Step |
Hyp |
Ref |
Expression |
1 |
|
pser.g |
⊢ 𝐺 = ( 𝑥 ∈ ℂ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑛 ) · ( 𝑥 ↑ 𝑛 ) ) ) ) |
2 |
1
|
pserval |
⊢ ( 𝑋 ∈ ℂ → ( 𝐺 ‘ 𝑋 ) = ( 𝑦 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑦 ) · ( 𝑋 ↑ 𝑦 ) ) ) ) |
3 |
2
|
fveq1d |
⊢ ( 𝑋 ∈ ℂ → ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑁 ) = ( ( 𝑦 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑦 ) · ( 𝑋 ↑ 𝑦 ) ) ) ‘ 𝑁 ) ) |
4 |
|
fveq2 |
⊢ ( 𝑦 = 𝑁 → ( 𝐴 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑁 ) ) |
5 |
|
oveq2 |
⊢ ( 𝑦 = 𝑁 → ( 𝑋 ↑ 𝑦 ) = ( 𝑋 ↑ 𝑁 ) ) |
6 |
4 5
|
oveq12d |
⊢ ( 𝑦 = 𝑁 → ( ( 𝐴 ‘ 𝑦 ) · ( 𝑋 ↑ 𝑦 ) ) = ( ( 𝐴 ‘ 𝑁 ) · ( 𝑋 ↑ 𝑁 ) ) ) |
7 |
|
eqid |
⊢ ( 𝑦 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑦 ) · ( 𝑋 ↑ 𝑦 ) ) ) = ( 𝑦 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑦 ) · ( 𝑋 ↑ 𝑦 ) ) ) |
8 |
|
ovex |
⊢ ( ( 𝐴 ‘ 𝑁 ) · ( 𝑋 ↑ 𝑁 ) ) ∈ V |
9 |
6 7 8
|
fvmpt |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑦 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑦 ) · ( 𝑋 ↑ 𝑦 ) ) ) ‘ 𝑁 ) = ( ( 𝐴 ‘ 𝑁 ) · ( 𝑋 ↑ 𝑁 ) ) ) |
10 |
3 9
|
sylan9eq |
⊢ ( ( 𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑁 ) = ( ( 𝐴 ‘ 𝑁 ) · ( 𝑋 ↑ 𝑁 ) ) ) |