Step |
Hyp |
Ref |
Expression |
1 |
|
pser.g |
⊢ 𝐺 = ( 𝑥 ∈ ℂ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑛 ) · ( 𝑥 ↑ 𝑛 ) ) ) ) |
2 |
|
radcnv.a |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ ) |
3 |
|
psergf.x |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
4 |
1
|
pserval |
⊢ ( 𝑋 ∈ ℂ → ( 𝐺 ‘ 𝑋 ) = ( 𝑚 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑚 ) · ( 𝑋 ↑ 𝑚 ) ) ) ) |
5 |
4
|
adantl |
⊢ ( ( 𝐴 : ℕ0 ⟶ ℂ ∧ 𝑋 ∈ ℂ ) → ( 𝐺 ‘ 𝑋 ) = ( 𝑚 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑚 ) · ( 𝑋 ↑ 𝑚 ) ) ) ) |
6 |
|
ffvelrn |
⊢ ( ( 𝐴 : ℕ0 ⟶ ℂ ∧ 𝑚 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑚 ) ∈ ℂ ) |
7 |
6
|
adantlr |
⊢ ( ( ( 𝐴 : ℕ0 ⟶ ℂ ∧ 𝑋 ∈ ℂ ) ∧ 𝑚 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑚 ) ∈ ℂ ) |
8 |
|
expcl |
⊢ ( ( 𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ0 ) → ( 𝑋 ↑ 𝑚 ) ∈ ℂ ) |
9 |
8
|
adantll |
⊢ ( ( ( 𝐴 : ℕ0 ⟶ ℂ ∧ 𝑋 ∈ ℂ ) ∧ 𝑚 ∈ ℕ0 ) → ( 𝑋 ↑ 𝑚 ) ∈ ℂ ) |
10 |
7 9
|
mulcld |
⊢ ( ( ( 𝐴 : ℕ0 ⟶ ℂ ∧ 𝑋 ∈ ℂ ) ∧ 𝑚 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑚 ) · ( 𝑋 ↑ 𝑚 ) ) ∈ ℂ ) |
11 |
5 10
|
fmpt3d |
⊢ ( ( 𝐴 : ℕ0 ⟶ ℂ ∧ 𝑋 ∈ ℂ ) → ( 𝐺 ‘ 𝑋 ) : ℕ0 ⟶ ℂ ) |
12 |
2 3 11
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) : ℕ0 ⟶ ℂ ) |