Step |
Hyp |
Ref |
Expression |
1 |
|
taylpfval.s |
⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } ) |
2 |
|
taylpfval.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ ) |
3 |
|
taylpfval.a |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 ) |
4 |
|
taylpfval.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
5 |
|
taylpfval.b |
⊢ ( 𝜑 → 𝐵 ∈ dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) |
6 |
|
0z |
⊢ 0 ∈ ℤ |
7 |
4
|
nn0zd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
8 |
|
fzval2 |
⊢ ( ( 0 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 0 ... 𝑁 ) = ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) |
9 |
6 7 8
|
sylancr |
⊢ ( 𝜑 → ( 0 ... 𝑁 ) = ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) |
10 |
9
|
eleq2d |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) ) |
11 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) ) |
12 |
11
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) |
13 |
4
|
orcd |
⊢ ( 𝜑 → ( 𝑁 ∈ ℕ0 ∨ 𝑁 = +∞ ) ) |
14 |
1 2 3 4 5
|
taylplem1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐵 ∈ dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ) |
15 |
1 2 3 13 14
|
taylfvallem1 |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ‘ 𝐵 ) / ( ! ‘ 𝑘 ) ) · ( ( 𝑋 − 𝐵 ) ↑ 𝑘 ) ) ∈ ℂ ) |
16 |
12 15
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → ( ( ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ‘ 𝐵 ) / ( ! ‘ 𝑘 ) ) · ( ( 𝑋 − 𝐵 ) ↑ 𝑘 ) ) ∈ ℂ ) |