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 |
|
2cn |
|- 2 e. CC |
12 |
|
mulcl |
|- ( ( 2 e. CC /\ S e. CC ) -> ( 2 x. S ) e. CC ) |
13 |
11 4 12
|
sylancr |
|- ( ph -> ( 2 x. S ) e. CC ) |
14 |
13
|
sqcld |
|- ( ph -> ( ( 2 x. S ) ^ 2 ) e. CC ) |
15 |
5 14
|
eqeltrd |
|- ( ph -> M e. CC ) |
16 |
15 1
|
addcld |
|- ( ph -> ( M + B ) e. CC ) |
17 |
11
|
a1i |
|- ( ph -> 2 e. CC ) |
18 |
|
2ne0 |
|- 2 =/= 0 |
19 |
18
|
a1i |
|- ( ph -> 2 =/= 0 ) |
20 |
16 17 19
|
sqdivd |
|- ( ph -> ( ( ( M + B ) / 2 ) ^ 2 ) = ( ( ( M + B ) ^ 2 ) / ( 2 ^ 2 ) ) ) |
21 |
|
sq2 |
|- ( 2 ^ 2 ) = 4 |
22 |
21
|
oveq2i |
|- ( ( ( M + B ) ^ 2 ) / ( 2 ^ 2 ) ) = ( ( ( M + B ) ^ 2 ) / 4 ) |
23 |
20 22
|
eqtrdi |
|- ( ph -> ( ( ( M + B ) / 2 ) ^ 2 ) = ( ( ( M + B ) ^ 2 ) / 4 ) ) |
24 |
23
|
oveq1d |
|- ( ph -> ( ( ( ( M + B ) / 2 ) ^ 2 ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) = ( ( ( ( M + B ) ^ 2 ) / 4 ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) ) |
25 |
16
|
sqcld |
|- ( ph -> ( ( M + B ) ^ 2 ) e. CC ) |
26 |
|
4cn |
|- 4 e. CC |
27 |
26
|
a1i |
|- ( ph -> 4 e. CC ) |
28 |
|
4ne0 |
|- 4 =/= 0 |
29 |
28
|
a1i |
|- ( ph -> 4 =/= 0 ) |
30 |
25 27 29
|
divcld |
|- ( ph -> ( ( ( M + B ) ^ 2 ) / 4 ) e. CC ) |
31 |
2
|
sqcld |
|- ( ph -> ( C ^ 2 ) e. CC ) |
32 |
31 27 29
|
divcld |
|- ( ph -> ( ( C ^ 2 ) / 4 ) e. CC ) |
33 |
32 15 6
|
divcld |
|- ( ph -> ( ( ( C ^ 2 ) / 4 ) / M ) e. CC ) |
34 |
30 33
|
subcld |
|- ( ph -> ( ( ( ( M + B ) ^ 2 ) / 4 ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) e. CC ) |
35 |
30 33 15
|
subdird |
|- ( ph -> ( ( ( ( ( M + B ) ^ 2 ) / 4 ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) x. M ) = ( ( ( ( ( M + B ) ^ 2 ) / 4 ) x. M ) - ( ( ( ( C ^ 2 ) / 4 ) / M ) x. M ) ) ) |
36 |
25 15 27 29
|
div23d |
|- ( ph -> ( ( ( ( M + B ) ^ 2 ) x. M ) / 4 ) = ( ( ( ( M + B ) ^ 2 ) / 4 ) x. M ) ) |
37 |
36
|
eqcomd |
|- ( ph -> ( ( ( ( M + B ) ^ 2 ) / 4 ) x. M ) = ( ( ( ( M + B ) ^ 2 ) x. M ) / 4 ) ) |
38 |
32 15 6
|
divcan1d |
|- ( ph -> ( ( ( ( C ^ 2 ) / 4 ) / M ) x. M ) = ( ( C ^ 2 ) / 4 ) ) |
39 |
37 38
|
oveq12d |
|- ( ph -> ( ( ( ( ( M + B ) ^ 2 ) / 4 ) x. M ) - ( ( ( ( C ^ 2 ) / 4 ) / M ) x. M ) ) = ( ( ( ( ( M + B ) ^ 2 ) x. M ) / 4 ) - ( ( C ^ 2 ) / 4 ) ) ) |
40 |
|
binom2 |
|- ( ( M e. CC /\ B e. CC ) -> ( ( M + B ) ^ 2 ) = ( ( ( M ^ 2 ) + ( 2 x. ( M x. B ) ) ) + ( B ^ 2 ) ) ) |
