C / 2 ) ) ) ) |
99 |
2 29 32
|
divcan2d |
|- ( ph -> ( 2 x. ( C / 2 ) ) = C ) |
100 |
99
|
oveq2d |
|- ( ph -> ( ( M x. X ) x. ( 2 x. ( C / 2 ) ) ) = ( ( M x. X ) x. C ) ) |
101 |
98 100
|
eqtrd |
|- ( ph -> ( 2 x. ( ( M x. X ) x. ( C / 2 ) ) ) = ( ( M x. X ) x. C ) ) |
102 |
97 101
|
oveq12d |
|- ( ph -> ( ( ( M x. X ) ^ 2 ) - ( 2 x. ( ( M x. X ) x. ( C / 2 ) ) ) ) = ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) ) |
103 |
2 29 32
|
sqdivd |
|- ( ph -> ( ( C / 2 ) ^ 2 ) = ( ( C ^ 2 ) / ( 2 ^ 2 ) ) ) |
104 |
|
sq2 |
|- ( 2 ^ 2 ) = 4 |
105 |
104
|
oveq2i |
|- ( ( C ^ 2 ) / ( 2 ^ 2 ) ) = ( ( C ^ 2 ) / 4 ) |
106 |
103 105
|
eqtrdi |
|- ( ph -> ( ( C / 2 ) ^ 2 ) = ( ( C ^ 2 ) / 4 ) ) |
107 |
102 106
|
oveq12d |
|- ( ph -> ( ( ( ( M x. X ) ^ 2 ) - ( 2 x. ( ( M x. X ) x. ( C / 2 ) ) ) ) + ( ( C / 2 ) ^ 2 ) ) = ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) + ( ( C ^ 2 ) / 4 ) ) ) |
108 |
96 107
|
eqtr2d |
|- ( ph -> ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) + ( ( C ^ 2 ) / 4 ) ) = ( ( ( M x. X ) - ( C / 2 ) ) ^ 2 ) ) |
109 |
108 5
|
oveq12d |
|- ( ph -> ( ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) + ( ( C ^ 2 ) / 4 ) ) / M ) = ( ( ( ( M x. X ) - ( C / 2 ) ) ^ 2 ) / ( ( 2 x. S ) ^ 2 ) ) ) |
110 |
89 90 18 6
|
divsubdird |
|- ( ph -> ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) / M ) = ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) / M ) - ( ( ( M x. X ) x. C ) / M ) ) ) |
111 |
88 13 18 6
|
div23d |
|- ( ph -> ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) / M ) = ( ( ( M ^ 2 ) / M ) x. ( X ^ 2 ) ) ) |
112 |
18
|
sqvald |
|- ( ph -> ( M ^ 2 ) = ( M x. M ) ) |
113 |
112
|
oveq1d |
|- ( ph -> ( ( M ^ 2 ) / M ) = ( ( M x. M ) / M ) ) |
114 |
18 18 6
|
divcan3d |
|- ( ph -> ( ( M x. M ) / M ) = M ) |
115 |
113 114
|
eqtrd |
|- ( ph -> ( ( M ^ 2 ) / M ) = M ) |
116 |
115
|
oveq1d |
|- ( ph -> ( ( ( M ^ 2 ) / M ) x. ( X ^ 2 ) ) = ( M x. ( X ^ 2 ) ) ) |
117 |
111 116
|
eqtrd |
|- ( ph -> ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) / M ) = ( M x. ( X ^ 2 ) ) ) |
118 |
18 3 2
|
mul32d |
|- ( ph -> ( ( M x. X ) x. C ) = ( ( M x. C ) x. X ) ) |
119 |
18 2 3
|
mulassd |
|- ( ph -> ( ( M x. C ) x. X ) = ( M x. ( C x. X ) ) ) |
120 |
118 119
|
eqtrd |
|- ( ph -> ( ( M x. X ) x. C ) = ( M x. ( C x. X ) ) ) |
121 |
120
|
oveq1d |
|- ( ph -> ( ( ( M x. X ) x. C ) / M ) = ( ( M x. ( C x. X ) ) / M ) ) |
122 |
2 3
|
mulcld |
|- ( ph -> ( C x. X ) e. CC ) |
123 |
122 18 6
|
divcan3d |
|- ( ph -> ( ( M x. ( C x. X ) ) / M ) = ( C x. X ) ) |
124 |
121 123
|
eqtrd |
|- ( ph -> ( ( ( M x. X ) x. C ) / M ) = ( C x. X ) ) |
125 |
117 124
|
oveq12d |
|- ( ph -> ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) / M ) - ( ( ( M x. X ) x. C ) / M ) ) = ( ( M x. ( X ^ 2 ) ) - ( C x. X ) ) ) |
126 |
110 125
|
eqtrd |
|- ( ph -> ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) / M ) = ( ( M x. ( X ^ 2 ) ) - ( C x. X ) ) ) |
127 |
126
|
oveq1d |
|- ( ph -> ( ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) / M ) + ( ( ( C ^ 2 ) / 4 ) / M ) ) = ( ( ( M x. ( X ^ 2 ) ) - ( C x. X ) ) + ( ( ( C ^ 2 ) / 4 ) / M ) ) ) |
128 |
93 18 6
|
divcld |
|- ( ph -> ( ( ( C ^ 2 ) / 4 ) / M ) e. CC ) |
