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 |
3
|
sqcld |
|- ( ph -> ( X ^ 2 ) e. CC ) |
10 |
|
2cn |
|- 2 e. CC |
11 |
|
mulcl |
|- ( ( 2 e. CC /\ S e. CC ) -> ( 2 x. S ) e. CC ) |
12 |
10 4 11
|
sylancr |
|- ( ph -> ( 2 x. S ) e. CC ) |
13 |
12
|
sqcld |
|- ( ph -> ( ( 2 x. S ) ^ 2 ) e. CC ) |
14 |
5 13
|
eqeltrd |
|- ( ph -> M e. CC ) |
15 |
14 1
|
addcld |
|- ( ph -> ( M + B ) e. CC ) |
16 |
15
|
halfcld |
|- ( ph -> ( ( M + B ) / 2 ) e. CC ) |
17 |
9 16
|
addcld |
|- ( ph -> ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) e. CC ) |
18 |
14
|
halfcld |
|- ( ph -> ( M / 2 ) e. CC ) |
19 |
18 3
|
mulcld |
|- ( ph -> ( ( M / 2 ) x. X ) e. CC ) |
20 |
|
4cn |
|- 4 e. CC |
21 |
20
|
a1i |
|- ( ph -> 4 e. CC ) |
22 |
|
4ne0 |
|- 4 =/= 0 |
23 |
22
|
a1i |
|- ( ph -> 4 =/= 0 ) |
24 |
2 21 23
|
divcld |
|- ( ph -> ( C / 4 ) e. CC ) |
25 |
19 24
|
subcld |
|- ( ph -> ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) e. CC ) |
26 |
5 6
|
eqnetrrd |
|- ( ph -> ( ( 2 x. S ) ^ 2 ) =/= 0 ) |
27 |
|
sqne0 |
|- ( ( 2 x. S ) e. CC -> ( ( ( 2 x. S ) ^ 2 ) =/= 0 <-> ( 2 x. S ) =/= 0 ) ) |
28 |
12 27
|
syl |
|- ( ph -> ( ( ( 2 x. S ) ^ 2 ) =/= 0 <-> ( 2 x. S ) =/= 0 ) ) |
29 |
26 28
|
mpbid |
|- ( ph -> ( 2 x. S ) =/= 0 ) |
30 |
|
mulne0b |
|- ( ( 2 e. CC /\ S e. CC ) -> ( ( 2 =/= 0 /\ S =/= 0 ) <-> ( 2 x. S ) =/= 0 ) ) |
31 |
10 4 30
|
sylancr |
|- ( ph -> ( ( 2 =/= 0 /\ S =/= 0 ) <-> ( 2 x. S ) =/= 0 ) ) |
32 |
29 31
|
mpbird |
|- ( ph -> ( 2 =/= 0 /\ S =/= 0 ) ) |
33 |
32
|
simprd |
|- ( ph -> S =/= 0 ) |
34 |
25 4 33
|
divcld |
|- ( ph -> ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) e. CC ) |
35 |
17 34
|
addcld |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) e. CC ) |
36 |
10
|
a1i |
|- ( ph -> 2 e. CC ) |
37 |
|
2ne0 |
|- 2 =/= 0 |
38 |
37
|
a1i |
|- ( ph -> 2 =/= 0 ) |
39 |
35 36 38
|
diveq0ad |
|- ( ph -> ( ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) / 2 ) = 0 <-> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 ) ) |
40 |
9 16 34
|
addassd |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = ( ( X ^ 2 ) + ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) ) |
41 |
40
|
oveq1d |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) / 2 ) = ( ( ( X ^ 2 ) + ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) / 2 ) ) |