41 |
15 1 40
|
syl2anc |
|- ( ph -> ( ( M + B ) ^ 2 ) = ( ( ( M ^ 2 ) + ( 2 x. ( M x. B ) ) ) + ( B ^ 2 ) ) ) |
42 |
41
|
oveq1d |
|- ( ph -> ( ( ( M + B ) ^ 2 ) x. M ) = ( ( ( ( M ^ 2 ) + ( 2 x. ( M x. B ) ) ) + ( B ^ 2 ) ) x. M ) ) |
43 |
15
|
sqcld |
|- ( ph -> ( M ^ 2 ) e. CC ) |
44 |
15 1
|
mulcld |
|- ( ph -> ( M x. B ) e. CC ) |
45 |
|
mulcl |
|- ( ( 2 e. CC /\ ( M x. B ) e. CC ) -> ( 2 x. ( M x. B ) ) e. CC ) |
46 |
11 44 45
|
sylancr |
|- ( ph -> ( 2 x. ( M x. B ) ) e. CC ) |
47 |
43 46
|
addcld |
|- ( ph -> ( ( M ^ 2 ) + ( 2 x. ( M x. B ) ) ) e. CC ) |
48 |
1
|
sqcld |
|- ( ph -> ( B ^ 2 ) e. CC ) |
49 |
47 48 15
|
adddird |
|- ( ph -> ( ( ( ( M ^ 2 ) + ( 2 x. ( M x. B ) ) ) + ( B ^ 2 ) ) x. M ) = ( ( ( ( M ^ 2 ) + ( 2 x. ( M x. B ) ) ) x. M ) + ( ( B ^ 2 ) x. M ) ) ) |
50 |
43 46 15
|
adddird |
|- ( ph -> ( ( ( M ^ 2 ) + ( 2 x. ( M x. B ) ) ) x. M ) = ( ( ( M ^ 2 ) x. M ) + ( ( 2 x. ( M x. B ) ) x. M ) ) ) |
51 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
52 |
51
|
oveq2i |
|- ( M ^ 3 ) = ( M ^ ( 2 + 1 ) ) |
53 |
|
2nn0 |
|- 2 e. NN0 |
54 |
|
expp1 |
|- ( ( M e. CC /\ 2 e. NN0 ) -> ( M ^ ( 2 + 1 ) ) = ( ( M ^ 2 ) x. M ) ) |
55 |
15 53 54
|
sylancl |
|- ( ph -> ( M ^ ( 2 + 1 ) ) = ( ( M ^ 2 ) x. M ) ) |
56 |
52 55
|
eqtr2id |
|- ( ph -> ( ( M ^ 2 ) x. M ) = ( M ^ 3 ) ) |
57 |
|
mulcl |
|- ( ( 2 e. CC /\ B e. CC ) -> ( 2 x. B ) e. CC ) |
58 |
11 1 57
|
sylancr |
|- ( ph -> ( 2 x. B ) e. CC ) |
59 |
58 15 15
|
mulassd |
|- ( ph -> ( ( ( 2 x. B ) x. M ) x. M ) = ( ( 2 x. B ) x. ( M x. M ) ) ) |
60 |
17 15 1
|
mulassd |
|- ( ph -> ( ( 2 x. M ) x. B ) = ( 2 x. ( M x. B ) ) ) |
61 |
17 15 1
|
mul32d |
|- ( ph -> ( ( 2 x. M ) x. B ) = ( ( 2 x. B ) x. M ) ) |
62 |
60 61
|
eqtr3d |
|- ( ph -> ( 2 x. ( M x. B ) ) = ( ( 2 x. B ) x. M ) ) |
63 |
62
|
oveq1d |
|- ( ph -> ( ( 2 x. ( M x. B ) ) x. M ) = ( ( ( 2 x. B ) x. M ) x. M ) ) |
64 |
15
|
sqvald |
|- ( ph -> ( M ^ 2 ) = ( M x. M ) ) |
65 |
64
|
oveq2d |
|- ( ph -> ( ( 2 x. B ) x. ( M ^ 2 ) ) = ( ( 2 x. B ) x. ( M x. M ) ) ) |
66 |
59 63 65
|
3eqtr4d |
|- ( ph -> ( ( 2 x. ( M x. B ) ) x. M ) = ( ( 2 x. B ) x. ( M ^ 2 ) ) ) |
67 |
56 66
|
oveq12d |
|- ( ph -> ( ( ( M ^ 2 ) x. M ) + ( ( 2 x. ( M x. B ) ) x. M ) ) = ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) ) |
68 |
50 67
|
eqtrd |
|- ( ph -> ( ( ( M ^ 2 ) + ( 2 x. ( M x. B ) ) ) x. M ) = ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) ) |
69 |
68
|
oveq1d |
|- ( ph -> ( ( ( ( M ^ 2 ) + ( 2 x. ( M x. B ) ) ) x. M ) + ( ( B ^ 2 ) x. M ) ) = ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( B ^ 2 ) x. M ) ) ) |
70 |
42 49 69
|
3eqtrd |
|- ( ph -> ( ( ( M + B ) ^ 2 ) x. M ) = ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( B ^ 2 ) x. M ) ) ) |
71 |
70
|
oveq1d |
|- ( ph -> ( ( ( ( M + B ) ^ 2 ) x. M ) - ( ( 4 x. D ) x. M ) ) = ( ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( B ^ 2 ) x. M ) ) - ( ( 4 x. D ) x. M ) ) ) |
72 |
|
3nn0 |
|- 3 e. NN0 |
73 |
|
expcl |
|- ( ( M e. CC /\ 3 e. NN0 ) -> ( M ^ 3 ) e. CC ) |
74 |
15 72 73
|
sylancl |
|- ( ph -> ( M ^ 3 ) e. CC ) |
75 |
58 43
|
mulcld |
|- ( ph -> ( ( 2 x. B ) x. ( M ^ 2 ) ) e. CC ) |
76 |
74 75
|
addcld |
|- ( ph -> ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) e. CC ) |
77 |
48 15
|
mulcld |
|- ( ph -> ( ( B ^ 2 ) x. M ) e. CC ) |
78 |
|
mulcl |
|- ( ( 4 e. CC /\ D e. CC ) -> ( 4 x. D ) e. CC ) |
79 |
26 9 78
|
sylancr |
|- ( ph -> ( 4 x. D ) e. CC ) |
80 |
79 15
|
mulcld |
|- ( ph -> ( ( 4 x. D ) x. M ) e. CC ) |
81 |
76 77 80
|
addsubassd |
|- ( ph -> ( ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( B ^ 2 ) x. M ) ) - ( ( 4 x. D ) x. M ) ) = ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) x. M ) - ( ( 4 x. D ) x. M ) ) ) ) |
82 |
48 79 15
|
subdird |
|- ( ph -> ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) = ( ( ( B ^ 2 ) x. M ) - ( ( 4 x. D ) x. M ) ) ) |
83 |
82
|
oveq2d |
|- ( ph -> ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) ) = ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) x. M ) - ( ( 4 x. D ) x. M ) ) ) ) |
84 |
81 83
|
eqtr4d |
|- ( ph -> ( ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( B ^ 2 ) x. M ) ) - ( ( 4 x. D ) x. M ) ) = ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) ) ) |
85 |
48 79
|
subcld |
|- ( ph -> ( ( B ^ 2 ) - ( 4 x. D ) ) e. CC ) |
86 |
85 15
|
mulcld |
|- ( ph -> ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) e. CC ) |
87 |
76 86
|
addcld |
|- ( ph -> ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) ) e. CC ) |
88 |
31
|
negcld |
|- ( ph -> -u ( C ^ 2 ) e. CC ) |
89 |
76 86 88
|
addassd |