129 |
43 122 128
|
subsubd |
|- ( ph -> ( ( M x. ( X ^ 2 ) ) - ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) = ( ( ( M x. ( X ^ 2 ) ) - ( C x. X ) ) + ( ( ( C ^ 2 ) / 4 ) / M ) ) ) |
130 |
127 129
|
eqtr4d |
|- ( ph -> ( ( ( ( ( M ^ 2 ) x. ( X ^ 2 ) ) - ( ( M x. X ) x. C ) ) / M ) + ( ( ( C ^ 2 ) / 4 ) / M ) ) = ( ( M x. ( X ^ 2 ) ) - ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) ) |
131 |
94 109 130
|
3eqtr3d |
|- ( ph -> ( ( ( ( M x. X ) - ( C / 2 ) ) ^ 2 ) / ( ( 2 x. S ) ^ 2 ) ) = ( ( M x. ( X ^ 2 ) ) - ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) ) |
132 |
83 87 131
|
3eqtrd |
|- ( ph -> ( ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ^ 2 ) = ( ( M x. ( X ^ 2 ) ) - ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) ) |
133 |
51 132
|
oveq12d |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) ^ 2 ) - ( ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ^ 2 ) ) = ( ( ( M x. ( X ^ 2 ) ) + ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) ) - ( ( M x. ( X ^ 2 ) ) - ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) ) ) |
134 |
48 49
|
addcld |
|- ( ph -> ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) e. CC ) |
135 |
122 128
|
subcld |
|- ( ph -> ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) e. CC ) |
136 |
43 134 135
|
pnncand |
|- ( ph -> ( ( ( M x. ( X ^ 2 ) ) + ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) ) - ( ( M x. ( X ^ 2 ) ) - ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) ) = ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) + ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) ) |
137 |
128
|
negcld |
|- ( ph -> -u ( ( ( C ^ 2 ) / 4 ) / M ) e. CC ) |
138 |
48 49 122 137
|
add4d |
|- ( ph -> ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) + ( ( C x. X ) + -u ( ( ( C ^ 2 ) / 4 ) / M ) ) ) = ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( C x. X ) ) + ( ( ( ( M + B ) / 2 ) ^ 2 ) + -u ( ( ( C ^ 2 ) / 4 ) / M ) ) ) ) |
139 |
122 128
|
negsubd |
|- ( ph -> ( ( C x. X ) + -u ( ( ( C ^ 2 ) / 4 ) / M ) ) = ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) |
140 |
139
|
oveq2d |
|- ( ph -> ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) + ( ( C x. X ) + -u ( ( ( C ^ 2 ) / 4 ) / M ) ) ) = ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) + ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) ) |
141 |
49 128
|
negsubd |
|- ( ph -> ( ( ( ( M + B ) / 2 ) ^ 2 ) + -u ( ( ( C ^ 2 ) / 4 ) / M ) ) = ( ( ( ( M + B ) / 2 ) ^ 2 ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) |
142 |
1 2 3 4 5 6 7 8 9 10
|
dquartlem2 |
|- ( ph -> ( ( ( ( M + B ) / 2 ) ^ 2 ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) = D ) |
143 |
141 142
|
eqtrd |
|- ( ph -> ( ( ( ( M + B ) / 2 ) ^ 2 ) + -u ( ( ( C ^ 2 ) / 4 ) / M ) ) = D ) |
144 |
143
|
oveq2d |
|- ( ph -> ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( C x. X ) ) + ( ( ( ( M + B ) / 2 ) ^ 2 ) + -u ( ( ( C ^ 2 ) / 4 ) / M ) ) ) = ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( C x. X ) ) + D ) ) |
145 |
48 122 9
|
addassd |
|- ( ph -> ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( C x. X ) ) + D ) = ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) ) |