42 |
16 34
|
addcld |
|- ( ph -> ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) e. CC ) |
43 |
9 42 36 38
|
divdird |
|- ( ph -> ( ( ( X ^ 2 ) + ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) ) / 2 ) = ( ( ( X ^ 2 ) / 2 ) + ( ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) / 2 ) ) ) |
44 |
9 36 38
|
divrec2d |
|- ( ph -> ( ( X ^ 2 ) / 2 ) = ( ( 1 / 2 ) x. ( X ^ 2 ) ) ) |
45 |
19 24 4 33
|
divsubdird |
|- ( ph -> ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) = ( ( ( ( M / 2 ) x. X ) / S ) - ( ( C / 4 ) / S ) ) ) |
46 |
18 3 4 33
|
div23d |
|- ( ph -> ( ( ( M / 2 ) x. X ) / S ) = ( ( ( M / 2 ) / S ) x. X ) ) |
47 |
4
|
sqvald |
|- ( ph -> ( S ^ 2 ) = ( S x. S ) ) |
48 |
47
|
oveq2d |
|- ( ph -> ( 2 x. ( S ^ 2 ) ) = ( 2 x. ( S x. S ) ) ) |
49 |
|
sqmul |
|- ( ( 2 e. CC /\ S e. CC ) -> ( ( 2 x. S ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( S ^ 2 ) ) ) |
50 |
10 4 49
|
sylancr |
|- ( ph -> ( ( 2 x. S ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( S ^ 2 ) ) ) |
51 |
10
|
sqvali |
|- ( 2 ^ 2 ) = ( 2 x. 2 ) |
52 |
51
|
oveq1i |
|- ( ( 2 ^ 2 ) x. ( S ^ 2 ) ) = ( ( 2 x. 2 ) x. ( S ^ 2 ) ) |
53 |
50 52
|
eqtrdi |
|- ( ph -> ( ( 2 x. S ) ^ 2 ) = ( ( 2 x. 2 ) x. ( S ^ 2 ) ) ) |
54 |
4
|
sqcld |
|- ( ph -> ( S ^ 2 ) e. CC ) |
55 |
36 36 54
|
mulassd |
|- ( ph -> ( ( 2 x. 2 ) x. ( S ^ 2 ) ) = ( 2 x. ( 2 x. ( S ^ 2 ) ) ) ) |
56 |
5 53 55
|
3eqtrd |
|- ( ph -> M = ( 2 x. ( 2 x. ( S ^ 2 ) ) ) ) |
57 |
56
|
oveq1d |
|- ( ph -> ( M / 2 ) = ( ( 2 x. ( 2 x. ( S ^ 2 ) ) ) / 2 ) ) |
58 |
|
mulcl |
|- ( ( 2 e. CC /\ ( S ^ 2 ) e. CC ) -> ( 2 x. ( S ^ 2 ) ) e. CC ) |
59 |
10 54 58
|
sylancr |
|- ( ph -> ( 2 x. ( S ^ 2 ) ) e. CC ) |
60 |
59 36 38
|
divcan3d |
|- ( ph -> ( ( 2 x. ( 2 x. ( S ^ 2 ) ) ) / 2 ) = ( 2 x. ( S ^ 2 ) ) ) |
61 |
57 60
|
eqtrd |
|- ( ph -> ( M / 2 ) = ( 2 x. ( S ^ 2 ) ) ) |
62 |
36 4 4
|
mulassd |
|- ( ph -> ( ( 2 x. S ) x. S ) = ( 2 x. ( S x. S ) ) ) |
63 |
48 61 62
|
3eqtr4d |
|- ( ph -> ( M / 2 ) = ( ( 2 x. S ) x. S ) ) |
64 |
63
|
oveq1d |
|- ( ph -> ( ( M / 2 ) / S ) = ( ( ( 2 x. S ) x. S ) / S ) ) |
65 |
12 4 33
|
divcan4d |
|- ( ph -> ( ( ( 2 x. S ) x. S ) / S ) = ( 2 x. S ) ) |
66 |
64 65
|
eqtrd |
|- ( ph -> ( ( M / 2 ) / S ) = ( 2 x. S ) ) |
67 |
66
|
oveq1d |
|- ( ph -> ( ( ( M / 2 ) / S ) x. X ) = ( ( 2 x. S ) x. X ) ) |
