Step |
Hyp |
Ref |
Expression |
1 |
|
df-4 |
|- 4 = ( 3 + 1 ) |
2 |
1
|
oveq2i |
|- ( ( A + B ) ^ 4 ) = ( ( A + B ) ^ ( 3 + 1 ) ) |
3 |
|
addcl |
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) |
4 |
|
3nn0 |
|- 3 e. NN0 |
5 |
|
expp1 |
|- ( ( ( A + B ) e. CC /\ 3 e. NN0 ) -> ( ( A + B ) ^ ( 3 + 1 ) ) = ( ( ( A + B ) ^ 3 ) x. ( A + B ) ) ) |
6 |
3 4 5
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ ( 3 + 1 ) ) = ( ( ( A + B ) ^ 3 ) x. ( A + B ) ) ) |
7 |
2 6
|
syl5eq |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 4 ) = ( ( ( A + B ) ^ 3 ) x. ( A + B ) ) ) |
8 |
|
binom3 |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 3 ) = ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) |
9 |
8
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) ^ 3 ) x. ( A + B ) ) = ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. ( A + B ) ) ) |
10 |
|
simpl |
|- ( ( A e. CC /\ B e. CC ) -> A e. CC ) |
11 |
|
expcl |
|- ( ( A e. CC /\ 3 e. NN0 ) -> ( A ^ 3 ) e. CC ) |
12 |
10 4 11
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ 3 ) e. CC ) |
13 |
|
3cn |
|- 3 e. CC |
14 |
10
|
sqcld |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) e. CC ) |
15 |
|
simpr |
|- ( ( A e. CC /\ B e. CC ) -> B e. CC ) |
16 |
14 15
|
mulcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) x. B ) e. CC ) |
17 |
|
mulcl |
|- ( ( 3 e. CC /\ ( ( A ^ 2 ) x. B ) e. CC ) -> ( 3 x. ( ( A ^ 2 ) x. B ) ) e. CC ) |
18 |
13 16 17
|
sylancr |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( ( A ^ 2 ) x. B ) ) e. CC ) |
19 |
12 18
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) e. CC ) |
20 |
15
|
sqcld |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ 2 ) e. CC ) |
21 |
10 20
|
mulcld |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. ( B ^ 2 ) ) e. CC ) |
22 |
|
mulcl |
|- ( ( 3 e. CC /\ ( A x. ( B ^ 2 ) ) e. CC ) -> ( 3 x. ( A x. ( B ^ 2 ) ) ) e. CC ) |
23 |
13 21 22
|
sylancr |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( A x. ( B ^ 2 ) ) ) e. CC ) |
24 |
|
expcl |
|- ( ( B e. CC /\ 3 e. NN0 ) -> ( B ^ 3 ) e. CC ) |
25 |
15 4 24
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ 3 ) e. CC ) |
26 |
23 25
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) e. CC ) |
27 |
19 26
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) e. CC ) |
28 |
27 10 15
|
adddid |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. ( A + B ) ) = ( ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. A ) + ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. B ) ) ) |
29 |
1
|
oveq2i |
|- ( A ^ 4 ) = ( A ^ ( 3 + 1 ) ) |
30 |
|
expp1 |
|- ( ( A e. CC /\ 3 e. NN0 ) -> ( A ^ ( 3 + 1 ) ) = ( ( A ^ 3 ) x. A ) ) |
31 |
10 4 30
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ ( 3 + 1 ) ) = ( ( A ^ 3 ) x. A ) ) |
32 |
29 31
|
eqtr2id |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 3 ) x. A ) = ( A ^ 4 ) ) |
33 |
13
|
a1i |
|- ( ( A e. CC /\ B e. CC ) -> 3 e. CC ) |
34 |
33 16 10
|
mulassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( ( A ^ 2 ) x. B ) ) x. A ) = ( 3 x. ( ( ( A ^ 2 ) x. B ) x. A ) ) ) |
35 |
14 15 10
|
mul32d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) x. B ) x. A ) = ( ( ( A ^ 2 ) x. A ) x. B ) ) |
36 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
37 |
36
|
oveq2i |
|- ( A ^ 3 ) = ( A ^ ( 2 + 1 ) ) |
38 |
|
2nn0 |
|- 2 e. NN0 |
39 |
|
expp1 |
|- ( ( A e. CC /\ 2 e. NN0 ) -> ( A ^ ( 2 + 1 ) ) = ( ( A ^ 2 ) x. A ) ) |
40 |
10 38 39
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ ( 2 + 1 ) ) = ( ( A ^ 2 ) x. A ) ) |
41 |
37 40
|
eqtr2id |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) x. A ) = ( A ^ 3 ) ) |
42 |
41
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) x. A ) x. B ) = ( ( A ^ 3 ) x. B ) ) |
43 |
35 42
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) x. B ) x. A ) = ( ( A ^ 3 ) x. B ) ) |
44 |
43
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( ( ( A ^ 2 ) x. B ) x. A ) ) = ( 3 x. ( ( A ^ 3 ) x. B ) ) ) |
45 |
34 44
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( ( A ^ 2 ) x. B ) ) x. A ) = ( 3 x. ( ( A ^ 3 ) x. B ) ) ) |
46 |
32 45
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 3 ) x. A ) + ( ( 3 x. ( ( A ^ 2 ) x. B ) ) x. A ) ) = ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) ) |