|- ( ph -> ( ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) ) + -u ( C ^ 2 ) ) = ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) + -u ( C ^ 2 ) ) ) ) |
90 |
87 31
|
negsubd |
|- ( ph -> ( ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) ) + -u ( C ^ 2 ) ) = ( ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) ) - ( C ^ 2 ) ) ) |
91 |
89 90 10
|
3eqtr3d |
|- ( ph -> ( ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) ) - ( C ^ 2 ) ) = 0 ) |
92 |
87 31 91
|
subeq0d |
|- ( ph -> ( ( ( M ^ 3 ) + ( ( 2 x. B ) x. ( M ^ 2 ) ) ) + ( ( ( B ^ 2 ) - ( 4 x. D ) ) x. M ) ) = ( C ^ 2 ) ) |
93 |
71 84 92
|
3eqtrd |
|- ( ph -> ( ( ( ( M + B ) ^ 2 ) x. M ) - ( ( 4 x. D ) x. M ) ) = ( C ^ 2 ) ) |
94 |
25 15
|
mulcld |
|- ( ph -> ( ( ( M + B ) ^ 2 ) x. M ) e. CC ) |
95 |
|
subsub23 |
|- ( ( ( ( ( M + B ) ^ 2 ) x. M ) e. CC /\ ( ( 4 x. D ) x. M ) e. CC /\ ( C ^ 2 ) e. CC ) -> ( ( ( ( ( M + B ) ^ 2 ) x. M ) - ( ( 4 x. D ) x. M ) ) = ( C ^ 2 ) <-> ( ( ( ( M + B ) ^ 2 ) x. M ) - ( C ^ 2 ) ) = ( ( 4 x. D ) x. M ) ) ) |
96 |
94 80 31 95
|
syl3anc |
|- ( ph -> ( ( ( ( ( M + B ) ^ 2 ) x. M ) - ( ( 4 x. D ) x. M ) ) = ( C ^ 2 ) <-> ( ( ( ( M + B ) ^ 2 ) x. M ) - ( C ^ 2 ) ) = ( ( 4 x. D ) x. M ) ) ) |
97 |
93 96
|
mpbid |
|- ( ph -> ( ( ( ( M + B ) ^ 2 ) x. M ) - ( C ^ 2 ) ) = ( ( 4 x. D ) x. M ) ) |
98 |
27 9 15
|
mulassd |
|- ( ph -> ( ( 4 x. D ) x. M ) = ( 4 x. ( D x. M ) ) ) |
99 |
97 98
|
eqtrd |
|- ( ph -> ( ( ( ( M + B ) ^ 2 ) x. M ) - ( C ^ 2 ) ) = ( 4 x. ( D x. M ) ) ) |
100 |
99
|
oveq1d |
|- ( ph -> ( ( ( ( ( M + B ) ^ 2 ) x. M ) - ( C ^ 2 ) ) / 4 ) = ( ( 4 x. ( D x. M ) ) / 4 ) ) |
101 |
94 31 27 29
|
divsubdird |
|- ( ph -> ( ( ( ( ( M + B ) ^ 2 ) x. M ) - ( C ^ 2 ) ) / 4 ) = ( ( ( ( ( M + B ) ^ 2 ) x. M ) / 4 ) - ( ( C ^ 2 ) / 4 ) ) ) |
102 |
9 15
|
mulcld |
|- ( ph -> ( D x. M ) e. CC ) |
103 |
102 27 29
|
divcan3d |
|- ( ph -> ( ( 4 x. ( D x. M ) ) / 4 ) = ( D x. M ) ) |
104 |
100 101 103
|
3eqtr3d |
|- ( ph -> ( ( ( ( ( M + B ) ^ 2 ) x. M ) / 4 ) - ( ( C ^ 2 ) / 4 ) ) = ( D x. M ) ) |
105 |
35 39 104
|
3eqtrd |
|- ( ph -> ( ( ( ( ( M + B ) ^ 2 ) / 4 ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) x. M ) = ( D x. M ) ) |
106 |
34 9 15 6 105
|
mulcan2ad |
|- ( ph -> ( ( ( ( M + B ) ^ 2 ) / 4 ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) = D ) |
107 |
24 106
|
eqtrd |
|- ( ph -> ( ( ( ( M + B ) / 2 ) ^ 2 ) - ( ( ( C ^ 2 ) / 4 ) / M ) ) = D ) |