Step |
Hyp |
Ref |
Expression |
1 |
|
3nn0 |
|- 3 e. NN0 |
2 |
|
fsumkthpow |
|- ( ( 3 e. NN0 /\ T e. NN0 ) -> sum_ k e. ( 0 ... T ) ( k ^ 3 ) = ( ( ( ( 3 + 1 ) BernPoly ( T + 1 ) ) - ( ( 3 + 1 ) BernPoly 0 ) ) / ( 3 + 1 ) ) ) |
3 |
1 2
|
mpan |
|- ( T e. NN0 -> sum_ k e. ( 0 ... T ) ( k ^ 3 ) = ( ( ( ( 3 + 1 ) BernPoly ( T + 1 ) ) - ( ( 3 + 1 ) BernPoly 0 ) ) / ( 3 + 1 ) ) ) |
4 |
|
df-4 |
|- 4 = ( 3 + 1 ) |
5 |
4
|
oveq1i |
|- ( 4 BernPoly ( T + 1 ) ) = ( ( 3 + 1 ) BernPoly ( T + 1 ) ) |
6 |
4
|
oveq1i |
|- ( 4 BernPoly 0 ) = ( ( 3 + 1 ) BernPoly 0 ) |
7 |
5 6
|
oveq12i |
|- ( ( 4 BernPoly ( T + 1 ) ) - ( 4 BernPoly 0 ) ) = ( ( ( 3 + 1 ) BernPoly ( T + 1 ) ) - ( ( 3 + 1 ) BernPoly 0 ) ) |
8 |
7 4
|
oveq12i |
|- ( ( ( 4 BernPoly ( T + 1 ) ) - ( 4 BernPoly 0 ) ) / 4 ) = ( ( ( ( 3 + 1 ) BernPoly ( T + 1 ) ) - ( ( 3 + 1 ) BernPoly 0 ) ) / ( 3 + 1 ) ) |
9 |
|
nn0cn |
|- ( T e. NN0 -> T e. CC ) |
10 |
|
peano2cn |
|- ( T e. CC -> ( T + 1 ) e. CC ) |
11 |
9 10
|
syl |
|- ( T e. NN0 -> ( T + 1 ) e. CC ) |
12 |
|
bpoly4 |
|- ( ( T + 1 ) e. CC -> ( 4 BernPoly ( T + 1 ) ) = ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) ) |
13 |
11 12
|
syl |
|- ( T e. NN0 -> ( 4 BernPoly ( T + 1 ) ) = ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) ) |
14 |
|
4nn |
|- 4 e. NN |
15 |
|
0exp |
|- ( 4 e. NN -> ( 0 ^ 4 ) = 0 ) |
16 |
14 15
|
ax-mp |
|- ( 0 ^ 4 ) = 0 |
17 |
|
3nn |
|- 3 e. NN |
18 |
|
0exp |
|- ( 3 e. NN -> ( 0 ^ 3 ) = 0 ) |
19 |
17 18
|
ax-mp |
|- ( 0 ^ 3 ) = 0 |
20 |
19
|
oveq2i |
|- ( 2 x. ( 0 ^ 3 ) ) = ( 2 x. 0 ) |
21 |
|
2t0e0 |
|- ( 2 x. 0 ) = 0 |
22 |
20 21
|
eqtri |
|- ( 2 x. ( 0 ^ 3 ) ) = 0 |
23 |
16 22
|
oveq12i |
|- ( ( 0 ^ 4 ) - ( 2 x. ( 0 ^ 3 ) ) ) = ( 0 - 0 ) |
24 |
|
0m0e0 |
|- ( 0 - 0 ) = 0 |
25 |
23 24
|
eqtri |
|- ( ( 0 ^ 4 ) - ( 2 x. ( 0 ^ 3 ) ) ) = 0 |
26 |
|
sq0 |
|- ( 0 ^ 2 ) = 0 |
27 |
25 26
|
oveq12i |
|- ( ( ( 0 ^ 4 ) - ( 2 x. ( 0 ^ 3 ) ) ) + ( 0 ^ 2 ) ) = ( 0 + 0 ) |
28 |
|
00id |
|- ( 0 + 0 ) = 0 |
29 |
27 28
|
eqtri |
|- ( ( ( 0 ^ 4 ) - ( 2 x. ( 0 ^ 3 ) ) ) + ( 0 ^ 2 ) ) = 0 |
30 |
29
|
oveq1i |
|- ( ( ( ( 0 ^ 4 ) - ( 2 x. ( 0 ^ 3 ) ) ) + ( 0 ^ 2 ) ) - ( 1 / ; 3 0 ) ) = ( 0 - ( 1 / ; 3 0 ) ) |
31 |
|
0cn |
|- 0 e. CC |
32 |
|
bpoly4 |
|- ( 0 e. CC -> ( 4 BernPoly 0 ) = ( ( ( ( 0 ^ 4 ) - ( 2 x. ( 0 ^ 3 ) ) ) + ( 0 ^ 2 ) ) - ( 1 / ; 3 0 ) ) ) |
33 |
31 32
|
ax-mp |
|- ( 4 BernPoly 0 ) = ( ( ( ( 0 ^ 4 ) - ( 2 x. ( 0 ^ 3 ) ) ) + ( 0 ^ 2 ) ) - ( 1 / ; 3 0 ) ) |
34 |
|
df-neg |
|- -u ( 1 / ; 3 0 ) = ( 0 - ( 1 / ; 3 0 ) ) |
35 |
30 33 34
|
3eqtr4i |
|- ( 4 BernPoly 0 ) = -u ( 1 / ; 3 0 ) |
36 |
35
|
a1i |
|- ( T e. NN0 -> ( 4 BernPoly 0 ) = -u ( 1 / ; 3 0 ) ) |
37 |
13 36
|
oveq12d |
|- ( T e. NN0 -> ( ( 4 BernPoly ( T + 1 ) ) - ( 4 BernPoly 0 ) ) = ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) - -u ( 1 / ; 3 0 ) ) ) |
38 |
|
4nn0 |
|- 4 e. NN0 |
39 |
|
expcl |
|- ( ( ( T + 1 ) e. CC /\ 4 e. NN0 ) -> ( ( T + 1 ) ^ 4 ) e. CC ) |
40 |
38 39
|
mpan2 |
|- ( ( T + 1 ) e. CC -> ( ( T + 1 ) ^ 4 ) e. CC ) |
41 |
|
2cn |
|- 2 e. CC |
42 |
|
expcl |
|- ( ( ( T + 1 ) e. CC /\ 3 e. NN0 ) -> ( ( T + 1 ) ^ 3 ) e. CC ) |
43 |
1 42
|
mpan2 |
|- ( ( T + 1 ) e. CC -> ( ( T + 1 ) ^ 3 ) e. CC ) |
44 |
|
mulcl |
|- ( ( 2 e. CC /\ ( ( T + 1 ) ^ 3 ) e. CC ) -> ( 2 x. ( ( T + 1 ) ^ 3 ) ) e. CC ) |
45 |
41 43 44
|
sylancr |
|- ( ( T + 1 ) e. CC -> ( 2 x. ( ( T + 1 ) ^ 3 ) ) e. CC ) |
46 |
40 45
|
subcld |
|- ( ( T + 1 ) e. CC -> ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) e. CC ) |
47 |
|
sqcl |
|- ( ( T + 1 ) e. CC -> ( ( T + 1 ) ^ 2 ) e. CC ) |
48 |
46 47
|
addcld |
|- ( ( T + 1 ) e. CC -> ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) e. CC ) |
49 |
10 48
|
syl |
|- ( T e. CC -> ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) e. CC ) |
50 |
9 49
|
syl |
|- ( T e. NN0 -> ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) e. CC ) |
51 |
|
0nn0 |
|- 0 e. NN0 |
52 |
1 51
|
deccl |
|- ; 3 0 e. NN0 |
53 |
52
|
nn0cni |
|- ; 3 0 e. CC |
54 |
52
|
nn0rei |
|- ; 3 0 e. RR |
55 |
|
10pos |
|- 0 < ; 1 0 |
56 |
17 51 51 55
|
declti |
|- 0 < ; 3 0 |
57 |
54 56
|
gt0ne0ii |
|- ; 3 0 =/= 0 |
58 |
53 57
|
reccli |
|- ( 1 / ; 3 0 ) e. CC |
59 |
|
subcl |
|- ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) e. CC /\ ( 1 / ; 3 0 ) e. CC ) -> ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) e. CC ) |
60 |
50 58 59
|
sylancl |
|- ( T e. NN0 -> ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) e. CC ) |
61 |
|
subneg |
|- ( ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) e. CC /\ ( 1 / ; 3 0 ) e. CC ) -> ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) - -u ( 1 / ; 3 0 ) ) = ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) + ( 1 / ; 3 0 ) ) ) |
62 |
60 58 61
|
sylancl |
|- ( T e. NN0 -> ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) - -u ( 1 / ; 3 0 ) ) = ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) + ( 1 / ; 3 0 ) ) ) |
63 |
|
npcan |
|- ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) e. CC /\ ( 1 / ; 3 0 ) e. CC ) -> ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) + ( 1 / ; 3 0 ) ) = ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) ) |
64 |
49 58 63
|
sylancl |
|- ( T e. CC -> ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) + ( 1 / ; 3 0 ) ) = ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) ) |
65 |
9 64
|
syl |
|- ( T e. NN0 -> ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) + ( 1 / ; 3 0 ) ) = ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) ) |
66 |
|
2p2e4 |
|- ( 2 + 2 ) = 4 |
67 |
66
|
eqcomi |
|- 4 = ( 2 + 2 ) |
68 |
67
|
oveq2i |
|- ( ( T + 1 ) ^ 4 ) = ( ( T + 1 ) ^ ( 2 + 2 ) ) |
69 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
70 |
69
|
oveq2i |
|- ( ( T + 1 ) ^ 3 ) = ( ( T + 1 ) ^ ( 2 + 1 ) ) |
71 |
70
|
oveq2i |
|- ( 2 x. ( ( T + 1 ) ^ 3 ) ) = ( 2 x. ( ( T + 1 ) ^ ( 2 + 1 ) ) ) |
72 |
68 71
|
oveq12i |
|- ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) = ( ( ( T + 1 ) ^ ( 2 + 2 ) ) - ( 2 x. ( ( T + 1 ) ^ ( 2 + 1 ) ) ) ) |
73 |
72
|
oveq1i |
|- ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) = ( ( ( ( T + 1 ) ^ ( 2 + 2 ) ) - ( 2 x. ( ( T + 1 ) ^ ( 2 + 1 ) ) ) ) + ( ( T + 1 ) ^ 2 ) ) |
74 |
|
2nn0 |
|- 2 e. NN0 |
75 |
|
expadd |
|- ( ( ( T + 1 ) e. CC /\ 2 e. NN0 /\ 2 e. NN0 ) -> ( ( T + 1 ) ^ ( 2 + 2 ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) ) |
76 |
74 74 75
|
mp3an23 |
|- ( ( T + 1 ) e. CC -> ( ( T + 1 ) ^ ( 2 + 2 ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) ) |
77 |
|
1nn0 |
|- 1 e. NN0 |
78 |
|
expadd |
|- ( ( ( T + 1 ) e. CC /\ 2 e. NN0 /\ 1 e. NN0 ) -> ( ( T + 1 ) ^ ( 2 + 1 ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) |
79 |
74 77 78
|
mp3an23 |
|- ( ( T + 1 ) e. CC -> ( ( T + 1 ) ^ ( 2 + 1 ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) |
80 |
79
|
oveq2d |
|- ( ( T + 1 ) e. CC -> ( 2 x. ( ( T + 1 ) ^ ( 2 + 1 ) ) ) = ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) ) |
81 |
76 80
|
oveq12d |
|- ( ( T + 1 ) e. CC -> ( ( ( T + 1 ) ^ ( 2 + 2 ) ) - ( 2 x. ( ( T + 1 ) ^ ( 2 + 1 ) ) ) ) = ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) ) ) |
82 |
10 81
|
syl |
|- ( T e. CC -> ( ( ( T + 1 ) ^ ( 2 + 2 ) ) - ( 2 x. ( ( T + 1 ) ^ ( 2 + 1 ) ) ) ) = ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) ) ) |
83 |
10
|
sqcld |
|- ( T e. CC -> ( ( T + 1 ) ^ 2 ) e. CC ) |
84 |
83
|
mulid1d |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) x. 1 ) = ( ( T + 1 ) ^ 2 ) ) |
85 |
84
|
eqcomd |
|- ( T e. CC -> ( ( T + 1 ) ^ 2 ) = ( ( ( T + 1 ) ^ 2 ) x. 1 ) ) |
86 |
82 85
|
oveq12d |
|- ( T e. CC -> ( ( ( ( T + 1 ) ^ ( 2 + 2 ) ) - ( 2 x. ( ( T + 1 ) ^ ( 2 + 1 ) ) ) ) + ( ( T + 1 ) ^ 2 ) ) = ( ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) ) + ( ( ( T + 1 ) ^ 2 ) x. 1 ) ) ) |
87 |
10
|
exp1d |
|- ( T e. CC -> ( ( T + 1 ) ^ 1 ) = ( T + 1 ) ) |
88 |
87
|
oveq2d |
|- ( T e. CC -> ( 2 x. ( ( T + 1 ) ^ 1 ) ) = ( 2 x. ( T + 1 ) ) ) |
89 |
88
|
oveq2d |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) x. ( 2 x. ( ( T + 1 ) ^ 1 ) ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( 2 x. ( T + 1 ) ) ) ) |