68 |
46 67
|
eqtrd |
|- ( ph -> ( ( ( M / 2 ) x. X ) / S ) = ( ( 2 x. S ) x. X ) ) |
69 |
68
|
oveq1d |
|- ( ph -> ( ( ( ( M / 2 ) x. X ) / S ) - ( ( C / 4 ) / S ) ) = ( ( ( 2 x. S ) x. X ) - ( ( C / 4 ) / S ) ) ) |
70 |
45 69
|
eqtrd |
|- ( ph -> ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) = ( ( ( 2 x. S ) x. X ) - ( ( C / 4 ) / S ) ) ) |
71 |
70
|
oveq2d |
|- ( ph -> ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = ( ( ( M + B ) / 2 ) + ( ( ( 2 x. S ) x. X ) - ( ( C / 4 ) / S ) ) ) ) |
72 |
12 3
|
mulcld |
|- ( ph -> ( ( 2 x. S ) x. X ) e. CC ) |
73 |
24 4 33
|
divcld |
|- ( ph -> ( ( C / 4 ) / S ) e. CC ) |
74 |
16 72 73
|
addsub12d |
|- ( ph -> ( ( ( M + B ) / 2 ) + ( ( ( 2 x. S ) x. X ) - ( ( C / 4 ) / S ) ) ) = ( ( ( 2 x. S ) x. X ) + ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) ) ) |
75 |
71 74
|
eqtrd |
|- ( ph -> ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = ( ( ( 2 x. S ) x. X ) + ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) ) ) |
76 |
75
|
oveq1d |
|- ( ph -> ( ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) / 2 ) = ( ( ( ( 2 x. S ) x. X ) + ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) ) / 2 ) ) |
77 |
16 73
|
subcld |
|- ( ph -> ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) e. CC ) |
78 |
72 77 36 38
|
divdird |
|- ( ph -> ( ( ( ( 2 x. S ) x. X ) + ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) ) / 2 ) = ( ( ( ( 2 x. S ) x. X ) / 2 ) + ( ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) / 2 ) ) ) |
79 |
36 4 3
|
mulassd |
|- ( ph -> ( ( 2 x. S ) x. X ) = ( 2 x. ( S x. X ) ) ) |
80 |
79
|
oveq1d |
|- ( ph -> ( ( ( 2 x. S ) x. X ) / 2 ) = ( ( 2 x. ( S x. X ) ) / 2 ) ) |
81 |
4 3
|
mulcld |
|- ( ph -> ( S x. X ) e. CC ) |
82 |
81 36 38
|
divcan3d |
|- ( ph -> ( ( 2 x. ( S x. X ) ) / 2 ) = ( S x. X ) ) |
83 |
80 82
|
eqtrd |
|- ( ph -> ( ( ( 2 x. S ) x. X ) / 2 ) = ( S x. X ) ) |
84 |
54
|
negcld |
|- ( ph -> -u ( S ^ 2 ) e. CC ) |
85 |
1
|
halfcld |
|- ( ph -> ( B / 2 ) e. CC ) |
86 |
84 85
|
subcld |
|- ( ph -> ( -u ( S ^ 2 ) - ( B / 2 ) ) e. CC ) |
87 |
54 86 73
|
subsub4d |
|- ( ph -> ( ( ( S ^ 2 ) - ( -u ( S ^ 2 ) - ( B / 2 ) ) ) - ( ( C / 4 ) / S ) ) = ( ( S ^ 2 ) - ( ( -u ( S ^ 2 ) - ( B / 2 ) ) + ( ( C / 4 ) / S ) ) ) ) |
88 |
14 1 36 38
|
divdird |
