Step |
Hyp |
Ref |
Expression |
1 |
|
knoppndv.t |
⊢ 𝑇 = ( 𝑥 ∈ ℝ ↦ ( abs ‘ ( ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) − 𝑥 ) ) ) |
2 |
|
knoppndv.f |
⊢ 𝐹 = ( 𝑦 ∈ ℝ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐶 ↑ 𝑛 ) · ( 𝑇 ‘ ( ( ( 2 · 𝑁 ) ↑ 𝑛 ) · 𝑦 ) ) ) ) ) |
3 |
|
knoppndv.w |
⊢ 𝑊 = ( 𝑤 ∈ ℝ ↦ Σ 𝑖 ∈ ℕ0 ( ( 𝐹 ‘ 𝑤 ) ‘ 𝑖 ) ) |
4 |
|
knoppndv.c |
⊢ ( 𝜑 → 𝐶 ∈ ( - 1 (,) 1 ) ) |
5 |
|
knoppndv.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
6 |
|
knoppndv.1 |
⊢ ( 𝜑 → 1 < ( 𝑁 · ( abs ‘ 𝐶 ) ) ) |
7 |
|
simpl |
⊢ ( ( 𝜑 ∧ ℎ ∈ dom ( ℝ D 𝑊 ) ) → 𝜑 ) |
8 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
9 |
8
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
10 |
4
|
knoppndvlem3 |
⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ∧ ( abs ‘ 𝐶 ) < 1 ) ) |
11 |
10
|
simpld |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
12 |
10
|
simprd |
⊢ ( 𝜑 → ( abs ‘ 𝐶 ) < 1 ) |
13 |
1 2 3 5 11 12
|
knoppcn |
⊢ ( 𝜑 → 𝑊 ∈ ( ℝ –cn→ ℂ ) ) |
14 |
|
cncff |
⊢ ( 𝑊 ∈ ( ℝ –cn→ ℂ ) → 𝑊 : ℝ ⟶ ℂ ) |
15 |
13 14
|
syl |
⊢ ( 𝜑 → 𝑊 : ℝ ⟶ ℂ ) |
16 |
|
ssidd |
⊢ ( 𝜑 → ℝ ⊆ ℝ ) |
17 |
9 15 16
|
dvbss |
⊢ ( 𝜑 → dom ( ℝ D 𝑊 ) ⊆ ℝ ) |
18 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ℎ ∈ dom ( ℝ D 𝑊 ) ) → dom ( ℝ D 𝑊 ) ⊆ ℝ ) |
19 |
|
simpr |
⊢ ( ( 𝜑 ∧ ℎ ∈ dom ( ℝ D 𝑊 ) ) → ℎ ∈ dom ( ℝ D 𝑊 ) ) |
20 |
18 19
|
sseldd |
⊢ ( ( 𝜑 ∧ ℎ ∈ dom ( ℝ D 𝑊 ) ) → ℎ ∈ ℝ ) |
21 |
7 20
|
jca |
⊢ ( ( 𝜑 ∧ ℎ ∈ dom ( ℝ D 𝑊 ) ) → ( 𝜑 ∧ ℎ ∈ ℝ ) ) |
22 |
|
ssidd |
⊢ ( ( 𝜑 ∧ ℎ ∈ ℝ ) → ℝ ⊆ ℝ ) |
23 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ ℎ ∈ ℝ ) → 𝑊 : ℝ ⟶ ℂ ) |
24 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ℎ ∈ ℝ ) ∧ ( 𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+ ) ) → 𝐶 ∈ ( - 1 (,) 1 ) ) |
25 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ ℎ ∈ ℝ ) ∧ ( 𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+ ) ) → 𝑑 ∈ ℝ+ ) |
26 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ ℎ ∈ ℝ ) ∧ ( 𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+ ) ) → 𝑒 ∈ ℝ+ ) |
27 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ℎ ∈ ℝ ) ∧ ( 𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+ ) ) → ℎ ∈ ℝ ) |
28 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ℎ ∈ ℝ ) ∧ ( 𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+ ) ) → 𝑁 ∈ ℕ ) |
29 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ℎ ∈ ℝ ) ∧ ( 𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+ ) ) → 1 < ( 𝑁 · ( abs ‘ 𝐶 ) ) ) |
30 |
1 2 3 24 25 26 27 28 29
|
knoppndvlem22 |
⊢ ( ( ( 𝜑 ∧ ℎ ∈ ℝ ) ∧ ( 𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+ ) ) → ∃ 𝑎 ∈ ℝ ∃ 𝑏 ∈ ℝ ( ( 𝑎 ≤ ℎ ∧ ℎ ≤ 𝑏 ) ∧ ( ( 𝑏 − 𝑎 ) < 𝑑 ∧ 𝑎 ≠ 𝑏 ) ∧ 𝑒 ≤ ( ( abs ‘ ( ( 𝑊 ‘ 𝑏 ) − ( 𝑊 ‘ 𝑎 ) ) ) / ( 𝑏 − 𝑎 ) ) ) ) |
31 |
30
|
ralrimivva |
⊢ ( ( 𝜑 ∧ ℎ ∈ ℝ ) → ∀ 𝑒 ∈ ℝ+ ∀ 𝑑 ∈ ℝ+ ∃ 𝑎 ∈ ℝ ∃ 𝑏 ∈ ℝ ( ( 𝑎 ≤ ℎ ∧ ℎ ≤ 𝑏 ) ∧ ( ( 𝑏 − 𝑎 ) < 𝑑 ∧ 𝑎 ≠ 𝑏 ) ∧ 𝑒 ≤ ( ( abs ‘ ( ( 𝑊 ‘ 𝑏 ) − ( 𝑊 ‘ 𝑎 ) ) ) / ( 𝑏 − 𝑎 ) ) ) ) |
32 |
22 23 31
|
unbdqndv2 |
⊢ ( ( 𝜑 ∧ ℎ ∈ ℝ ) → ¬ ℎ ∈ dom ( ℝ D 𝑊 ) ) |
33 |
21 32
|
syl |
⊢ ( ( 𝜑 ∧ ℎ ∈ dom ( ℝ D 𝑊 ) ) → ¬ ℎ ∈ dom ( ℝ D 𝑊 ) ) |
34 |
33
|
pm2.01da |
⊢ ( 𝜑 → ¬ ℎ ∈ dom ( ℝ D 𝑊 ) ) |
35 |
34
|
alrimiv |
⊢ ( 𝜑 → ∀ ℎ ¬ ℎ ∈ dom ( ℝ D 𝑊 ) ) |
36 |
|
eq0 |
⊢ ( dom ( ℝ D 𝑊 ) = ∅ ↔ ∀ ℎ ¬ ℎ ∈ dom ( ℝ D 𝑊 ) ) |
37 |
35 36
|
sylibr |
⊢ ( 𝜑 → dom ( ℝ D 𝑊 ) = ∅ ) |