47 |
12 10 18 46
|
joinlmuladdmuld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) x. A ) = ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) ) |
48 |
33 21 10
|
mulassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( A x. ( B ^ 2 ) ) ) x. A ) = ( 3 x. ( ( A x. ( B ^ 2 ) ) x. A ) ) ) |
49 |
10 20 10
|
mul32d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( B ^ 2 ) ) x. A ) = ( ( A x. A ) x. ( B ^ 2 ) ) ) |
50 |
10
|
sqvald |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) = ( A x. A ) ) |
51 |
50
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) x. ( B ^ 2 ) ) = ( ( A x. A ) x. ( B ^ 2 ) ) ) |
52 |
49 51
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( B ^ 2 ) ) x. A ) = ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) |
53 |
52
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( ( A x. ( B ^ 2 ) ) x. A ) ) = ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) |
54 |
48 53
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( A x. ( B ^ 2 ) ) ) x. A ) = ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) |
55 |
25 10
|
mulcomd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( B ^ 3 ) x. A ) = ( A x. ( B ^ 3 ) ) ) |
56 |
54 55
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( 3 x. ( A x. ( B ^ 2 ) ) ) x. A ) + ( ( B ^ 3 ) x. A ) ) = ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) ) |
57 |
23 10 25 56
|
joinlmuladdmuld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) x. A ) = ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) ) |
58 |
47 57
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) x. A ) + ( ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) x. A ) ) = ( ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) ) ) |
59 |
19 10 26 58
|
joinlmuladdmuld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. A ) = ( ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) ) ) |
60 |
19 26 15
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. B ) = ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) x. B ) + ( ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) x. B ) ) ) |
61 |
33 16 15
|
mulassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( ( A ^ 2 ) x. B ) ) x. B ) = ( 3 x. ( ( ( A ^ 2 ) x. B ) x. B ) ) ) |
62 |
14 15 15
|
mulassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) x. B ) x. B ) = ( ( A ^ 2 ) x. ( B x. B ) ) ) |
63 |
15
|
sqvald |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ 2 ) = ( B x. B ) ) |
64 |
63
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) x. ( B ^ 2 ) ) = ( ( A ^ 2 ) x. ( B x. B ) ) ) |
65 |
62 64
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) x. B ) x. B ) = ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) |
66 |
65
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( ( ( A ^ 2 ) x. B ) x. B ) ) = ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) |
67 |
61 66
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( ( A ^ 2 ) x. B ) ) x. B ) = ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) |
68 |
67
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 3 ) x. B ) + ( ( 3 x. ( ( A ^ 2 ) x. B ) ) x. B ) ) = ( ( ( A ^ 3 ) x. B ) + ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) ) |
69 |
12 15 18 68
|
joinlmuladdmuld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) x. B ) = ( ( ( A ^ 3 ) x. B ) + ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) ) |
70 |
33 21 15
|
mulassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( A x. ( B ^ 2 ) ) ) x. B ) = ( 3 x. ( ( A x. ( B ^ 2 ) ) x. B ) ) ) |
71 |
10 20 15
|
mulassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( B ^ 2 ) ) x. B ) = ( A x. ( ( B ^ 2 ) x. B ) ) ) |
72 |
36
|
oveq2i |
|- ( B ^ 3 ) = ( B ^ ( 2 + 1 ) ) |
73 |
|
expp1 |
|- ( ( B e. CC /\ 2 e. NN0 ) -> ( B ^ ( 2 + 1 ) ) = ( ( B ^ 2 ) x. B ) ) |
74 |
15 38 73
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ ( 2 + 1 ) ) = ( ( B ^ 2 ) x. B ) ) |
75 |
72 74
|
eqtr2id |
|- ( ( A e. CC /\ B e. CC ) -> ( ( B ^ 2 ) x. B ) = ( B ^ 3 ) ) |
76 |
75
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. ( ( B ^ 2 ) x. B ) ) = ( A x. ( B ^ 3 ) ) ) |
77 |
71 76
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( B ^ 2 ) ) x. B ) = ( A x. ( B ^ 3 ) ) ) |
78 |
77
