| Step | Hyp | Ref
| Expression |
| 1 | | caushft.8 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) |
| 2 | | caures.1 |
. . . . . 6
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 3 | | caures.4 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
| 4 | | metxmet 24344 |
. . . . . . 7
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
| 5 | 3, 4 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
| 6 | | caures.3 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 7 | | caushft.7 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝑁))) |
| 8 | 7 | ralrimiva 3146 |
. . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝑁))) |
| 9 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) |
| 10 | | fvoveq1 7454 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → (𝐺‘(𝑘 + 𝑁)) = (𝐺‘(𝑗 + 𝑁))) |
| 11 | 9, 10 | eqeq12d 2753 |
. . . . . . . 8
⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝑁)) ↔ (𝐹‘𝑗) = (𝐺‘(𝑗 + 𝑁)))) |
| 12 | 11 | rspccva 3621 |
. . . . . . 7
⊢
((∀𝑘 ∈
𝑍 (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝑁)) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘(𝑗 + 𝑁))) |
| 13 | 8, 12 | sylan 580 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘(𝑗 + 𝑁))) |
| 14 | 2, 5, 6, 7, 13 | iscau4 25313 |
. . . . 5
⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧
∀𝑥 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥)))) |
| 15 | 1, 14 | mpbid 232 |
. . . 4
⊢ (𝜑 → (𝐹 ∈ (𝑋 ↑pm ℂ) ∧
∀𝑥 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥))) |
| 16 | 15 | simprd 495 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥)) |
| 17 | 2 | eleq2i 2833 |
. . . . . . . . 9
⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀)) |
| 18 | 17 | biimpi 216 |
. . . . . . . 8
⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀)) |
| 19 | | caushft.5 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 20 | | eluzadd 12907 |
. . . . . . . 8
⊢ ((𝑗 ∈
(ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝑗 + 𝑁) ∈
(ℤ≥‘(𝑀 + 𝑁))) |
| 21 | 18, 19, 20 | syl2anr 597 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 𝑁) ∈
(ℤ≥‘(𝑀 + 𝑁))) |
| 22 | | caushft.4 |
. . . . . . 7
⊢ 𝑊 =
(ℤ≥‘(𝑀 + 𝑁)) |
| 23 | 21, 22 | eleqtrrdi 2852 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 𝑁) ∈ 𝑊) |
| 24 | | simplr 769 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑗 ∈ 𝑍) |
| 25 | 24, 2 | eleqtrdi 2851 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑗 ∈ (ℤ≥‘𝑀)) |
| 26 | | eluzelz 12888 |
. . . . . . . . . . 11
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑗 ∈ ℤ) |
| 27 | 25, 26 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑗 ∈ ℤ) |
| 28 | 19 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑁 ∈ ℤ) |
| 29 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) |
| 30 | | eluzsub 12908 |
. . . . . . . . . 10
⊢ ((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑚 ∈
(ℤ≥‘(𝑗 + 𝑁))) → (𝑚 − 𝑁) ∈ (ℤ≥‘𝑗)) |
| 31 | 27, 28, 29, 30 | syl3anc 1373 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → (𝑚 − 𝑁) ∈ (ℤ≥‘𝑗)) |
| 32 | | simp3 1139 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) → ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) |
| 33 | 32 | ralimi 3083 |
. . . . . . . . 9
⊢
(∀𝑘 ∈
(ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) → ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) |
| 34 | | fvoveq1 7454 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑚 − 𝑁) → (𝐺‘(𝑘 + 𝑁)) = (𝐺‘((𝑚 − 𝑁) + 𝑁))) |
| 35 | 34 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑚 − 𝑁) → ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) = ((𝐺‘((𝑚 − 𝑁) + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁)))) |
| 36 | 35 | breq1d 5153 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑚 − 𝑁) → (((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥 ↔ ((𝐺‘((𝑚 − 𝑁) + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥)) |
| 37 | 36 | rspcv 3618 |
. . . . . . . . 9
⊢ ((𝑚 − 𝑁) ∈ (ℤ≥‘𝑗) → (∀𝑘 ∈
(ℤ≥‘𝑗)((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥 → ((𝐺‘((𝑚 − 𝑁) + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥)) |
| 38 | 31, 33, 37 | syl2im 40 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) → ((𝐺‘((𝑚 − 𝑁) + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥)) |
| 39 | | eluzelz 12888 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈
(ℤ≥‘(𝑗 + 𝑁)) → 𝑚 ∈ ℤ) |
| 40 | 39 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑚 ∈ ℤ) |
| 41 | 40 | zcnd 12723 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑚 ∈ ℂ) |
