Step |
Hyp |
Ref |
Expression |
1 |
|
climabs0.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
climabs0.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
climabs0.3 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
4 |
|
climabs0.4 |
⊢ ( 𝜑 → 𝐺 ∈ 𝑊 ) |
5 |
|
climabs0.5 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
6 |
|
climabs0.6 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) = ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
7 |
1
|
uztrn2 |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
8 |
|
absidm |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ → ( abs ‘ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) = ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
9 |
5 8
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) = ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
10 |
9
|
breq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( abs ‘ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
11 |
7 10
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( abs ‘ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
12 |
11
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( abs ‘ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
13 |
12
|
ralbidva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
14 |
13
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
15 |
14
|
ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
16 |
5
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
17 |
16
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
18 |
1 2 4 6 17
|
clim0c |
⊢ ( 𝜑 → ( 𝐺 ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ) ) |
19 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
20 |
1 2 3 19 5
|
clim0c |
⊢ ( 𝜑 → ( 𝐹 ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
21 |
15 18 20
|
3bitr4rd |
⊢ ( 𝜑 → ( 𝐹 ⇝ 0 ↔ 𝐺 ⇝ 0 ) ) |