Step |
Hyp |
Ref |
Expression |
1 |
|
9nn0 |
⊢ 9 ∈ ℕ0 |
2 |
|
sumcubes |
⊢ ( 9 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 9 ) ( 𝑘 ↑ 3 ) = ( Σ 𝑘 ∈ ( 1 ... 9 ) 𝑘 ↑ 2 ) ) |
3 |
1 2
|
ax-mp |
⊢ Σ 𝑘 ∈ ( 1 ... 9 ) ( 𝑘 ↑ 3 ) = ( Σ 𝑘 ∈ ( 1 ... 9 ) 𝑘 ↑ 2 ) |
4 |
|
arisum |
⊢ ( 9 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 9 ) 𝑘 = ( ( ( 9 ↑ 2 ) + 9 ) / 2 ) ) |
5 |
1 4
|
ax-mp |
⊢ Σ 𝑘 ∈ ( 1 ... 9 ) 𝑘 = ( ( ( 9 ↑ 2 ) + 9 ) / 2 ) |
6 |
|
8nn0 |
⊢ 8 ∈ ℕ0 |
7 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
8 |
|
sq9 |
⊢ ( 9 ↑ 2 ) = ; 8 1 |
9 |
|
8p1e9 |
⊢ ( 8 + 1 ) = 9 |
10 |
|
9cn |
⊢ 9 ∈ ℂ |
11 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
12 |
|
9p1e10 |
⊢ ( 9 + 1 ) = ; 1 0 |
13 |
10 11 12
|
addcomli |
⊢ ( 1 + 9 ) = ; 1 0 |
14 |
6 7 1 8 9 13
|
decaddci2 |
⊢ ( ( 9 ↑ 2 ) + 9 ) = ; 9 0 |
15 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
16 |
|
4nn0 |
⊢ 4 ∈ ℕ0 |
17 |
|
5nn0 |
⊢ 5 ∈ ℕ0 |
18 |
|
eqid |
⊢ ; 4 5 = ; 4 5 |
19 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
20 |
|
4t2e8 |
⊢ ( 4 · 2 ) = 8 |
21 |
20
|
oveq1i |
⊢ ( ( 4 · 2 ) + 1 ) = ( 8 + 1 ) |
22 |
21 9
|
eqtri |
⊢ ( ( 4 · 2 ) + 1 ) = 9 |
23 |
|
5t2e10 |
⊢ ( 5 · 2 ) = ; 1 0 |
24 |
15 16 17 18 19 7 22 23
|
decmul1c |
⊢ ( ; 4 5 · 2 ) = ; 9 0 |
25 |
14 24
|
eqtr4i |
⊢ ( ( 9 ↑ 2 ) + 9 ) = ( ; 4 5 · 2 ) |
26 |
25
|
oveq1i |
⊢ ( ( ( 9 ↑ 2 ) + 9 ) / 2 ) = ( ( ; 4 5 · 2 ) / 2 ) |
27 |
16 17
|
deccl |
⊢ ; 4 5 ∈ ℕ0 |
28 |
27
|
nn0cni |
⊢ ; 4 5 ∈ ℂ |
29 |
|
2cn |
⊢ 2 ∈ ℂ |
30 |
|
2ne0 |
⊢ 2 ≠ 0 |
31 |
28 29 30
|
divcan4i |
⊢ ( ( ; 4 5 · 2 ) / 2 ) = ; 4 5 |
32 |
5 26 31
|
3eqtri |
⊢ Σ 𝑘 ∈ ( 1 ... 9 ) 𝑘 = ; 4 5 |
33 |
32
|
oveq1i |
⊢ ( Σ 𝑘 ∈ ( 1 ... 9 ) 𝑘 ↑ 2 ) = ( ; 4 5 ↑ 2 ) |
34 |
|
sq45 |
⊢ ( ; 4 5 ↑ 2 ) = ; ; ; 2 0 2 5 |
35 |
3 33 34
|
3eqtri |
⊢ Σ 𝑘 ∈ ( 1 ... 9 ) ( 𝑘 ↑ 3 ) = ; ; ; 2 0 2 5 |