|- ( ph -> ( ( M + B ) / 2 ) = ( ( M / 2 ) + ( B / 2 ) ) ) |
89 |
54
|
2timesd |
|- ( ph -> ( 2 x. ( S ^ 2 ) ) = ( ( S ^ 2 ) + ( S ^ 2 ) ) ) |
90 |
61 89
|
eqtrd |
|- ( ph -> ( M / 2 ) = ( ( S ^ 2 ) + ( S ^ 2 ) ) ) |
91 |
90
|
oveq1d |
|- ( ph -> ( ( M / 2 ) + ( B / 2 ) ) = ( ( ( S ^ 2 ) + ( S ^ 2 ) ) + ( B / 2 ) ) ) |
92 |
88 91
|
eqtrd |
|- ( ph -> ( ( M + B ) / 2 ) = ( ( ( S ^ 2 ) + ( S ^ 2 ) ) + ( B / 2 ) ) ) |
93 |
54 54 85
|
addassd |
|- ( ph -> ( ( ( S ^ 2 ) + ( S ^ 2 ) ) + ( B / 2 ) ) = ( ( S ^ 2 ) + ( ( S ^ 2 ) + ( B / 2 ) ) ) ) |
94 |
54 85
|
addcld |
|- ( ph -> ( ( S ^ 2 ) + ( B / 2 ) ) e. CC ) |
95 |
54 94
|
subnegd |
|- ( ph -> ( ( S ^ 2 ) - -u ( ( S ^ 2 ) + ( B / 2 ) ) ) = ( ( S ^ 2 ) + ( ( S ^ 2 ) + ( B / 2 ) ) ) ) |
96 |
54 85
|
negdi2d |
|- ( ph -> -u ( ( S ^ 2 ) + ( B / 2 ) ) = ( -u ( S ^ 2 ) - ( B / 2 ) ) ) |
97 |
96
|
oveq2d |
|- ( ph -> ( ( S ^ 2 ) - -u ( ( S ^ 2 ) + ( B / 2 ) ) ) = ( ( S ^ 2 ) - ( -u ( S ^ 2 ) - ( B / 2 ) ) ) ) |
98 |
95 97
|
eqtr3d |
|- ( ph -> ( ( S ^ 2 ) + ( ( S ^ 2 ) + ( B / 2 ) ) ) = ( ( S ^ 2 ) - ( -u ( S ^ 2 ) - ( B / 2 ) ) ) ) |
99 |
92 93 98
|
3eqtrd |
|- ( ph -> ( ( M + B ) / 2 ) = ( ( S ^ 2 ) - ( -u ( S ^ 2 ) - ( B / 2 ) ) ) ) |
100 |
99
|
oveq1d |
|- ( ph -> ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) = ( ( ( S ^ 2 ) - ( -u ( S ^ 2 ) - ( B / 2 ) ) ) - ( ( C / 4 ) / S ) ) ) |
101 |
8
|
oveq2d |
|- ( ph -> ( ( S ^ 2 ) - ( I ^ 2 ) ) = ( ( S ^ 2 ) - ( ( -u ( S ^ 2 ) - ( B / 2 ) ) + ( ( C / 4 ) / S ) ) ) ) |
102 |
87 100 101
|
3eqtr4d |
|- ( ph -> ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) = ( ( S ^ 2 ) - ( I ^ 2 ) ) ) |
103 |
102
|
oveq1d |
|- ( ph -> ( ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) / 2 ) = ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) |
104 |
83 103
|
oveq12d |
|- ( ph -> ( ( ( ( 2 x. S ) x. X ) / 2 ) + ( ( ( ( M + B ) / 2 ) - ( ( C / 4 ) / S ) ) / 2 ) ) = ( ( S x. X ) + ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) |
105 |
76 78 104
|
3eqtrd |
|- ( ph -> ( ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) / 2 ) = ( ( S x. X ) + ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) |
106 |
44 105
|
oveq12d |
|- ( ph -> ( ( ( X ^ 2 ) / 2 ) + ( ( ( ( M + B ) / 2 ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) / 2 ) ) = ( ( ( 1 / 2 ) x. ( X ^ 2 ) ) + ( ( S x. X ) + ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) ) |
