Step |
Hyp |
Ref |
Expression |
1 |
|
abscvgcvg.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
abscvgcvg.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
abscvgcvg.3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( abs ‘ ( 𝐺 ‘ 𝑘 ) ) ) |
4 |
|
abscvgcvg.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
5 |
|
abscvgcvg.5 |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
6 |
|
uzid |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
7 |
2 6
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
8 |
7 1
|
eleqtrrdi |
⊢ ( 𝜑 → 𝑀 ∈ 𝑍 ) |
9 |
4
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( 𝐺 ‘ 𝑘 ) ) ∈ ℝ ) |
10 |
3 9
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
11 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
12 |
1
|
eleq2i |
⊢ ( 𝑘 ∈ 𝑍 ↔ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
13 |
3
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( 𝐺 ‘ 𝑘 ) ) = ( 𝐹 ‘ 𝑘 ) ) |
14 |
9 13
|
eqled |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( 𝐺 ‘ 𝑘 ) ) ≤ ( 𝐹 ‘ 𝑘 ) ) |
15 |
10
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
16 |
15
|
mulid2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 1 · ( 𝐹 ‘ 𝑘 ) ) = ( 𝐹 ‘ 𝑘 ) ) |
17 |
14 16
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( 𝐺 ‘ 𝑘 ) ) ≤ ( 1 · ( 𝐹 ‘ 𝑘 ) ) ) |
18 |
12 17
|
sylan2br |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( abs ‘ ( 𝐺 ‘ 𝑘 ) ) ≤ ( 1 · ( 𝐹 ‘ 𝑘 ) ) ) |
19 |
1 8 10 4 5 11 18
|
cvgcmpce |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐺 ) ∈ dom ⇝ ) |