Metamath Proof Explorer


Theorem dquartlem1

Description: Lemma for dquart . (Contributed by Mario Carneiro, 6-May-2015)

Ref Expression
Hypotheses dquart.b
|- ( ph -> B e. CC )
dquart.c
|- ( ph -> C e. CC )
dquart.x
|- ( ph -> X e. CC )
dquart.s
|- ( ph -> S e. CC )
dquart.m
|- ( ph -> M = ( ( 2 x. S ) ^ 2 ) )
dquart.m0
|- ( ph -> M =/= 0 )
dquart.i
|- ( ph -> I e. CC )
dquart.i2
|- ( ph -> ( I ^ 2 ) = ( ( -u ( S ^ 2 ) - ( B / 2 ) ) + ( ( C / 4 ) / S ) ) )
Assertion dquartlem1
|- ( ph -> ( ( ( ( X ^ 2 ) + ( ( M + B ) / 2 ) ) + ( ( ( ( M / 2 ) x. X ) - ( C / 4 ) ) / S ) ) = 0 <-> ( X = ( -u S + I ) \/ X = ( -u S - I ) ) ) )

Proof

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 ) ) ) )