| 42 | 19 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 43 | 42 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑁 ∈ ℂ) |
| 44 | 41, 43 | npcand 11624 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → ((𝑚 − 𝑁) + 𝑁) = 𝑚) |
| 45 | 44 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → (𝐺‘((𝑚 − 𝑁) + 𝑁)) = (𝐺‘𝑚)) |
| 46 | 45 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → ((𝐺‘((𝑚 − 𝑁) + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) = ((𝐺‘𝑚)𝐷(𝐺‘(𝑗 + 𝑁)))) |
| 47 | 3 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝐷 ∈ (Met‘𝑋)) |
| 48 | | caushft.9 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺:𝑊⟶𝑋) |
| 49 | 48 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝐺:𝑊⟶𝑋) |
| 50 | 22 | uztrn2 12897 |
. . . . . . . . . . . . 13
⊢ (((𝑗 + 𝑁) ∈ 𝑊 ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑚 ∈ 𝑊) |
| 51 | 23, 50 | sylan 580 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → 𝑚 ∈ 𝑊) |
| 52 | 49, 51 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → (𝐺‘𝑚) ∈ 𝑋) |
| 53 | 48 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐺:𝑊⟶𝑋) |
| 54 | 53, 23 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘(𝑗 + 𝑁)) ∈ 𝑋) |
| 55 | 54 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → (𝐺‘(𝑗 + 𝑁)) ∈ 𝑋) |
| 56 | | metsym 24360 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐺‘𝑚) ∈ 𝑋 ∧ (𝐺‘(𝑗 + 𝑁)) ∈ 𝑋) → ((𝐺‘𝑚)𝐷(𝐺‘(𝑗 + 𝑁))) = ((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚))) |
| 57 | 47, 52, 55, 56 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → ((𝐺‘𝑚)𝐷(𝐺‘(𝑗 + 𝑁))) = ((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚))) |
| 58 | 46, 57 | eqtrd 2777 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → ((𝐺‘((𝑚 − 𝑁) + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) = ((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚))) |
| 59 | 58 | breq1d 5153 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → (((𝐺‘((𝑚 − 𝑁) + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥 ↔ ((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚)) < 𝑥)) |
| 60 | 38, 59 | sylibd 239 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) → ((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚)) < 𝑥)) |
| 61 | 60 | ralrimdva 3154 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) → ∀𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚)) < 𝑥)) |
| 62 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑛 = (𝑗 + 𝑁) → (ℤ≥‘𝑛) =
(ℤ≥‘(𝑗 + 𝑁))) |
| 63 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑗 + 𝑁) → (𝐺‘𝑛) = (𝐺‘(𝑗 + 𝑁))) |
| 64 | 63 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝑛 = (𝑗 + 𝑁) → ((𝐺‘𝑛)𝐷(𝐺‘𝑚)) = ((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚))) |
| 65 | 64 | breq1d 5153 |
. . . . . . . 8
⊢ (𝑛 = (𝑗 + 𝑁) → (((𝐺‘𝑛)𝐷(𝐺‘𝑚)) < 𝑥 ↔ ((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚)) < 𝑥)) |
| 66 | 62, 65 | raleqbidv 3346 |
. . . . . . 7
⊢ (𝑛 = (𝑗 + 𝑁) → (∀𝑚 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛)𝐷(𝐺‘𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚)) < 𝑥)) |
| 67 | 66 | rspcev 3622 |
. . . . . 6
⊢ (((𝑗 + 𝑁) ∈ 𝑊 ∧ ∀𝑚 ∈ (ℤ≥‘(𝑗 + 𝑁))((𝐺‘(𝑗 + 𝑁))𝐷(𝐺‘𝑚)) < 𝑥) → ∃𝑛 ∈ 𝑊 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛)𝐷(𝐺‘𝑚)) < 𝑥) |
| 68 | 23, 61, 67 | syl6an 684 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) → ∃𝑛 ∈ 𝑊 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛)𝐷(𝐺‘𝑚)) < 𝑥)) |
| 69 | 68 | rexlimdva 3155 |
. . . 4
⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) → ∃𝑛 ∈ 𝑊 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛)𝐷(𝐺‘𝑚)) < 𝑥)) |
| 70 | 69 | ralimdv 3169 |
. . 3
⊢ (𝜑 → (∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐺‘(𝑘 + 𝑁)) ∈ 𝑋 ∧ ((𝐺‘(𝑘 + 𝑁))𝐷(𝐺‘(𝑗 + 𝑁))) < 𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑊 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛)𝐷(𝐺‘𝑚)) < 𝑥)) |
| 71 | 16, 70 | mpd 15 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑊 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛)𝐷(𝐺‘𝑚)) < 𝑥) |
| 72 | 6, 19 | zaddcld 12726 |
. . 3
⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℤ) |
| 73 | | eqidd 2738 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐺‘𝑚) = (𝐺‘𝑚)) |
| 74 | | eqidd 2738 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑊) → (𝐺‘𝑛) = (𝐺‘𝑛)) |
| 75 | 22, 5, 72, 73, 74, 48 | iscauf 25314 |
. 2
⊢ (𝜑 → (𝐺 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑊 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛)𝐷(𝐺‘𝑚)) < 𝑥)) |
| 76 | 71, 75 | mpbird 257 |
1
⊢ (𝜑 → 𝐺 ∈ (Cau‘𝐷)) |