Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem19.s |
⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } ) |
2 |
|
etransclem19.x |
⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) ) |
3 |
|
etransclem19.p |
⊢ ( 𝜑 → 𝑃 ∈ ℕ ) |
4 |
|
etransclem19.1 |
⊢ 𝐻 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 − 𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) |
5 |
|
etransclem19.J |
⊢ ( 𝜑 → 𝐽 ∈ ( 0 ... 𝑀 ) ) |
6 |
|
etransclem19.n |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
7 |
|
etransclem19.7 |
⊢ ( 𝜑 → if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) < 𝑁 ) |
8 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
9 |
6
|
zred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
10 |
|
nnm1nn0 |
⊢ ( 𝑃 ∈ ℕ → ( 𝑃 − 1 ) ∈ ℕ0 ) |
11 |
3 10
|
syl |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ℕ0 ) |
12 |
11
|
nn0red |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ℝ ) |
13 |
3
|
nnred |
⊢ ( 𝜑 → 𝑃 ∈ ℝ ) |
14 |
12 13
|
ifcld |
⊢ ( 𝜑 → if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ∈ ℝ ) |
15 |
11
|
nn0ge0d |
⊢ ( 𝜑 → 0 ≤ ( 𝑃 − 1 ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = 0 ) → 0 ≤ ( 𝑃 − 1 ) ) |
17 |
|
iftrue |
⊢ ( 𝐽 = 0 → if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) = ( 𝑃 − 1 ) ) |
18 |
17
|
eqcomd |
⊢ ( 𝐽 = 0 → ( 𝑃 − 1 ) = if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐽 = 0 ) → ( 𝑃 − 1 ) = if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) |
20 |
16 19
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝐽 = 0 ) → 0 ≤ if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) |
21 |
3
|
nnnn0d |
⊢ ( 𝜑 → 𝑃 ∈ ℕ0 ) |
22 |
21
|
nn0ge0d |
⊢ ( 𝜑 → 0 ≤ 𝑃 ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐽 = 0 ) → 0 ≤ 𝑃 ) |
24 |
|
iffalse |
⊢ ( ¬ 𝐽 = 0 → if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) = 𝑃 ) |
25 |
24
|
eqcomd |
⊢ ( ¬ 𝐽 = 0 → 𝑃 = if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) |
26 |
25
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝐽 = 0 ) → 𝑃 = if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) |
27 |
23 26
|
breqtrd |
⊢ ( ( 𝜑 ∧ ¬ 𝐽 = 0 ) → 0 ≤ if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) |
28 |
20 27
|
pm2.61dan |
⊢ ( 𝜑 → 0 ≤ if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) |
29 |
8 14 9 28 7
|
lelttrd |
⊢ ( 𝜑 → 0 < 𝑁 ) |
30 |
8 9 29
|
ltled |
⊢ ( 𝜑 → 0 ≤ 𝑁 ) |
31 |
|
elnn0z |
⊢ ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℤ ∧ 0 ≤ 𝑁 ) ) |
32 |
6 30 31
|
sylanbrc |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
33 |
1 2 3 4 5 32
|
etransclem17 |
⊢ ( 𝜑 → ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝐽 ) ) ‘ 𝑁 ) = ( 𝑥 ∈ 𝑋 ↦ if ( if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) < 𝑁 , 0 , ( ( ( ! ‘ if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) / ( ! ‘ ( if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) − 𝑁 ) ) ) · ( ( 𝑥 − 𝐽 ) ↑ ( if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) − 𝑁 ) ) ) ) ) ) |
34 |
7
|
iftrued |
⊢ ( 𝜑 → if ( if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) < 𝑁 , 0 , ( ( ( ! ‘ if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) / ( ! ‘ ( if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) − 𝑁 ) ) ) · ( ( 𝑥 − 𝐽 ) ↑ ( if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) − 𝑁 ) ) ) ) = 0 ) |
35 |
34
|
mpteq2dv |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ if ( if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) < 𝑁 , 0 , ( ( ( ! ‘ if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) / ( ! ‘ ( if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) − 𝑁 ) ) ) · ( ( 𝑥 − 𝐽 ) ↑ ( if ( 𝐽 = 0 , ( 𝑃 − 1 ) , 𝑃 ) − 𝑁 ) ) ) ) ) = ( 𝑥 ∈ 𝑋 ↦ 0 ) ) |
36 |
33 35
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝐽 ) ) ‘ 𝑁 ) = ( 𝑥 ∈ 𝑋 ↦ 0 ) ) |