Step |
Hyp |
Ref |
Expression |
1 |
|
0dp2dp.a |
⊢ 𝐴 ∈ ℕ0 |
2 |
|
0dp2dp.b |
⊢ 𝐵 ∈ ℝ+ |
3 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
4 |
|
0z |
⊢ 0 ∈ ℤ |
5 |
|
1z |
⊢ 1 ∈ ℤ |
6 |
1 2 3 4 5
|
dpexpp1 |
⊢ ( ( 𝐴 . 𝐵 ) · ( ; 1 0 ↑ 0 ) ) = ( ( 0 . _ 𝐴 𝐵 ) · ( ; 1 0 ↑ 1 ) ) |
7 |
|
10nn0 |
⊢ ; 1 0 ∈ ℕ0 |
8 |
7
|
nn0cni |
⊢ ; 1 0 ∈ ℂ |
9 |
|
exp0 |
⊢ ( ; 1 0 ∈ ℂ → ( ; 1 0 ↑ 0 ) = 1 ) |
10 |
8 9
|
ax-mp |
⊢ ( ; 1 0 ↑ 0 ) = 1 |
11 |
10
|
oveq2i |
⊢ ( ( 𝐴 . 𝐵 ) · ( ; 1 0 ↑ 0 ) ) = ( ( 𝐴 . 𝐵 ) · 1 ) |
12 |
|
exp1 |
⊢ ( ; 1 0 ∈ ℂ → ( ; 1 0 ↑ 1 ) = ; 1 0 ) |
13 |
8 12
|
ax-mp |
⊢ ( ; 1 0 ↑ 1 ) = ; 1 0 |
14 |
13
|
oveq2i |
⊢ ( ( 0 . _ 𝐴 𝐵 ) · ( ; 1 0 ↑ 1 ) ) = ( ( 0 . _ 𝐴 𝐵 ) · ; 1 0 ) |
15 |
6 11 14
|
3eqtr3ri |
⊢ ( ( 0 . _ 𝐴 𝐵 ) · ; 1 0 ) = ( ( 𝐴 . 𝐵 ) · 1 ) |
16 |
1 2
|
rpdpcl |
⊢ ( 𝐴 . 𝐵 ) ∈ ℝ+ |
17 |
|
rpcn |
⊢ ( ( 𝐴 . 𝐵 ) ∈ ℝ+ → ( 𝐴 . 𝐵 ) ∈ ℂ ) |
18 |
16 17
|
ax-mp |
⊢ ( 𝐴 . 𝐵 ) ∈ ℂ |
19 |
|
mulid1 |
⊢ ( ( 𝐴 . 𝐵 ) ∈ ℂ → ( ( 𝐴 . 𝐵 ) · 1 ) = ( 𝐴 . 𝐵 ) ) |
20 |
18 19
|
ax-mp |
⊢ ( ( 𝐴 . 𝐵 ) · 1 ) = ( 𝐴 . 𝐵 ) |
21 |
15 20
|
eqtri |
⊢ ( ( 0 . _ 𝐴 𝐵 ) · ; 1 0 ) = ( 𝐴 . 𝐵 ) |