Step |
Hyp |
Ref |
Expression |
1 |
|
cvgcmp.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
cvgcmp.2 |
⊢ ( 𝜑 → 𝑁 ∈ 𝑍 ) |
3 |
|
cvgcmp.3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
4 |
|
cvgcmp.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) ∈ ℝ ) |
5 |
|
cvgcmp.5 |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
6 |
|
cvgcmp.6 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 0 ≤ ( 𝐺 ‘ 𝑘 ) ) |
7 |
|
cvgcmp.7 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑘 ) ) |
8 |
|
seqex |
⊢ seq 𝑀 ( + , 𝐺 ) ∈ V |
9 |
8
|
a1i |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐺 ) ∈ V ) |
10 |
2 1
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
11 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
12 |
10 11
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
13 |
1
|
climcau |
⊢ ( ( 𝑀 ∈ ℤ ∧ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) |
14 |
12 5 13
|
syl2anc |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) |
15 |
1 12 3
|
serfre |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℝ ) |
16 |
15
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℝ ) |
17 |
16
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ) |
18 |
17
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ 𝑍 ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ) |
19 |
1
|
r19.29uz |
⊢ ( ( ∀ 𝑛 ∈ 𝑍 ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
20 |
19
|
ex |
⊢ ( ∀ 𝑛 ∈ 𝑍 ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ → ( ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 → ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
21 |
18 20
|
syl |
⊢ ( 𝜑 → ( ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 → ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
22 |
21
|
ralimdv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
23 |
14 22
|
mpd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
24 |
1
|
uztrn2 |
⊢ ( ( 𝑁 ∈ 𝑍 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑛 ∈ 𝑍 ) |
25 |
2 24
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑛 ∈ 𝑍 ) |
26 |
1 12 4
|
serfre |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐺 ) : 𝑍 ⟶ ℝ ) |
27 |
26
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ) |
28 |
27
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ) |
29 |
25 28
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ) |
30 |
29
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ) |
31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ) |
32 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝜑 ) |
33 |
32 15
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℝ ) |
34 |
32 2
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝑁 ∈ 𝑍 ) |
35 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
36 |
1
|
uztrn2 |
⊢ ( ( 𝑁 ∈ 𝑍 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑚 ∈ 𝑍 ) |
37 |
34 35 36
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝑚 ∈ 𝑍 ) |
38 |
33 37
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ∈ ℝ ) |
39 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑁 ) = ( ℤ≥ ‘ 𝑁 ) |
40 |
39
|
uztrn2 |
⊢ ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
41 |
40
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
42 |
34 41 24
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝑛 ∈ 𝑍 ) |
43 |
32 42 16
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℝ ) |
44 |
32 42 27
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ) |
45 |
32 26
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → seq 𝑀 ( + , 𝐺 ) : 𝑍 ⟶ ℝ ) |
46 |
45 37
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ∈ ℝ ) |
47 |
44 46
|
resubcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ∈ ℝ ) |
48 |
37 1
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
49 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) |
50 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑛 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
51 |
50 1
|
eleqtrrdi |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑛 ) → 𝑘 ∈ 𝑍 ) |
52 |
|
fveq2 |
⊢ ( 𝑚 = 𝑘 → ( 𝐹 ‘ 𝑚 ) = ( 𝐹 ‘ 𝑘 ) ) |
53 |
|
fveq2 |
⊢ ( 𝑚 = 𝑘 → ( 𝐺 ‘ 𝑚 ) = ( 𝐺 ‘ 𝑘 ) ) |
54 |
52 53
|
oveq12d |
⊢ ( 𝑚 = 𝑘 → ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) = ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
55 |
|
eqid |
⊢ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) = ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) |
56 |
|
ovex |
⊢ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ∈ V |
57 |
54 55 56
|
fvmpt |
⊢ ( 𝑘 ∈ 𝑍 → ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
58 |
57
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
59 |
3 4
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ∈ ℝ ) |
60 |
58 59
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) ∈ ℝ ) |
61 |
32 51 60
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) → ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) ∈ ℝ ) |
62 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( ( 𝑚 + 1 ) ... 𝑛 ) → 𝑘 ∈ ( ℤ≥ ‘ ( 𝑚 + 1 ) ) ) |
63 |
|
peano2uz |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) → ( 𝑚 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
64 |
35 63
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( 𝑚 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
65 |
39
|
uztrn2 |
⊢ ( ( ( 𝑚 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑚 + 1 ) ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
66 |
64 65
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑚 + 1 ) ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
67 |
1
|
uztrn2 |
⊢ ( ( 𝑁 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑘 ∈ 𝑍 ) |
68 |
2 67
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑘 ∈ 𝑍 ) |
69 |
3 4
|
subge0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 0 ≤ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ↔ ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
70 |
68 69
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 0 ≤ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ↔ ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
71 |
7 70
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 0 ≤ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
72 |
68 57
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
73 |
71 72
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 0 ≤ ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) ) |
74 |
32 66 73
|
syl2an2r |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑚 + 1 ) ) ) → 0 ≤ ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) ) |
75 |
62 74
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( ( 𝑚 + 1 ) ... 𝑛 ) ) → 0 ≤ ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) ) |
76 |
48 49 61 75
|
sermono |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ) ‘ 𝑚 ) ≤ ( seq 𝑀 ( + , ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ) ‘ 𝑛 ) ) |
77 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑚 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
78 |
77 1
|
eleqtrrdi |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑚 ) → 𝑘 ∈ 𝑍 ) |
79 |
3
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
80 |
32 78 79
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑚 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
81 |
4
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
82 |
32 78 81
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑚 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
83 |
32 78 58
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑚 ) ) → ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
84 |
48 80 82 83
|
sersub |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ) ‘ 𝑚 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) |
85 |
42 1
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
86 |
32 51 79
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
87 |
32 51 81
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
88 |
32 51 58
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) → ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
89 |
85 86 87 88
|
sersub |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) − ( 𝐺 ‘ 𝑚 ) ) ) ) ‘ 𝑛 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) ) |
90 |
76 84 89
|
3brtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ≤ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) ) |
91 |
43 44
|
resubcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) ∈ ℝ ) |
92 |
38 46 91
|
lesubaddd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ≤ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) ↔ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ≤ ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) ) |
93 |
90 92
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ≤ ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) |
94 |
43
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ) |
95 |
44
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ) |
96 |
46
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ∈ ℂ ) |
97 |
94 95 96
|
subsubd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) |
98 |
93 97
|
breqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ≤ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) ) |
99 |
38 43 47 98
|
lesubd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ≤ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) |
100 |
43 38
|
resubcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ∈ ℝ ) |
101 |
|
rpre |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ ) |
102 |
101
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → 𝑥 ∈ ℝ ) |
103 |
|
lelttr |
⊢ ( ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ∈ ℝ ∧ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ≤ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ∧ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) < 𝑥 ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) < 𝑥 ) ) |
104 |
47 100 102 103
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ≤ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ∧ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) < 𝑥 ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) < 𝑥 ) ) |
105 |
99 104
|
mpand |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) < 𝑥 → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) < 𝑥 ) ) |
106 |
32 51 3
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
107 |
62 66
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( ( 𝑚 + 1 ) ... 𝑛 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
108 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 0 ∈ ℝ ) |
109 |
68 4
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℝ ) |
110 |
68 3
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
111 |
108 109 110 6 7
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) ) |
112 |
32 107 111
|
syl2an2r |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( ( 𝑚 + 1 ) ... 𝑛 ) ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) ) |
113 |
48 49 106 112
|
sermono |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ≤ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ) |
114 |
38 43 113
|
abssubge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) |
115 |
114
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ↔ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) < 𝑥 ) ) |
116 |
32 51 4
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℝ ) |
117 |
32 66 6
|
syl2an2r |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑚 + 1 ) ) ) → 0 ≤ ( 𝐺 ‘ 𝑘 ) ) |
118 |
62 117
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ ( ( 𝑚 + 1 ) ... 𝑛 ) ) → 0 ≤ ( 𝐺 ‘ 𝑘 ) ) |
119 |
48 49 116 118
|
sermono |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ≤ ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) |
120 |
46 44 119
|
abssubge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) |
121 |
120
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ↔ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) < 𝑥 ) ) |
122 |
105 115 121
|
3imtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 → ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
123 |
122
|
anassrs |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 → ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
124 |
123
|
adantld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
125 |
124
|
ralimdva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
126 |
125
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
127 |
39
|
r19.29uz |
⊢ ( ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
128 |
31 126 127
|
syl6an |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
129 |
128
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
130 |
1 39
|
cau4 |
⊢ ( 𝑁 ∈ 𝑍 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
131 |
2 130
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
132 |
1 39
|
cau4 |
⊢ ( 𝑁 ∈ 𝑍 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
133 |
2 132
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
134 |
129 131 133
|
3imtr4d |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
135 |
23 134
|
mpd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
136 |
1
|
uztrn2 |
⊢ ( ( 𝑚 ∈ 𝑍 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → 𝑛 ∈ 𝑍 ) |
137 |
|
simpr |
⊢ ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) |
138 |
27
|
biantrurd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ↔ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
139 |
137 138
|
syl5ib |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
140 |
136 139
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑍 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) → ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
141 |
140
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
142 |
141
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
143 |
142
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
144 |
143
|
ralimdv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℂ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
145 |
135 144
|
mpd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ ℝ ∧ ( abs ‘ ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑚 ) ) ) < 𝑥 ) ) |
146 |
1 9 145
|
caurcvg2 |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐺 ) ∈ dom ⇝ ) |