146 |
144 145
|
eqtrd |
|- ( ph -> ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( C x. X ) ) + ( ( ( ( M + B ) / 2 ) ^ 2 ) + -u ( ( ( C ^ 2 ) / 4 ) / M ) ) ) = ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) ) |
147 |
138 140 146
|
3eqtr3d |
|- ( ph -> ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( ( M + B ) / 2 ) ^ 2 ) ) + ( ( C x. X ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) = ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) ) |
148 |
133 136 147
|
3eqtrd |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) ^ 2 ) - ( ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ^ 2 ) ) = ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) ) |
149 |
13 20
|
addcld |
|- ( ph -> ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) e. CC ) |
150 |
59 4 67
|
divcld |
|- ( ph -> ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) e. CC ) |
151 |
|
subsq |
|- ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) e. CC /\ ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) e. CC ) -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) ^ 2 ) - ( ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ^ 2 ) ) = ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) x. ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) ) |
152 |
149 150 151
|
syl2anc |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) ^ 2 ) - ( ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ^ 2 ) ) = ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) x. ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) ) |
153 |
148 152
|
eqtr3d |
|- ( ph -> ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) = ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) x. ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) ) |
154 |
153
|
eqeq1d |
|- ( ph -> ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) = 0 <-> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) x. ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) = 0 ) ) |
155 |
149 150
|
addcld |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) e. CC ) |
156 |
149 150
|
subcld |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) e. CC ) |
157 |
155 156
|
mul0ord |
|- ( ph -> ( ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) x. ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) = 0 <-> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 \/ ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 ) ) ) |
158 |
1 2 3 4 5 6 7 8
|
dquartlem1 |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 <-> ( X = ( -u S + I ) \/ X = ( -u S - I ) ) ) ) |
159 |
4
|
negcld |
|- ( ph -> -u S e. CC ) |
160 |
|
sqneg |
|- ( ( 2 x. S ) e. CC -> ( -u ( 2 x. S ) ^ 2 ) = ( ( 2 x. S ) ^ 2 ) ) |
161 |
16 160
|
syl |
|- ( ph -> ( -u ( 2 x. S ) ^ 2 ) = ( ( 2 x. S ) ^ 2 ) ) |
162 |
5 161
|
eqtr4d |
|- ( ph -> M = ( -u ( 2 x. S ) ^ 2 ) ) |
163 |
|
mulneg2 |
|- ( ( 2 e. CC /\ S e. CC ) -> ( 2 x. -u S ) = -u ( 2 x. S ) ) |
164 |
14 4 163
|
sylancr |
|- ( ph -> ( 2 x. -u S ) = -u ( 2 x. S ) ) |
165 |
164
|
oveq1d |
|- ( ph -> ( ( 2 x. -u S ) ^ 2 ) = ( -u ( 2 x. S ) ^ 2 ) ) |
166 |
162 165
|
eqtr4d |
