Step |
Hyp |
Ref |
Expression |
1 |
|
clim2ser.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
climserle.2 |
⊢ ( 𝜑 → 𝑁 ∈ 𝑍 ) |
3 |
|
climserle.3 |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) ⇝ 𝐴 ) |
4 |
|
climserle.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
5 |
|
climserle.5 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) ) |
6 |
2 1
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
7 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
8 |
6 7
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
9 |
1 8 4
|
serfre |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℝ ) |
10 |
9
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ∈ ℝ ) |
11 |
1
|
peano2uzs |
⊢ ( 𝑗 ∈ 𝑍 → ( 𝑗 + 1 ) ∈ 𝑍 ) |
12 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) |
13 |
12
|
breq2d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( 0 ≤ ( 𝐹 ‘ 𝑘 ) ↔ 0 ≤ ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) ) |
14 |
13
|
imbi2d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( ( 𝜑 → 0 ≤ ( 𝐹 ‘ 𝑘 ) ) ↔ ( 𝜑 → 0 ≤ ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) ) ) |
15 |
5
|
expcom |
⊢ ( 𝑘 ∈ 𝑍 → ( 𝜑 → 0 ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
16 |
14 15
|
vtoclga |
⊢ ( ( 𝑗 + 1 ) ∈ 𝑍 → ( 𝜑 → 0 ≤ ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) ) |
17 |
16
|
impcom |
⊢ ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ 𝑍 ) → 0 ≤ ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) |
18 |
11 17
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 0 ≤ ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) |
19 |
12
|
eleq1d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℝ ↔ ( 𝐹 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) ) |
20 |
19
|
imbi2d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( ( 𝜑 → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) ↔ ( 𝜑 → ( 𝐹 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) ) ) |
21 |
4
|
expcom |
⊢ ( 𝑘 ∈ 𝑍 → ( 𝜑 → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) ) |
22 |
20 21
|
vtoclga |
⊢ ( ( 𝑗 + 1 ) ∈ 𝑍 → ( 𝜑 → ( 𝐹 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) ) |
23 |
22
|
impcom |
⊢ ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ 𝑍 ) → ( 𝐹 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) |
24 |
11 23
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) |
25 |
10 24
|
addge01d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 0 ≤ ( 𝐹 ‘ ( 𝑗 + 1 ) ) ↔ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ≤ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) + ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) ) ) |
26 |
18 25
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ≤ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) + ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) ) |
27 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ 𝑍 ) |
28 |
27 1
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
29 |
|
seqp1 |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑗 + 1 ) ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) + ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) ) |
30 |
28 29
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑗 + 1 ) ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) + ( 𝐹 ‘ ( 𝑗 + 1 ) ) ) ) |
31 |
26 30
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ≤ ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑗 + 1 ) ) ) |
32 |
1 2 3 10 31
|
climub |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) ≤ 𝐴 ) |