| Step | Hyp | Ref
| Expression |
| 1 | | cvgcmpce.1 |
. 2
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 2 | | cvgcmpce.2 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ 𝑍) |
| 3 | 2, 1 | eleqtrdi 2851 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 4 | | eluzel2 12883 |
. . . . 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 14071 |
. . 3
⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ) |
| 8 | 7 | ffvelcdmda 7104 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ) |
| 9 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) |
| 10 | 9 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑚 = 𝑘 → (𝐶 · (𝐹‘𝑚)) = (𝐶 · (𝐹‘𝑘))) |
| 11 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚))) |
| 12 | | ovex 7464 |
. . . . . . . 8
⊢ (𝐶 · (𝐹‘𝑘)) ∈ V |
| 13 | 10, 11, 12 | fvmpt 7016 |
. . . . . . 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 11291 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐶 · (𝐹‘𝑘)) ∈ ℝ) |
| 19 | 14, 18 | eqeltrd 2841 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) ∈ ℝ) |
| 20 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑚 = 𝑘 → (abs‘(𝐺‘𝑚)) = (abs‘(𝐺‘𝑘))) |
| 21 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))) |
| 22 | | fvex 6919 |
. . . . . . . 8
⊢
(abs‘(𝐺‘𝑘)) ∈ V |
| 23 | 20, 21, 22 | fvmpt 7016 |
. . . . . . 7
⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
| 24 | 23 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
| 25 | 6 | abscld 15475 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐺‘𝑘)) ∈ ℝ) |
| 26 | 24, 25 | eqeltrd 2841 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) ∈ ℝ) |
| 27 | 15 | recnd 11289 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 28 | | cvgcmpce.5 |
. . . . . . . 8
⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
| 29 | | climdm 15590 |
. . . . . . . 8
⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹))) |
| 30 | 28, 29 | sylib 218 |
. . . . . . 7
⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹))) |
| 31 | 17 | recnd 11289 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
| 32 | 1, 5, 27, 30, 31, 14 | isermulc2 15694 |
. . . . . 6
⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹)))) |
| 33 | | climrel 15528 |
. . . . . . 7
⊢ Rel
⇝ |
| 34 | 33 | releldmi 5959 |
. . . . . 6
⊢ (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))) → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ∈ dom ⇝ ) |
| 35 | 32, 34 | syl 17 |
. . . . 5
⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ∈ dom ⇝ ) |
| 36 | 1 | uztrn2 12897 |
. . . . . . 7
⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
| 37 | 2, 36 | sylan 580 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
| 38 | 6 | absge0d 15483 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (abs‘(𝐺‘𝑘))) |
| 39 | 38, 24 | breqtrrd 5171 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘)) |
| 40 | 37, 39 | syldan 591 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘)) |
| 41 | | cvgcmpce.7 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (abs‘(𝐺‘𝑘)) ≤ (𝐶 · (𝐹‘𝑘))) |
| 42 | 37, 23 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
| 43 | 37, 13 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
| 44 | 41, 42, 43 | 3brtr4d 5175 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) ≤ ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘)) |
| 45 | 1, 2, 19, 26, 35, 40, 44 | cvgcmp 15852 |
. . . 4
⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))) ∈ dom ⇝ ) |
| 46 | 1 | climcau 15707 |
. . . 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) |
| 47 | 5, 45, 46 | syl2anc 584 |
. . 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) |
| 48 | 1, 5, 26 | serfre 14072 |
. . . . . . . . . . . . 13
⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))):𝑍⟶ℝ) |
| 49 | 48 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))):𝑍⟶ℝ) |
| 50 | 1 | uztrn2 12897 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → 𝑛 ∈ 𝑍) |
| 51 | 50 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑛 ∈ 𝑍) |
| 52 | 49, 51 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) ∈ ℝ) |
| 53 | | simprl 771 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑗 ∈ 𝑍) |
| 54 | 49, 53 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗) ∈ ℝ) |
| 55 | 52, 54 | resubcld 11691 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) ∈ ℝ) |
| 56 | | 0red 11264 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ∈
ℝ) |
| 57 | 7 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → seq𝑀( + , 𝐺):𝑍⟶ℂ) |
| 58 | 57, 51 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ) |
| 59 | 57, 53 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ) |
| 60 | 58, 59 | subcld 11620 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) ∈ ℂ) |
| 61 | 60 | abscld 15475 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ) |
| 62 | 60 | absge0d 15483 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ≤
(abs‘((seq𝑀( + ,
𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)))) |
| 63 | | fzfid 14014 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑛) ∈ Fin) |
| 64 | | difss 4136 |
. . . . . . . . . . . . . 14
⊢ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛) |
| 65 | | ssfi 9213 |
. . . . . . . . . . . . . 14
⊢ (((𝑀...𝑛) ∈ Fin ∧ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin) |
| 66 | 63, 64, 65 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin) |
| 67 | | eldifi 4131 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗)) → 𝑘 ∈ (𝑀...𝑛)) |
| 68 | | simpll 767 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝜑) |
| 69 | | elfzuz 13560 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀)) |
| 70 | 69, 1 | eleqtrrdi 2852 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍) |
| 71 | 68, 70, 6 | syl2an 596 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℂ) |
| 72 | 67, 71 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (𝐺‘𝑘) ∈ ℂ) |
| 73 | 66, 72 | fsumabs 15837 |
. . . . . . . . . . . 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))) |
| 74 | | eqidd 2738 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) = (𝐺‘𝑘)) |
| 75 | 51, 1 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑛 ∈ (ℤ≥‘𝑀)) |
| 76 | 74, 75, 71 | fsumser 15766 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) = (seq𝑀( + , 𝐺)‘𝑛)) |
| 77 | | eqidd 2738 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) = (𝐺‘𝑘)) |
| 78 | 53, 1 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑗 ∈ (ℤ≥‘𝑀)) |
| 79 | | elfzuz 13560 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) |
| 80 | 79, 1 | eleqtrrdi 2852 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) |
| 81 | 68, 80, 6 | syl2an 596 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) ∈ ℂ) |
| 82 | 77, 78, 81 | fsumser 15766 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) = (seq𝑀( + , 𝐺)‘𝑗)) |
| 83 | 76, 82 | oveq12d 7449 |
. . . . . . . . . . . . . 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))) |
| 84 | | fzfid 14014 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑗) ∈ Fin) |
| 85 | 84, 81 | fsumcl 15769 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) ∈ ℂ) |
| 86 | 66, 72 | fsumcl 15769 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘) ∈ ℂ) |
| 87 | | disjdif 4472 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅ |
| 88 | 87 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅) |
| 89 | | undif2 4477 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ((𝑀...𝑗) ∪ (𝑀...𝑛)) |
| 90 | | fzss2 13604 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈
(ℤ≥‘𝑗) → (𝑀...𝑗) ⊆ (𝑀...𝑛)) |
| 91 | 90 | ad2antll 729 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑗) ⊆ (𝑀...𝑛)) |
| 92 | | ssequn1 4186 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀...𝑗) ⊆ (𝑀...𝑛) ↔ ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛)) |
| 93 | 91, 92 | sylib 218 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛)) |
| 94 | 89, 93 | eqtr2id 2790 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑛) = ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗)))) |
| 95 | 88, 94, 63, 71 | fsumsplit 15777 |
. . . . . . . . . . . . . . 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))) |
| 96 | 85, 86, 95 | mvrladdd 11676 |
. . . . . . . . . . . . . 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)) |
| 97 | 83, 96 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘)) |
| 98 | 97 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘))) |
| 99 | 70 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ 𝑍) |
| 100 | 99, 23 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
| 101 | | abscl 15317 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺‘𝑘) ∈ ℂ → (abs‘(𝐺‘𝑘)) ∈ ℝ) |
| 102 | 101 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺‘𝑘) ∈ ℂ → (abs‘(𝐺‘𝑘)) ∈ ℂ) |
| 103 | 71, 102 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (abs‘(𝐺‘𝑘)) ∈ ℂ) |
| 104 | 100, 75, 103 | fsumser 15766 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) = (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛)) |
| 105 | 80 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘 ∈ 𝑍) |
| 106 | 105, 23 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘))) |
| 107 | 81, 102 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (abs‘(𝐺‘𝑘)) ∈ ℂ) |
| 108 | 106, 78, 107 | fsumser 15766 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) = (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) |
| 109 | 104, 108 | oveq12d 7449 |
. . . . . . . . . . . . 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))) |
| 110 | 84, 107 | fsumcl 15769 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) ∈ ℂ) |
| 111 | 72, 102 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (abs‘(𝐺‘𝑘)) ∈ ℂ) |
| 112 | 66, 111 | fsumcl 15769 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘)) ∈ ℂ) |
| 113 | 88, 94, 63, 103 | fsumsplit 15777 |
. . . . . . . . . . . . . 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)))) |
| 114 | 110, 112,
113 | mvrladdd 11676 |
. . . . . . . . . . . . 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))) |
| 115 | 109, 114 | eqtr3d 2779 |
. . . . . . . . . . . 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))) |
| 116 | 73, 98, 115 | 3brtr4d 5175 |
. . . . . . . . . . 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))) |
| 117 | 56, 61, 55, 62, 116 | letrd 11418 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ≤ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) |
| 118 | 55, 117 | absidd 15461 |
. . . . . . . . 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))) |
| 119 | 118 | breq1d 5153 |
. . . . . . . 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)) |
| 120 | | rpre 13043 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
| 121 | 120 | ad2antlr 727 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑥 ∈ ℝ) |
| 122 | | lelttr 11351 |
. . . . . . . . . 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)) |
| 123 | 61, 55, 121, 122 | syl3anc 1373 |
. . . . . . . . 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)) |
| 124 | 116, 123 | mpand 695 |
. . . . . . . 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)) |
| 125 | 119, 124 | sylbid 240 |
. . . . . . 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)) |
| 126 | 125 | anassrs 467 |
. . . . . 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)) |
| 127 | 126 | ralimdva 3167 |
. . . . 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)) |
| 128 | 127 | reximdva 3168 |
. . . 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)) |
| 129 | 128 | ralimdva 3167 |
. . 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)) |
| 130 | 47, 129 | mpd 15 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥) |
| 131 | | seqex 14044 |
. . 3
⊢ seq𝑀( + , 𝐺) ∈ V |
| 132 | 131 | a1i 11 |
. 2
⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V) |
| 133 | 1, 8, 130, 132 | caucvg 15715 |
1
⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ ) |