Step |
Hyp |
Ref |
Expression |
1 |
|
caucvgb.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
serf0.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
serf0.3 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
4 |
|
serf0.4 |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
5 |
|
serf0.5 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
6 |
1
|
caucvgb |
⊢ ( ( 𝑀 ∈ ℤ ∧ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) → ( seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
7 |
2 4 6
|
syl2anc |
⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
8 |
4 7
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ) ) < 𝑥 ) ) |
9 |
1
|
cau3 |
⊢ ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 ) ) |
10 |
8 9
|
sylib |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 ) ) |
11 |
1
|
peano2uzs |
⊢ ( 𝑗 ∈ 𝑍 → ( 𝑗 + 1 ) ∈ 𝑍 ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝑗 + 1 ) ∈ 𝑍 ) |
13 |
|
eluzelz |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) → 𝑚 ∈ ℤ ) |
14 |
|
uzid |
⊢ ( 𝑚 ∈ ℤ → 𝑚 ∈ ( ℤ≥ ‘ 𝑚 ) ) |
15 |
|
peano2uz |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑚 ) → ( 𝑚 + 1 ) ∈ ( ℤ≥ ‘ 𝑚 ) ) |
16 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) = ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) |
17 |
16
|
oveq2d |
⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) |
18 |
17
|
fveq2d |
⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) = ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) ) |
19 |
18
|
breq1d |
⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 ↔ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 ) ) |
20 |
19
|
rspcv |
⊢ ( ( 𝑚 + 1 ) ∈ ( ℤ≥ ‘ 𝑚 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 ) ) |
21 |
13 14 15 20
|
4syl |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 ) ) |
22 |
21
|
adantld |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 ) ) |
23 |
22
|
ralimia |
⊢ ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 ) |
24 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ 𝑍 ) |
25 |
24 1
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
26 |
|
eluzelz |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑗 ∈ ℤ ) |
27 |
25 26
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ ℤ ) |
28 |
|
eluzp1m1 |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( 𝑘 − 1 ) ∈ ( ℤ≥ ‘ 𝑗 ) ) |
29 |
27 28
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( 𝑘 − 1 ) ∈ ( ℤ≥ ‘ 𝑗 ) ) |
30 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑘 − 1 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) = ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) ) |
31 |
|
fvoveq1 |
⊢ ( 𝑚 = ( 𝑘 − 1 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) = ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) |
32 |
30 31
|
oveq12d |
⊢ ( 𝑚 = ( 𝑘 − 1 ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) ) |
33 |
32
|
fveq2d |
⊢ ( 𝑚 = ( 𝑘 − 1 ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) = ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) ) ) |
34 |
33
|
breq1d |
⊢ ( 𝑚 = ( 𝑘 − 1 ) → ( ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 ↔ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) ) < 𝑥 ) ) |
35 |
34
|
rspcv |
⊢ ( ( 𝑘 − 1 ) ∈ ( ℤ≥ ‘ 𝑗 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) ) < 𝑥 ) ) |
36 |
29 35
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) ) < 𝑥 ) ) |
37 |
1 2 5
|
serf |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℂ ) |
38 |
37
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℂ ) |
39 |
1
|
uztrn2 |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ ( 𝑘 − 1 ) ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝑘 − 1 ) ∈ 𝑍 ) |
40 |
24 29 39
|
syl2an2r |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( 𝑘 − 1 ) ∈ 𝑍 ) |
41 |
38 40
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) ∈ ℂ ) |
42 |
1
|
uztrn2 |
⊢ ( ( ( 𝑗 + 1 ) ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → 𝑘 ∈ 𝑍 ) |
43 |
12 42
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → 𝑘 ∈ 𝑍 ) |
44 |
38 43
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ∈ ℂ ) |
45 |
41 44
|
abssubd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) = ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) ) ) ) |
46 |
|
eluzelz |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) → 𝑘 ∈ ℤ ) |
47 |
46
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → 𝑘 ∈ ℤ ) |
48 |
47
|
zcnd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → 𝑘 ∈ ℂ ) |
49 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
50 |
|
npcan |
⊢ ( ( 𝑘 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑘 − 1 ) + 1 ) = 𝑘 ) |
51 |
48 49 50
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( ( 𝑘 − 1 ) + 1 ) = 𝑘 ) |
52 |
51
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) |
53 |
52
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) |
54 |
53
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) ) = ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) ) |
55 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → 𝑀 ∈ ℤ ) |
56 |
|
eluzp1p1 |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
57 |
25 56
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
58 |
|
eqid |
⊢ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) = ( ℤ≥ ‘ ( 𝑀 + 1 ) ) |
59 |
58
|
uztrn2 |
⊢ ( ( ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
60 |
57 59
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
61 |
|
seqm1 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) + ( 𝐹 ‘ 𝑘 ) ) ) |
62 |
55 60 61
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) + ( 𝐹 ‘ 𝑘 ) ) ) |
63 |
62
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) + ( 𝐹 ‘ 𝑘 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) ) ) |
64 |
5
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
65 |
43 64
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
66 |
41 65
|
pncan2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) + ( 𝐹 ‘ 𝑘 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) ) = ( 𝐹 ‘ 𝑘 ) ) |
67 |
63 66
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) ) ) |
68 |
67
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) = ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) ) ) ) |
69 |
45 54 68
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) ) = ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
70 |
69
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑘 − 1 ) ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( ( 𝑘 − 1 ) + 1 ) ) ) ) < 𝑥 ↔ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
71 |
36 70
|
sylibd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
72 |
71
|
ralrimdva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ ( 𝑚 + 1 ) ) ) ) < 𝑥 → ∀ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
73 |
23 72
|
syl5 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
74 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑗 + 1 ) → ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) |
75 |
74
|
raleqdv |
⊢ ( 𝑛 = ( 𝑗 + 1 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
76 |
75
|
rspcev |
⊢ ( ( ( 𝑗 + 1 ) ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) → ∃ 𝑛 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) |
77 |
12 73 76
|
syl6an |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 ) → ∃ 𝑛 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
78 |
77
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 ) → ∃ 𝑛 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
79 |
78
|
ralimdv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℂ ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑘 ) ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑛 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
80 |
10 79
|
mpd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑛 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) |
81 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
82 |
1 2 3 81 5
|
clim0c |
⊢ ( 𝜑 → ( 𝐹 ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑛 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑛 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑥 ) ) |
83 |
80 82
|
mpbird |
⊢ ( 𝜑 → 𝐹 ⇝ 0 ) |