|- ( ph -> M = ( ( 2 x. -u S ) ^ 2 ) ) |
167 |
4
|
sqcld |
|- ( ph -> ( S ^ 2 ) e. CC ) |
168 |
167
|
negcld |
|- ( ph -> -u ( S ^ 2 ) e. CC ) |
169 |
1
|
halfcld |
|- ( ph -> ( B / 2 ) e. CC ) |
170 |
168 169
|
subcld |
|- ( ph -> ( -u ( S ^ 2 ) - ( B / 2 ) ) e. CC ) |
171 |
58 4 67
|
divcld |
|- ( ph -> ( ( C / 4 ) / S ) e. CC ) |
172 |
170 171
|
negsubd |
|- ( ph -> ( ( -u ( S ^ 2 ) - ( B / 2 ) ) + -u ( ( C / 4 ) / S ) ) = ( ( -u ( S ^ 2 ) - ( B / 2 ) ) - ( ( C / 4 ) / S ) ) ) |
173 |
|
sqneg |
|- ( S e. CC -> ( -u S ^ 2 ) = ( S ^ 2 ) ) |
174 |
4 173
|
syl |
|- ( ph -> ( -u S ^ 2 ) = ( S ^ 2 ) ) |
175 |
174
|
eqcomd |
|- ( ph -> ( S ^ 2 ) = ( -u S ^ 2 ) ) |
176 |
175
|
negeqd |
|- ( ph -> -u ( S ^ 2 ) = -u ( -u S ^ 2 ) ) |
177 |
176
|
oveq1d |
|- ( ph -> ( -u ( S ^ 2 ) - ( B / 2 ) ) = ( -u ( -u S ^ 2 ) - ( B / 2 ) ) ) |
178 |
58 4 67
|
divneg2d |
|- ( ph -> -u ( ( C / 4 ) / S ) = ( ( C / 4 ) / -u S ) ) |
179 |
177 178
|
oveq12d |
|- ( ph -> ( ( -u ( S ^ 2 ) - ( B / 2 ) ) + -u ( ( C / 4 ) / S ) ) = ( ( -u ( -u S ^ 2 ) - ( B / 2 ) ) + ( ( C / 4 ) / -u S ) ) ) |
180 |
12 172 179
|
3eqtr2d |
|- ( ph -> ( J ^ 2 ) = ( ( -u ( -u S ^ 2 ) - ( B / 2 ) ) + ( ( C / 4 ) / -u S ) ) ) |
181 |
1 2 3 159 166 6 11 180
|
dquartlem1 |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / -u S ) ) = 0 <-> ( X = ( -u -u S + J ) \/ X = ( -u -u S - J ) ) ) ) |
182 |
59 4 67
|
divneg2d |
|- ( ph -> -u ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) = ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / -u S ) ) |
183 |
182
|
oveq2d |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + -u ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / -u S ) ) ) |
184 |
149 150
|
negsubd |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + -u ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) |
185 |
183 184
|
eqtr3d |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / -u S ) ) = ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) |
186 |
185
|
eqeq1d |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / -u S ) ) = 0 <-> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 ) ) |
187 |
4
|
negnegd |
|- ( ph -> -u -u S = S ) |
188 |
187
|
oveq1d |
|- ( ph -> ( -u -u S + J ) = ( S + J ) ) |
189 |
188
|
eqeq2d |
|- ( ph -> ( X = ( -u -u S + J ) <-> X = ( S + J ) ) ) |
190 |
187
|
oveq1d |
|- ( ph -> ( -u -u S - J ) = ( S - J ) ) |
191 |
190
|
eqeq2d |
|- ( ph -> ( X = ( -u -u S - J ) <-> X = ( S - J ) ) ) |
192 |
189 191
|
orbi12d |
|- ( ph -> ( ( X = ( -u -u S + J ) \/ X = ( -u -u S - J ) ) <-> ( X = ( S + J ) \/ X = ( S - J ) ) ) ) |
193 |
181 186 192
|
3bitr3d |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 <-> ( X = ( S + J ) \/ X = ( S - J ) ) ) ) |
194 |
158 193
|
orbi12d |
|- ( ph -> ( ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 \/ ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) - ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 ) <-> ( ( X = ( -u S + I ) \/ X = ( -u S - I ) ) \/ ( X = ( S + J ) \/ X = ( S - J ) ) ) ) ) |
195 |
154 157 194
|
3bitrd |
|- ( ph -> ( ( ( ( X ^ 4 ) + ( B x. ( X ^ 2 ) ) ) + ( ( C x. X ) + D ) ) = 0 <-> ( ( X = ( -u S + I ) \/ X = ( -u S - I ) ) \/ ( X = ( S + J ) \/ X = ( S - J ) ) ) ) ) |