90 |
89
|
oveq2d |
|- ( T e. CC -> ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( ( ( T + 1 ) ^ 2 ) x. ( 2 x. ( ( T + 1 ) ^ 1 ) ) ) ) = ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( ( ( T + 1 ) ^ 2 ) x. ( 2 x. ( T + 1 ) ) ) ) ) |
91 |
87 10
|
eqeltrd |
|- ( T e. CC -> ( ( T + 1 ) ^ 1 ) e. CC ) |
92 |
|
mul12 |
|- ( ( 2 e. CC /\ ( ( T + 1 ) ^ 2 ) e. CC /\ ( ( T + 1 ) ^ 1 ) e. CC ) -> ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( 2 x. ( ( T + 1 ) ^ 1 ) ) ) ) |
93 |
41 83 91 92
|
mp3an2i |
|- ( T e. CC -> ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( 2 x. ( ( T + 1 ) ^ 1 ) ) ) ) |
94 |
93
|
oveq2d |
|- ( T e. CC -> ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) ) = ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( ( ( T + 1 ) ^ 2 ) x. ( 2 x. ( ( T + 1 ) ^ 1 ) ) ) ) ) |
95 |
|
mulcl |
|- ( ( 2 e. CC /\ ( T + 1 ) e. CC ) -> ( 2 x. ( T + 1 ) ) e. CC ) |
96 |
41 10 95
|
sylancr |
|- ( T e. CC -> ( 2 x. ( T + 1 ) ) e. CC ) |
97 |
83 83 96
|
subdid |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) x. ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) ) = ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( ( ( T + 1 ) ^ 2 ) x. ( 2 x. ( T + 1 ) ) ) ) ) |
98 |
90 94 97
|
3eqtr4d |
|- ( T e. CC -> ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) ) ) |
99 |
98
|
oveq1d |
|- ( T e. CC -> ( ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) ) + ( ( ( T + 1 ) ^ 2 ) x. 1 ) ) = ( ( ( ( T + 1 ) ^ 2 ) x. ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) ) + ( ( ( T + 1 ) ^ 2 ) x. 1 ) ) ) |
100 |
83 96
|
subcld |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) e. CC ) |
101 |
|
ax-1cn |
|- 1 e. CC |
102 |
|
adddi |
|- ( ( ( ( T + 1 ) ^ 2 ) e. CC /\ ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) e. CC /\ 1 e. CC ) -> ( ( ( T + 1 ) ^ 2 ) x. ( ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) + 1 ) ) = ( ( ( ( T + 1 ) ^ 2 ) x. ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) ) + ( ( ( T + 1 ) ^ 2 ) x. 1 ) ) ) |
103 |
101 102
|
mp3an3 |
|- ( ( ( ( T + 1 ) ^ 2 ) e. CC /\ ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) e. CC ) -> ( ( ( T + 1 ) ^ 2 ) x. ( ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) + 1 ) ) = ( ( ( ( T + 1 ) ^ 2 ) x. ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) ) + ( ( ( T + 1 ) ^ 2 ) x. 1 ) ) ) |
104 |
83 100 103
|
syl2anc |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) x. ( ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) + 1 ) ) = ( ( ( ( T + 1 ) ^ 2 ) x. ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) ) + ( ( ( T + 1 ) ^ 2 ) x. 1 ) ) ) |
