Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem43.s |
⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } ) |
2 |
|
etransclem43.x |
⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) ) |
3 |
|
etransclem43.p |
⊢ ( 𝜑 → 𝑃 ∈ ℕ ) |
4 |
|
etransclem43.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
5 |
|
etransclem43.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ) ) |
6 |
|
etransclem43.g |
⊢ 𝐺 = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑖 ∈ ( 0 ... 𝑅 ) ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑖 ) ‘ 𝑥 ) ) |
7 |
1 2
|
dvdmsscn |
⊢ ( 𝜑 → 𝑋 ⊆ ℂ ) |
8 |
|
fzfid |
⊢ ( 𝜑 → ( 0 ... 𝑅 ) ∈ Fin ) |
9 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑅 ) ) → 𝑆 ∈ { ℝ , ℂ } ) |
10 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑅 ) ) → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) ) |
11 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑅 ) ) → 𝑃 ∈ ℕ ) |
12 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑅 ) ) → 𝑀 ∈ ℕ0 ) |
13 |
|
elfznn0 |
⊢ ( 𝑖 ∈ ( 0 ... 𝑅 ) → 𝑖 ∈ ℕ0 ) |
14 |
13
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑅 ) ) → 𝑖 ∈ ℕ0 ) |
15 |
9 10 11 12 5 14
|
etransclem33 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑅 ) ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑖 ) : 𝑋 ⟶ ℂ ) |
16 |
15
|
feqmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑅 ) ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑖 ) = ( 𝑥 ∈ 𝑋 ↦ ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑖 ) ‘ 𝑥 ) ) ) |
17 |
9 10 11 12 5 14
|
etransclem40 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑅 ) ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑖 ) ∈ ( 𝑋 –cn→ ℂ ) ) |
18 |
16 17
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑅 ) ) → ( 𝑥 ∈ 𝑋 ↦ ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑖 ) ‘ 𝑥 ) ) ∈ ( 𝑋 –cn→ ℂ ) ) |
19 |
7 8 18
|
fsumcncf |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ Σ 𝑖 ∈ ( 0 ... 𝑅 ) ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑖 ) ‘ 𝑥 ) ) ∈ ( 𝑋 –cn→ ℂ ) ) |
20 |
6 19
|
eqeltrid |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝑋 –cn→ ℂ ) ) |