Step |
Hyp |
Ref |
Expression |
1 |
|
knoppndvlem8.t |
⊢ 𝑇 = ( 𝑥 ∈ ℝ ↦ ( abs ‘ ( ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) − 𝑥 ) ) ) |
2 |
|
knoppndvlem8.f |
⊢ 𝐹 = ( 𝑦 ∈ ℝ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐶 ↑ 𝑛 ) · ( 𝑇 ‘ ( ( ( 2 · 𝑁 ) ↑ 𝑛 ) · 𝑦 ) ) ) ) ) |
3 |
|
knoppndvlem8.a |
⊢ 𝐴 = ( ( ( ( 2 · 𝑁 ) ↑ - 𝐽 ) / 2 ) · 𝑀 ) |
4 |
|
knoppndvlem8.c |
⊢ ( 𝜑 → 𝐶 ∈ ( - 1 (,) 1 ) ) |
5 |
|
knoppndvlem8.j |
⊢ ( 𝜑 → 𝐽 ∈ ℕ0 ) |
6 |
|
knoppndvlem8.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
7 |
|
knoppndvlem8.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
8 |
|
knoppndvlem8.1 |
⊢ ( 𝜑 → 2 ∥ 𝑀 ) |
9 |
1 2 3 5 6 7
|
knoppndvlem7 |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝐽 ) = ( ( 𝐶 ↑ 𝐽 ) · ( 𝑇 ‘ ( 𝑀 / 2 ) ) ) ) |
10 |
|
2z |
⊢ 2 ∈ ℤ |
11 |
10
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℤ ) |
12 |
|
2ne0 |
⊢ 2 ≠ 0 |
13 |
12
|
a1i |
⊢ ( 𝜑 → 2 ≠ 0 ) |
14 |
11 13 6
|
3jca |
⊢ ( 𝜑 → ( 2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑀 ∈ ℤ ) ) |
15 |
|
dvdsval2 |
⊢ ( ( 2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑀 ∈ ℤ ) → ( 2 ∥ 𝑀 ↔ ( 𝑀 / 2 ) ∈ ℤ ) ) |
16 |
14 15
|
syl |
⊢ ( 𝜑 → ( 2 ∥ 𝑀 ↔ ( 𝑀 / 2 ) ∈ ℤ ) ) |
17 |
8 16
|
mpbid |
⊢ ( 𝜑 → ( 𝑀 / 2 ) ∈ ℤ ) |
18 |
1 17
|
dnizeq0 |
⊢ ( 𝜑 → ( 𝑇 ‘ ( 𝑀 / 2 ) ) = 0 ) |
19 |
18
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝐶 ↑ 𝐽 ) · ( 𝑇 ‘ ( 𝑀 / 2 ) ) ) = ( ( 𝐶 ↑ 𝐽 ) · 0 ) ) |
20 |
4
|
knoppndvlem3 |
⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ∧ ( abs ‘ 𝐶 ) < 1 ) ) |
21 |
20
|
simpld |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
22 |
21
|
recnd |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
23 |
22 5
|
expcld |
⊢ ( 𝜑 → ( 𝐶 ↑ 𝐽 ) ∈ ℂ ) |
24 |
23
|
mul01d |
⊢ ( 𝜑 → ( ( 𝐶 ↑ 𝐽 ) · 0 ) = 0 ) |
25 |
9 19 24
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝐽 ) = 0 ) |