Step | Hyp | Ref
| Expression |
1 | | cvgcmpce.1 |
. 2
⊢ 𝑍 =
(ℤ≥‘𝑀) |
2 | | cvgcmpce.2 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ 𝑍) |
3 | 2, 1 | syl6eleq 2698 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
4 | | eluzel2 11568 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
5 | 3, 4 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
6 | | cvgcmpce.4 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
7 | 1, 5, 6 | serf 12691 |
. . 3
⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ) |
8 | 7 | ffvelrnda 6267 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ) |
9 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) |
10 | 9 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑚 = 𝑘 → (𝐶 · (𝐹‘𝑚)) = (𝐶 · (𝐹‘𝑘))) |
11 | | eqid 2610 |
. . . . . . . 8
⊢ (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚))) |
12 | | ovex 6577 |
. . . . . . . 8
⊢ (𝐶 · (𝐹‘𝑘)) ∈ V |
13 | 10, 11, 12 | fvmpt 6191 |
. . . . . . 7
⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
14 | 13 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
15 | | cvgcmpce.6 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ ℝ) |
16 | 15 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℝ) |
17 | | cvgcmpce.3 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
18 | 16, 17 | remulcld 9949 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐶 · (𝐹‘𝑘)) ∈ ℝ) |
19 | 14, 18 | eqeltrd 2688 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) ∈ ℝ) |
20 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘)) |
21 | 20 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝑚 = 𝑘 → (abs‘(𝐺‘𝑚)) = (abs‘(𝐺‘𝑘))) |
22 | | eqid 2610 |
. . . . . . . 8
⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))) |
23 | | fvex 6113 |
. . . . . . . 8
⊢
(abs‘(𝐺‘𝑘)) ∈ V |
24 | 21, 22, 23 | fvmpt 6191 |
. . . . . . 7
⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
25 | 24 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
26 | 6 | abscld 14023 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐺‘𝑘)) ∈ ℝ) |
27 | 25, 26 | eqeltrd 2688 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) ∈ ℝ) |
28 | 15 | recnd 9947 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ ℂ) |
29 | | cvgcmpce.5 |
. . . . . . . 8
⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
30 | | climdm 14133 |
. . . . . . . 8
⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹))) |
31 | 29, 30 | sylib 207 |
. . . . . . 7
⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹))) |
32 | 17 | recnd 9947 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
33 | 1, 5, 28, 31, 32, 14 | isermulc2 14236 |
. . . . . 6
⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹)))) |
34 | | climrel 14071 |
. . . . . . 7
⊢ Rel
⇝ |
35 | 34 | releldmi 5283 |
. . . . . 6
⊢ (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))) → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ∈ dom ⇝ ) |
36 | 33, 35 | syl 17 |
. . . . 5
⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ∈ dom ⇝ ) |
37 | 1 | uztrn2 11581 |
. . . . . . 7
⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
38 | 2, 37 | sylan 487 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
39 | 6 | absge0d 14031 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (abs‘(𝐺‘𝑘))) |
40 | 39, 25 | breqtrrd 4611 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘)) |
41 | 38, 40 | syldan 486 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘)) |
42 | | cvgcmpce.7 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (abs‘(𝐺‘𝑘)) ≤ (𝐶 · (𝐹‘𝑘))) |
43 | 38, 24 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
44 | 38, 13 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
45 | 42, 43, 44 | 3brtr4d 4615 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) ≤ ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘)) |
46 | 1, 2, 19, 27, 36, 41, 45 | cvgcmp 14389 |
. . . 4
⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))) ∈ dom ⇝ ) |
47 | 1 | climcau 14249 |
. . . 4
⊢ ((𝑀 ∈ ℤ ∧ seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))) ∈ dom ⇝ ) →
∀𝑥 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥) |
48 | 5, 46, 47 | syl2anc 691 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥) |
49 | 1, 5, 27 | serfre 12692 |
. . . . . . . . . . . . 13
⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))):𝑍⟶ℝ) |
50 | 49 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))):𝑍⟶ℝ) |
51 | 1 | uztrn2 11581 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → 𝑛 ∈ 𝑍) |
52 | 51 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑛 ∈ 𝑍) |
53 | 50, 52 | ffvelrnd 6268 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) ∈ ℝ) |
54 | | simprl 790 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑗 ∈ 𝑍) |
55 | 50, 54 | ffvelrnd 6268 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗) ∈ ℝ) |
56 | 53, 55 | resubcld 10337 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) ∈ ℝ) |
57 | | 0red 9920 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ∈
ℝ) |
58 | 7 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → seq𝑀( + , 𝐺):𝑍⟶ℂ) |
59 | 58, 52 | ffvelrnd 6268 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ) |
60 | 58, 54 | ffvelrnd 6268 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ) |
61 | 59, 60 | subcld 10271 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) ∈ ℂ) |
62 | 61 | abscld 14023 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ) |
63 | 61 | absge0d 14031 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ≤
(abs‘((seq𝑀( + ,
𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)))) |
64 | | fzfid 12634 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑛) ∈ Fin) |
65 | | difss 3699 |
. . . . . . . . . . . . . 14
⊢ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛) |
66 | | ssfi 8065 |
. . . . . . . . . . . . . 14
⊢ (((𝑀...𝑛) ∈ Fin ∧ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin) |
67 | 64, 65, 66 | sylancl 693 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin) |
68 | | eldifi 3694 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗)) → 𝑘 ∈ (𝑀...𝑛)) |
69 | | simpll 786 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝜑) |
70 | | elfzuz 12209 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀)) |
71 | 70, 1 | syl6eleqr 2699 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍) |
72 | 69, 71, 6 | syl2an 493 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℂ) |
73 | 68, 72 | sylan2 490 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (𝐺‘𝑘) ∈ ℂ) |
74 | 67, 73 | fsumabs 14374 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) →
(abs‘Σ𝑘 ∈
((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘)) ≤ Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘))) |
75 | | eqidd 2611 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) = (𝐺‘𝑘)) |
76 | 52, 1 | syl6eleq 2698 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑛 ∈ (ℤ≥‘𝑀)) |
77 | 75, 76, 72 | fsumser 14308 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) = (seq𝑀( + , 𝐺)‘𝑛)) |
78 | | eqidd 2611 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) = (𝐺‘𝑘)) |
79 | 54, 1 | syl6eleq 2698 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑗 ∈ (ℤ≥‘𝑀)) |
80 | | elfzuz 12209 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) |
81 | 80, 1 | syl6eleqr 2699 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) |
82 | 69, 81, 6 | syl2an 493 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) ∈ ℂ) |
83 | 78, 79, 82 | fsumser 14308 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) = (seq𝑀( + , 𝐺)‘𝑗)) |
84 | 77, 83 | oveq12d 6567 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘)) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) |
85 | | disjdif 3992 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅ |
86 | 85 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅) |
87 | | undif2 3996 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ((𝑀...𝑗) ∪ (𝑀...𝑛)) |
88 | | fzss2 12252 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈
(ℤ≥‘𝑗) → (𝑀...𝑗) ⊆ (𝑀...𝑛)) |
89 | 88 | ad2antll 761 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑗) ⊆ (𝑀...𝑛)) |
90 | | ssequn1 3745 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀...𝑗) ⊆ (𝑀...𝑛) ↔ ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛)) |
91 | 89, 90 | sylib 207 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛)) |
92 | 87, 91 | syl5req 2657 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑛) = ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗)))) |
93 | 86, 92, 64, 72 | fsumsplit 14318 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) = (Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘))) |
94 | 93 | eqcomd 2616 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘)) |
95 | 64, 72 | fsumcl 14311 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) ∈ ℂ) |
96 | | fzfid 12634 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑗) ∈ Fin) |
97 | 96, 82 | fsumcl 14311 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) ∈ ℂ) |
98 | 67, 73 | fsumcl 14311 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘) ∈ ℂ) |
99 | 95, 97, 98 | subaddd 10289 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘) ↔ (Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘))) |
100 | 94, 99 | mpbird 246 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘)) |
101 | 84, 100 | eqtr3d 2646 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘)) |
102 | 101 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘))) |
103 | 71 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ 𝑍) |
104 | 103, 24 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
105 | | abscl 13866 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺‘𝑘) ∈ ℂ → (abs‘(𝐺‘𝑘)) ∈ ℝ) |
106 | 105 | recnd 9947 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺‘𝑘) ∈ ℂ → (abs‘(𝐺‘𝑘)) ∈ ℂ) |
107 | 72, 106 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (abs‘(𝐺‘𝑘)) ∈ ℂ) |
108 | 104, 76, 107 | fsumser 14308 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) = (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛)) |
109 | 81 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘 ∈ 𝑍) |
110 | 109, 24 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
111 | 82, 106 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (abs‘(𝐺‘𝑘)) ∈ ℂ) |
112 | 110, 79, 111 | fsumser 14308 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) = (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) |
113 | 108, 112 | oveq12d 6567 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘))) = ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) |
114 | 86, 92, 64, 107 | fsumsplit 14318 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) = (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘)))) |
115 | 114 | eqcomd 2616 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘))) = Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘))) |
116 | 64, 107 | fsumcl 14311 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) ∈ ℂ) |
117 | 96, 111 | fsumcl 14311 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) ∈ ℂ) |
118 | 73, 106 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (abs‘(𝐺‘𝑘)) ∈ ℂ) |
119 | 67, 118 | fsumcl 14311 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘)) ∈ ℂ) |
120 | 116, 117,
119 | subaddd 10289 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘))) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘)) ↔ (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘))) = Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)))) |
121 | 115, 120 | mpbird 246 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘))) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘))) |
122 | 113, 121 | eqtr3d 2646 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘))) |
123 | 74, 102, 122 | 3brtr4d 4615 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) |
124 | 57, 62, 56, 63, 123 | letrd 10073 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ≤ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) |
125 | 56, 124 | absidd 14009 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) = ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) |
126 | 125 | breq1d 4593 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) →
((abs‘((seq𝑀( + ,
(𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 ↔ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) < 𝑥)) |
127 | | rpre 11715 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
128 | 127 | ad2antlr 759 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑥 ∈ ℝ) |
129 | | lelttr 10007 |
. . . . . . . . . 10
⊢
(((abs‘((seq𝑀(
+ , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ ∧ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) ∈ ℝ ∧ 𝑥 ∈ ℝ) →
(((abs‘((seq𝑀( + ,
𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) ∧ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)) |
130 | 62, 56, 128, 129 | syl3anc 1318 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) →
(((abs‘((seq𝑀( + ,
𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) ∧ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)) |
131 | 123, 130 | mpand 707 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)) |
132 | 126, 131 | sylbid 229 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) →
((abs‘((seq𝑀( + ,
(𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)) |
133 | 132 | anassrs 678 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → ((abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)) |
134 | 133 | ralimdva 2945 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → (∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)) |
135 | 134 | reximdva 3000 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)) |
136 | 135 | ralimdva 2945 |
. . 3
⊢ (𝜑 → (∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)) |
137 | 48, 136 | mpd 15 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥) |
138 | | seqex 12665 |
. . 3
⊢ seq𝑀( + , 𝐺) ∈ V |
139 | 138 | a1i 11 |
. 2
⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V) |
140 | 1, 8, 137, 139 | caucvg 14257 |
1
⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ ) |