105 |
99 104
|
eqtr4d |
|- ( T e. CC -> ( ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) ) + ( ( ( T + 1 ) ^ 2 ) x. 1 ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) + 1 ) ) ) |
106 |
|
adddi |
|- ( ( 2 e. CC /\ T e. CC /\ 1 e. CC ) -> ( 2 x. ( T + 1 ) ) = ( ( 2 x. T ) + ( 2 x. 1 ) ) ) |
107 |
41 101 106
|
mp3an13 |
|- ( T e. CC -> ( 2 x. ( T + 1 ) ) = ( ( 2 x. T ) + ( 2 x. 1 ) ) ) |
108 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
109 |
108
|
oveq2i |
|- ( ( 2 x. T ) + ( 2 x. 1 ) ) = ( ( 2 x. T ) + 2 ) |
110 |
107 109
|
eqtrdi |
|- ( T e. CC -> ( 2 x. ( T + 1 ) ) = ( ( 2 x. T ) + 2 ) ) |
111 |
110
|
oveq1d |
|- ( T e. CC -> ( ( 2 x. ( T + 1 ) ) - 1 ) = ( ( ( 2 x. T ) + 2 ) - 1 ) ) |
112 |
|
mulcl |
|- ( ( 2 e. CC /\ T e. CC ) -> ( 2 x. T ) e. CC ) |
113 |
41 112
|
mpan |
|- ( T e. CC -> ( 2 x. T ) e. CC ) |
114 |
|
addsubass |
|- ( ( ( 2 x. T ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( ( 2 x. T ) + 2 ) - 1 ) = ( ( 2 x. T ) + ( 2 - 1 ) ) ) |
115 |
41 101 114
|
mp3an23 |
|- ( ( 2 x. T ) e. CC -> ( ( ( 2 x. T ) + 2 ) - 1 ) = ( ( 2 x. T ) + ( 2 - 1 ) ) ) |
116 |
113 115
|
syl |
|- ( T e. CC -> ( ( ( 2 x. T ) + 2 ) - 1 ) = ( ( 2 x. T ) + ( 2 - 1 ) ) ) |
117 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
118 |
117
|
oveq2i |
|- ( ( 2 x. T ) + ( 2 - 1 ) ) = ( ( 2 x. T ) + 1 ) |
119 |
116 118
|
eqtrdi |
|- ( T e. CC -> ( ( ( 2 x. T ) + 2 ) - 1 ) = ( ( 2 x. T ) + 1 ) ) |
120 |
111 119
|
eqtrd |
|- ( T e. CC -> ( ( 2 x. ( T + 1 ) ) - 1 ) = ( ( 2 x. T ) + 1 ) ) |
121 |
120
|
oveq2d |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) - ( ( 2 x. ( T + 1 ) ) - 1 ) ) = ( ( ( T + 1 ) ^ 2 ) - ( ( 2 x. T ) + 1 ) ) ) |
122 |
|
subsub |
|- ( ( ( ( T + 1 ) ^ 2 ) e. CC /\ ( 2 x. ( T + 1 ) ) e. CC /\ 1 e. CC ) -> ( ( ( T + 1 ) ^ 2 ) - ( ( 2 x. ( T + 1 ) ) - 1 ) ) = ( ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) + 1 ) ) |
123 |
101 122
|
mp3an3 |
|- ( ( ( ( T + 1 ) ^ 2 ) e. CC /\ ( 2 x. ( T + 1 ) ) e. CC ) -> ( ( ( T + 1 ) ^ 2 ) - ( ( 2 x. ( T + 1 ) ) - 1 ) ) = ( ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) + 1 ) ) |
124 |
83 96 123
|
syl2anc |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) - ( ( 2 x. ( T + 1 ) ) - 1 ) ) = ( ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) + 1 ) ) |
125 |
|
sqcl |
|- ( T e. CC -> ( T ^ 2 ) e. CC ) |
126 |
|
peano2cn |
|- ( ( 2 x. T ) e. CC -> ( ( 2 x. T ) + 1 ) e. CC ) |
127 |
113 126
|
syl |
|- ( T e. CC -> ( ( 2 x. T ) + 1 ) e. CC ) |
128 |
|
binom21 |
|- ( T e. CC -> ( ( T + 1 ) ^ 2 ) = ( ( ( T ^ 2 ) + ( 2 x. T ) ) + 1 ) ) |
129 |
|
addass |