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( ( A x. ( B ^ 2 ) ) x. B ) ) = ( 3 x. ( A x. ( B ^ 3 ) ) ) ) |
79 |
70 78
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( A x. ( B ^ 2 ) ) ) x. B ) = ( 3 x. ( A x. ( B ^ 3 ) ) ) ) |
80 |
1
|
oveq2i |
|- ( B ^ 4 ) = ( B ^ ( 3 + 1 ) ) |
81 |
|
expp1 |
|- ( ( B e. CC /\ 3 e. NN0 ) -> ( B ^ ( 3 + 1 ) ) = ( ( B ^ 3 ) x. B ) ) |
82 |
15 4 81
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ ( 3 + 1 ) ) = ( ( B ^ 3 ) x. B ) ) |
83 |
80 82
|
eqtr2id |
|- ( ( A e. CC /\ B e. CC ) -> ( ( B ^ 3 ) x. B ) = ( B ^ 4 ) ) |
84 |
79 83
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( 3 x. ( A x. ( B ^ 2 ) ) ) x. B ) + ( ( B ^ 3 ) x. B ) ) = ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) |
85 |
23 15 25 84
|
joinlmuladdmuld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) x. B ) = ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) |
86 |
69 85
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) x. B ) + ( ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) x. B ) ) = ( ( ( ( A ^ 3 ) x. B ) + ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) |
87 |
12 15
|
mulcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 3 ) x. B ) e. CC ) |
88 |
14 20
|
mulcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) x. ( B ^ 2 ) ) e. CC ) |
89 |
|
mulcl |
|- ( ( 3 e. CC /\ ( ( A ^ 2 ) x. ( B ^ 2 ) ) e. CC ) -> ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) e. CC ) |
90 |
13 88 89
|
sylancr |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) e. CC ) |
91 |
10 25
|
mulcld |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. ( B ^ 3 ) ) e. CC ) |
92 |
|
mulcl |
|- ( ( 3 e. CC /\ ( A x. ( B ^ 3 ) ) e. CC ) -> ( 3 x. ( A x. ( B ^ 3 ) ) ) e. CC ) |
93 |
13 91 92
|
sylancr |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( A x. ( B ^ 3 ) ) ) e. CC ) |
94 |
|
4nn0 |
|- 4 e. NN0 |
95 |
|
expcl |
|- ( ( B e. CC /\ 4 e. NN0 ) -> ( B ^ 4 ) e. CC ) |
96 |
15 94 95
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ 4 ) e. CC ) |
97 |
93 96
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) e. CC ) |
98 |
87 90 97
|
addassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) x. B ) + ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) = ( ( ( A ^ 3 ) x. B ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) |
99 |
60 86 98
|
3eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. B ) = ( ( ( A ^ 3 ) x. B ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) |
100 |
59 99
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. A ) + ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. B ) ) = ( ( ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) ) + ( ( ( A ^ 3 ) x. B ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) ) |
101 |
|
expcl |
|- ( ( A e. CC /\ 4 e. NN0 ) -> ( A ^ 4 ) e. CC ) |
102 |
10 94 101
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ 4 ) e. CC ) |
103 |
|
mulcl |
|- ( ( 3 e. CC /\ ( ( A ^ 3 ) x. B ) e. CC ) -> ( 3 x. ( ( A ^ 3 ) x. B ) ) e. CC ) |
104 |
13 87 103
|
sylancr |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( ( A ^ 3 ) x. B ) ) e. CC ) |
105 |
102 104
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) e. CC ) |
106 |
90 91
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) e. CC ) |
107 |
90 97
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) e. CC ) |
108 |
105 106 87 107
|
add4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) ) + ( ( ( A ^ 3 ) x. B ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) = ( ( ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( A ^ 3 ) x. B ) ) + ( ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) ) |
109 |
102 104 87
|
addassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( A ^ 3 ) x. B ) ) = ( ( A ^ 4 ) + ( ( 3 x. ( ( A ^ 3 ) x. B ) ) + ( ( A ^ 3 ) x. B ) ) ) ) |
110 |
1
|
oveq1i |
|- ( 4 x. ( ( A ^ 3 ) x. B ) ) = ( ( 3 + 1 ) x. ( ( A ^ 3 ) x. B ) ) |
111 |
|
ax-1cn |
|- 1 e. CC |
112 |
111
|
a1i |
|- ( ( A e. CC /\ B e. CC ) -> 1 e. CC ) |
113 |
33 112 87
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 + 1 ) x. ( ( A ^ 3 ) x. B ) ) = ( ( 3 x. ( ( A ^ 3 ) x. B ) ) + ( 1 x. ( ( A ^ 3 ) x. B ) ) ) ) |
114 |
110 113
|
syl5eq |
|- ( ( A e. CC /\ B e. CC ) -> ( 4 x. ( ( A ^ 3 ) x. B ) ) = ( ( 3 x. ( ( A ^ 3 ) x. B ) ) + ( 1 x. ( ( A ^ 3 ) x. B ) ) ) ) |