107 |
41 43 106
|
3eqtrd |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) / 2 ) = ( ( ( 1 / 2 ) x. ( X ^ 2 ) ) + ( ( S x. X ) + ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) ) |
108 |
107
|
eqeq1d |
|- ( ph -> ( ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) / 2 ) = 0 <-> ( ( ( 1 / 2 ) x. ( X ^ 2 ) ) + ( ( S x. X ) + ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) = 0 ) ) |
109 |
39 108
|
bitr3d |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 <-> ( ( ( 1 / 2 ) x. ( X ^ 2 ) ) + ( ( S x. X ) + ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) = 0 ) ) |
110 |
|
ax-1cn |
|- 1 e. CC |
111 |
|
halfcl |
|- ( 1 e. CC -> ( 1 / 2 ) e. CC ) |
112 |
110 111
|
mp1i |
|- ( ph -> ( 1 / 2 ) e. CC ) |
113 |
|
ax-1ne0 |
|- 1 =/= 0 |
114 |
110 10 113 37
|
divne0i |
|- ( 1 / 2 ) =/= 0 |
115 |
114
|
a1i |
|- ( ph -> ( 1 / 2 ) =/= 0 ) |
116 |
7
|
sqcld |
|- ( ph -> ( I ^ 2 ) e. CC ) |
117 |
54 116
|
subcld |
|- ( ph -> ( ( S ^ 2 ) - ( I ^ 2 ) ) e. CC ) |
118 |
117
|
halfcld |
|- ( ph -> ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) e. CC ) |
119 |
110
|
a1i |
|- ( ph -> 1 e. CC ) |
120 |
|
2cnne0 |
|- ( 2 e. CC /\ 2 =/= 0 ) |
121 |
120
|
a1i |
|- ( ph -> ( 2 e. CC /\ 2 =/= 0 ) ) |
122 |
|
divmuldiv |
|- ( ( ( 1 e. CC /\ ( ( S ^ 2 ) - ( I ^ 2 ) ) e. CC ) /\ ( ( 2 e. CC /\ 2 =/= 0 ) /\ ( 2 e. CC /\ 2 =/= 0 ) ) ) -> ( ( 1 / 2 ) x. ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) = ( ( 1 x. ( ( S ^ 2 ) - ( I ^ 2 ) ) ) / ( 2 x. 2 ) ) ) |
123 |
119 117 121 121 122
|
syl22anc |
|- ( ph -> ( ( 1 / 2 ) x. ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) = ( ( 1 x. ( ( S ^ 2 ) - ( I ^ 2 ) ) ) / ( 2 x. 2 ) ) ) |
124 |
117
|
mulid2d |
|- ( ph -> ( 1 x. ( ( S ^ 2 ) - ( I ^ 2 ) ) ) = ( ( S ^ 2 ) - ( I ^ 2 ) ) ) |
125 |
|
2t2e4 |
|- ( 2 x. 2 ) = 4 |
126 |
125
|
a1i |
|- ( ph -> ( 2 x. 2 ) = 4 ) |
127 |
124 126
|
oveq12d |
|- ( ph -> ( ( 1 x. ( ( S ^ 2 ) - ( I ^ 2 ) ) ) / ( 2 x. 2 ) ) = ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 4 ) ) |
128 |
123 127
|
eqtrd |
|- ( ph -> ( ( 1 / 2 ) x. ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) = ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 4 ) ) |
129 |
128
|
oveq2d |
