Step |
Hyp |
Ref |
Expression |
1 |
|
taylfval.s |
⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } ) |
2 |
|
taylfval.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ ) |
3 |
|
taylfval.a |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 ) |
4 |
|
taylfval.n |
⊢ ( 𝜑 → ( 𝑁 ∈ ℕ0 ∨ 𝑁 = +∞ ) ) |
5 |
|
taylfval.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐵 ∈ dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ) |
6 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑆 ∈ { ℝ , ℂ } ) |
7 |
|
cnex |
⊢ ℂ ∈ V |
8 |
7
|
a1i |
⊢ ( 𝜑 → ℂ ∈ V ) |
9 |
|
elpm2r |
⊢ ( ( ( ℂ ∈ V ∧ 𝑆 ∈ { ℝ , ℂ } ) ∧ ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ⊆ 𝑆 ) ) → 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) |
10 |
8 1 2 3 9
|
syl22anc |
⊢ ( 𝜑 → 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) |
11 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) |
12 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) |
13 |
12
|
elin2d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑘 ∈ ℤ ) |
14 |
12
|
elin1d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑘 ∈ ( 0 [,] 𝑁 ) ) |
15 |
|
0xr |
⊢ 0 ∈ ℝ* |
16 |
|
nn0re |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ ) |
17 |
16
|
rexrd |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ* ) |
18 |
|
id |
⊢ ( 𝑁 = +∞ → 𝑁 = +∞ ) |
19 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
20 |
18 19
|
eqeltrdi |
⊢ ( 𝑁 = +∞ → 𝑁 ∈ ℝ* ) |
21 |
17 20
|
jaoi |
⊢ ( ( 𝑁 ∈ ℕ0 ∨ 𝑁 = +∞ ) → 𝑁 ∈ ℝ* ) |
22 |
4 21
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℝ* ) |
23 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑁 ∈ ℝ* ) |
24 |
|
elicc1 |
⊢ ( ( 0 ∈ ℝ* ∧ 𝑁 ∈ ℝ* ) → ( 𝑘 ∈ ( 0 [,] 𝑁 ) ↔ ( 𝑘 ∈ ℝ* ∧ 0 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁 ) ) ) |
25 |
15 23 24
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( 𝑘 ∈ ( 0 [,] 𝑁 ) ↔ ( 𝑘 ∈ ℝ* ∧ 0 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁 ) ) ) |
26 |
14 25
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( 𝑘 ∈ ℝ* ∧ 0 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁 ) ) |
27 |
26
|
simp2d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 0 ≤ 𝑘 ) |
28 |
|
elnn0z |
⊢ ( 𝑘 ∈ ℕ0 ↔ ( 𝑘 ∈ ℤ ∧ 0 ≤ 𝑘 ) ) |
29 |
13 27 28
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑘 ∈ ℕ0 ) |
30 |
|
dvnf |
⊢ ( ( 𝑆 ∈ { ℝ , ℂ } ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) : dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⟶ ℂ ) |
31 |
6 11 29 30
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) : dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⟶ ℂ ) |
32 |
5
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐵 ∈ dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ) |
33 |
31 32
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ‘ 𝐵 ) ∈ ℂ ) |
34 |
29
|
faccld |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ! ‘ 𝑘 ) ∈ ℕ ) |
35 |
34
|
nncnd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ! ‘ 𝑘 ) ∈ ℂ ) |
36 |
34
|
nnne0d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ! ‘ 𝑘 ) ≠ 0 ) |
37 |
33 35 36
|
divcld |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ‘ 𝐵 ) / ( ! ‘ 𝑘 ) ) ∈ ℂ ) |
38 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝑋 ∈ ℂ ) |
39 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐹 : 𝐴 ⟶ ℂ ) |
40 |
|
dvnbss |
⊢ ( ( 𝑆 ∈ { ℝ , ℂ } ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑘 ∈ ℕ0 ) → dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⊆ dom 𝐹 ) |
41 |
6 11 29 40
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⊆ dom 𝐹 ) |
42 |
39 41
|
fssdmd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⊆ 𝐴 ) |
43 |
|
recnprss |
⊢ ( 𝑆 ∈ { ℝ , ℂ } → 𝑆 ⊆ ℂ ) |
44 |
1 43
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ℂ ) |
45 |
3 44
|
sstrd |
⊢ ( 𝜑 → 𝐴 ⊆ ℂ ) |
46 |
45
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐴 ⊆ ℂ ) |
47 |
42 46
|
sstrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → dom ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ⊆ ℂ ) |
48 |
47 32
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → 𝐵 ∈ ℂ ) |
49 |
38 48
|
subcld |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( 𝑋 − 𝐵 ) ∈ ℂ ) |
50 |
49 29
|
expcld |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( 𝑋 − 𝐵 ) ↑ 𝑘 ) ∈ ℂ ) |
51 |
37 50
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 [,] 𝑁 ) ∩ ℤ ) ) → ( ( ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑘 ) ‘ 𝐵 ) / ( ! ‘ 𝑘 ) ) · ( ( 𝑋 − 𝐵 ) ↑ 𝑘 ) ) ∈ ℂ ) |