|- ( ( ( T ^ 2 ) e. CC /\ ( 2 x. T ) e. CC /\ 1 e. CC ) -> ( ( ( T ^ 2 ) + ( 2 x. T ) ) + 1 ) = ( ( T ^ 2 ) + ( ( 2 x. T ) + 1 ) ) ) |
130 |
101 129
|
mp3an3 |
|- ( ( ( T ^ 2 ) e. CC /\ ( 2 x. T ) e. CC ) -> ( ( ( T ^ 2 ) + ( 2 x. T ) ) + 1 ) = ( ( T ^ 2 ) + ( ( 2 x. T ) + 1 ) ) ) |
131 |
125 113 130
|
syl2anc |
|- ( T e. CC -> ( ( ( T ^ 2 ) + ( 2 x. T ) ) + 1 ) = ( ( T ^ 2 ) + ( ( 2 x. T ) + 1 ) ) ) |
132 |
128 131
|
eqtrd |
|- ( T e. CC -> ( ( T + 1 ) ^ 2 ) = ( ( T ^ 2 ) + ( ( 2 x. T ) + 1 ) ) ) |
133 |
125 127 132
|
mvrraddd |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) - ( ( 2 x. T ) + 1 ) ) = ( T ^ 2 ) ) |
134 |
121 124 133
|
3eqtr3d |
|- ( T e. CC -> ( ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) + 1 ) = ( T ^ 2 ) ) |
135 |
134
|
oveq2d |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) x. ( ( ( ( T + 1 ) ^ 2 ) - ( 2 x. ( T + 1 ) ) ) + 1 ) ) = ( ( ( T + 1 ) ^ 2 ) x. ( T ^ 2 ) ) ) |
136 |
83 125
|
mulcomd |
|- ( T e. CC -> ( ( ( T + 1 ) ^ 2 ) x. ( T ^ 2 ) ) = ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) ) |
137 |
105 135 136
|
3eqtrd |
|- ( T e. CC -> ( ( ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) - ( 2 x. ( ( ( T + 1 ) ^ 2 ) x. ( ( T + 1 ) ^ 1 ) ) ) ) + ( ( ( T + 1 ) ^ 2 ) x. 1 ) ) = ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) ) |
138 |
86 137
|
eqtrd |
|- ( T e. CC -> ( ( ( ( T + 1 ) ^ ( 2 + 2 ) ) - ( 2 x. ( ( T + 1 ) ^ ( 2 + 1 ) ) ) ) + ( ( T + 1 ) ^ 2 ) ) = ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) ) |
139 |
9 138
|
syl |
|- ( T e. NN0 -> ( ( ( ( T + 1 ) ^ ( 2 + 2 ) ) - ( 2 x. ( ( T + 1 ) ^ ( 2 + 1 ) ) ) ) + ( ( T + 1 ) ^ 2 ) ) = ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) ) |
140 |
73 139
|
syl5eq |
|- ( T e. NN0 -> ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) = ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) ) |
141 |
65 140
|
eqtrd |
|- ( T e. NN0 -> ( ( ( ( ( ( T + 1 ) ^ 4 ) - ( 2 x. ( ( T + 1 ) ^ 3 ) ) ) + ( ( T + 1 ) ^ 2 ) ) - ( 1 / ; 3 0 ) ) + ( 1 / ; 3 0 ) ) = ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) ) |
142 |
37 62 141
|
3eqtrd |
|- ( T e. NN0 -> ( ( 4 BernPoly ( T + 1 ) ) - ( 4 BernPoly 0 ) ) = ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) ) |
143 |
142
|
oveq1d |
|- ( T e. NN0 -> ( ( ( 4 BernPoly ( T + 1 ) ) - ( 4 BernPoly 0 ) ) / 4 ) = ( ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) / 4 ) ) |
144 |
8 143
|
syl5eqr |
|- ( T e. NN0 -> ( ( ( ( 3 + 1 ) BernPoly ( T + 1 ) ) - ( ( 3 + 1 ) BernPoly 0 ) ) / ( 3 + 1 ) ) = ( ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) / 4 ) ) |
145 |
3 144
|
eqtrd |
|- ( T e. NN0 -> sum_ k e. ( 0 ... T ) ( k ^ 3 ) = ( ( ( T ^ 2 ) x. ( ( T + 1 ) ^ 2 ) ) / 4 ) ) |