|- ( ph -> ( 4 x. ( ( 1 / 2 ) x. ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) = ( 4 x. ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 4 ) ) ) |
130 |
117 21 23
|
divcan2d |
|- ( ph -> ( 4 x. ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 4 ) ) = ( ( S ^ 2 ) - ( I ^ 2 ) ) ) |
131 |
129 130
|
eqtrd |
|- ( ph -> ( 4 x. ( ( 1 / 2 ) x. ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) = ( ( S ^ 2 ) - ( I ^ 2 ) ) ) |
132 |
131
|
oveq2d |
|- ( ph -> ( ( S ^ 2 ) - ( 4 x. ( ( 1 / 2 ) x. ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) ) = ( ( S ^ 2 ) - ( ( S ^ 2 ) - ( I ^ 2 ) ) ) ) |
133 |
54 116
|
nncand |
|- ( ph -> ( ( S ^ 2 ) - ( ( S ^ 2 ) - ( I ^ 2 ) ) ) = ( I ^ 2 ) ) |
134 |
132 133
|
eqtr2d |
|- ( ph -> ( I ^ 2 ) = ( ( S ^ 2 ) - ( 4 x. ( ( 1 / 2 ) x. ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) ) ) |
135 |
112 115 4 118 3 7 134
|
quad2 |
|- ( ph -> ( ( ( ( 1 / 2 ) x. ( X ^ 2 ) ) + ( ( S x. X ) + ( ( ( S ^ 2 ) - ( I ^ 2 ) ) / 2 ) ) ) = 0 <-> ( X = ( ( -u S + I ) / ( 2 x. ( 1 / 2 ) ) ) \/ X = ( ( -u S - I ) / ( 2 x. ( 1 / 2 ) ) ) ) ) ) |
136 |
10 37
|
recidi |
|- ( 2 x. ( 1 / 2 ) ) = 1 |
137 |
136
|
oveq2i |
|- ( ( -u S + I ) / ( 2 x. ( 1 / 2 ) ) ) = ( ( -u S + I ) / 1 ) |
138 |
4
|
negcld |
|- ( ph -> -u S e. CC ) |
139 |
138 7
|
addcld |
|- ( ph -> ( -u S + I ) e. CC ) |
140 |
139
|
div1d |
|- ( ph -> ( ( -u S + I ) / 1 ) = ( -u S + I ) ) |
141 |
137 140
|
syl5eq |
|- ( ph -> ( ( -u S + I ) / ( 2 x. ( 1 / 2 ) ) ) = ( -u S + I ) ) |
142 |
141
|
eqeq2d |
|- ( ph -> ( X = ( ( -u S + I ) / ( 2 x. ( 1 / 2 ) ) ) <-> X = ( -u S + I ) ) ) |
143 |
136
|
oveq2i |
|- ( ( -u S - I ) / ( 2 x. ( 1 / 2 ) ) ) = ( ( -u S - I ) / 1 ) |
144 |
138 7
|
subcld |
|- ( ph -> ( -u S - I ) e. CC ) |
145 |
144
|
div1d |
|- ( ph -> ( ( -u S - I ) / 1 ) = ( -u S - I ) ) |
146 |
143 145
|
syl5eq |
|- ( ph -> ( ( -u S - I ) / ( 2 x. ( 1 / 2 ) ) ) = ( -u S - I ) ) |
147 |
146
|
eqeq2d |
|- ( ph -> ( X = ( ( -u S - I ) / ( 2 x. ( 1 / 2 ) ) ) <-> X = ( -u S - I ) ) ) |
148 |
142 147
|
orbi12d |
|- ( ph -> ( ( X = ( ( -u S + I ) / ( 2 x. ( 1 / 2 ) ) ) \/ X = ( ( -u S - I ) / ( 2 x. ( 1 / 2 ) ) ) ) <-> ( X = ( -u S + I ) \/ X = ( -u S - I ) ) ) ) |
149 |
109 135 148
|
3bitrd |
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 <-> ( X = ( -u S + I ) \/ X = ( -u S - I ) ) ) ) |