Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem8.x |
⊢ ( 𝜑 → 𝑋 ⊆ ℂ ) |
2 |
|
etransclem8.p |
⊢ ( 𝜑 → 𝑃 ∈ ℕ ) |
3 |
|
etransclem8.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ) ) |
4 |
1
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝑥 ∈ ℂ ) |
5 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝑃 ∈ ℕ ) |
6 |
|
nnm1nn0 |
⊢ ( 𝑃 ∈ ℕ → ( 𝑃 − 1 ) ∈ ℕ0 ) |
7 |
5 6
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝑃 − 1 ) ∈ ℕ0 ) |
8 |
4 7
|
expcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝑥 ↑ ( 𝑃 − 1 ) ) ∈ ℂ ) |
9 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 1 ... 𝑀 ) ∈ Fin ) |
10 |
4
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑥 ∈ ℂ ) |
11 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 𝑗 ∈ ℤ ) |
12 |
11
|
zcnd |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 𝑗 ∈ ℂ ) |
13 |
12
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑗 ∈ ℂ ) |
14 |
10 13
|
subcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑥 − 𝑗 ) ∈ ℂ ) |
15 |
2
|
nnnn0d |
⊢ ( 𝜑 → 𝑃 ∈ ℕ0 ) |
16 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑃 ∈ ℕ0 ) |
17 |
14 16
|
expcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ∈ ℂ ) |
18 |
9 17
|
fprodcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ∈ ℂ ) |
19 |
8 18
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ) ∈ ℂ ) |
20 |
19 3
|
fmptd |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℂ ) |