115 |
87
|
mulid2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 3 ) x. B ) ) = ( ( A ^ 3 ) x. B ) ) |
116 |
115
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( ( A ^ 3 ) x. B ) ) + ( 1 x. ( ( A ^ 3 ) x. B ) ) ) = ( ( 3 x. ( ( A ^ 3 ) x. B ) ) + ( ( A ^ 3 ) x. B ) ) ) |
117 |
114 116
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( 4 x. ( ( A ^ 3 ) x. B ) ) = ( ( 3 x. ( ( A ^ 3 ) x. B ) ) + ( ( A ^ 3 ) x. B ) ) ) |
118 |
117
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 4 ) + ( 4 x. ( ( A ^ 3 ) x. B ) ) ) = ( ( A ^ 4 ) + ( ( 3 x. ( ( A ^ 3 ) x. B ) ) + ( ( A ^ 3 ) x. B ) ) ) ) |
119 |
109 118
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( A ^ 3 ) x. B ) ) = ( ( A ^ 4 ) + ( 4 x. ( ( A ^ 3 ) x. B ) ) ) ) |
120 |
90 91 90 97
|
add4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) = ( ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) + ( ( A x. ( B ^ 3 ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) |
121 |
|
3p3e6 |
|- ( 3 + 3 ) = 6 |
122 |
121
|
oveq1i |
|- ( ( 3 + 3 ) x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) = ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) |
123 |
33 33 88
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 + 3 ) x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) = ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) ) |
124 |
122 123
|
eqtr3id |
|- ( ( A e. CC /\ B e. CC ) -> ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) = ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) ) |
125 |
|
3p1e4 |
|- ( 3 + 1 ) = 4 |
126 |
13 111 125
|
addcomli |
|- ( 1 + 3 ) = 4 |
127 |
126
|
oveq1i |
|- ( ( 1 + 3 ) x. ( A x. ( B ^ 3 ) ) ) = ( 4 x. ( A x. ( B ^ 3 ) ) ) |
128 |
112 33 91
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + 3 ) x. ( A x. ( B ^ 3 ) ) ) = ( ( 1 x. ( A x. ( B ^ 3 ) ) ) + ( 3 x. ( A x. ( B ^ 3 ) ) ) ) ) |
129 |
127 128
|
eqtr3id |
|- ( ( A e. CC /\ B e. CC ) -> ( 4 x. ( A x. ( B ^ 3 ) ) ) = ( ( 1 x. ( A x. ( B ^ 3 ) ) ) + ( 3 x. ( A x. ( B ^ 3 ) ) ) ) ) |
130 |
91
|
mulid2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( A x. ( B ^ 3 ) ) ) = ( A x. ( B ^ 3 ) ) ) |
131 |
130
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 1 x. ( A x. ( B ^ 3 ) ) ) + ( 3 x. ( A x. ( B ^ 3 ) ) ) ) = ( ( A x. ( B ^ 3 ) ) + ( 3 x. ( A x. ( B ^ 3 ) ) ) ) ) |
132 |
129 131
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( 4 x. ( A x. ( B ^ 3 ) ) ) = ( ( A x. ( B ^ 3 ) ) + ( 3 x. ( A x. ( B ^ 3 ) ) ) ) ) |
133 |
132
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) = ( ( ( A x. ( B ^ 3 ) ) + ( 3 x. ( A x. ( B ^ 3 ) ) ) ) + ( B ^ 4 ) ) ) |
134 |
91 93 96
|
addassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. ( B ^ 3 ) ) + ( 3 x. ( A x. ( B ^ 3 ) ) ) ) + ( B ^ 4 ) ) = ( ( A x. ( B ^ 3 ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) |
135 |
133 134
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) = ( ( A x. ( B ^ 3 ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) |
136 |
124 135
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) = ( ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) ) + ( ( A x. ( B ^ 3 ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) |
137 |
120 136
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) = ( ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) |
138 |
119 137
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( A ^ 3 ) x. B ) ) + ( ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) = ( ( ( A ^ 4 ) + ( 4 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) |
139 |
108 138
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 4 ) + ( 3 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( A x. ( B ^ 3 ) ) ) ) + ( ( ( A ^ 3 ) x. B ) + ( ( 3 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 3 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) = ( ( ( A ^ 4 ) + ( 4 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) |
140 |
28 100 139
|
3eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) x. ( A + B ) ) = ( ( ( A ^ 4 ) + ( 4 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) |
141 |
7 9 140
|
3eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 4 ) = ( ( ( A ^ 4 ) + ( 4 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) ) |