| Step | Hyp | Ref
| Expression |
| 1 | | inss1 4237 |
. . . . . . . . 9
⊢ (𝑋 ∩ 𝑌) ⊆ 𝑋 |
| 2 | | xpss2 5705 |
. . . . . . . . 9
⊢ ((𝑋 ∩ 𝑌) ⊆ 𝑋 → (ℂ × (𝑋 ∩ 𝑌)) ⊆ (ℂ × 𝑋)) |
| 3 | 1, 2 | ax-mp 5 |
. . . . . . . 8
⊢ (ℂ
× (𝑋 ∩ 𝑌)) ⊆ (ℂ ×
𝑋) |
| 4 | | sstr 3992 |
. . . . . . . 8
⊢ ((𝑓 ⊆ (ℂ × (𝑋 ∩ 𝑌)) ∧ (ℂ × (𝑋 ∩ 𝑌)) ⊆ (ℂ × 𝑋)) → 𝑓 ⊆ (ℂ × 𝑋)) |
| 5 | 3, 4 | mpan2 691 |
. . . . . . 7
⊢ (𝑓 ⊆ (ℂ × (𝑋 ∩ 𝑌)) → 𝑓 ⊆ (ℂ × 𝑋)) |
| 6 | 5 | anim2i 617 |
. . . . . 6
⊢ ((Fun
𝑓 ∧ 𝑓 ⊆ (ℂ × (𝑋 ∩ 𝑌))) → (Fun 𝑓 ∧ 𝑓 ⊆ (ℂ × 𝑋))) |
| 7 | 6 | a1i 11 |
. . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → ((Fun 𝑓 ∧ 𝑓 ⊆ (ℂ × (𝑋 ∩ 𝑌))) → (Fun 𝑓 ∧ 𝑓 ⊆ (ℂ × 𝑋)))) |
| 8 | | elfvdm 6943 |
. . . . . . 7
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) |
| 9 | | inex1g 5319 |
. . . . . . 7
⊢ (𝑋 ∈ dom ∞Met →
(𝑋 ∩ 𝑌) ∈ V) |
| 10 | 8, 9 | syl 17 |
. . . . . 6
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∩ 𝑌) ∈ V) |
| 11 | | cnex 11236 |
. . . . . 6
⊢ ℂ
∈ V |
| 12 | | elpmg 8883 |
. . . . . 6
⊢ (((𝑋 ∩ 𝑌) ∈ V ∧ ℂ ∈ V) →
(𝑓 ∈ ((𝑋 ∩ 𝑌) ↑pm ℂ) ↔ (Fun
𝑓 ∧ 𝑓 ⊆ (ℂ × (𝑋 ∩ 𝑌))))) |
| 13 | 10, 11, 12 | sylancl 586 |
. . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ ((𝑋 ∩ 𝑌) ↑pm ℂ) ↔ (Fun
𝑓 ∧ 𝑓 ⊆ (ℂ × (𝑋 ∩ 𝑌))))) |
| 14 | | elpmg 8883 |
. . . . . 6
⊢ ((𝑋 ∈ dom ∞Met ∧
ℂ ∈ V) → (𝑓
∈ (𝑋
↑pm ℂ) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (ℂ × 𝑋)))) |
| 15 | 8, 11, 14 | sylancl 586 |
. . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ (𝑋 ↑pm ℂ) ↔ (Fun
𝑓 ∧ 𝑓 ⊆ (ℂ × 𝑋)))) |
| 16 | 7, 13, 15 | 3imtr4d 294 |
. . . 4
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ ((𝑋 ∩ 𝑌) ↑pm ℂ) → 𝑓 ∈ (𝑋 ↑pm
ℂ))) |
| 17 | | uzid 12893 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℤ → 𝑦 ∈
(ℤ≥‘𝑦)) |
| 18 | 17 | adantl 481 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ≥‘𝑦)) |
| 19 | | simp2 1138 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌)) |
| 20 | 19 | ralimi 3083 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
(ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌)) |
| 21 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑦 → (𝑓‘𝑧) = (𝑓‘𝑦)) |
| 22 | 21 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑦 → ((𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ↔ (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌))) |
| 23 | 22 | rspcva 3620 |
. . . . . . . . 9
⊢ ((𝑦 ∈
(ℤ≥‘𝑦) ∧ ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌)) → (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌)) |
| 24 | 18, 20, 23 | syl2an 596 |
. . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ ∀𝑧 ∈
(ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥)) → (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌)) |
| 25 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌)) → (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌)) |
| 26 | 25 | elin2d 4205 |
. . . . . . . . . . . . 13
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌)) → (𝑓‘𝑦) ∈ 𝑌) |
| 27 | | inss2 4238 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑋 ∩ 𝑌) ⊆ 𝑌 |
| 28 | 27 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) → (𝑋 ∩ 𝑌) ⊆ 𝑌) |
| 29 | 28 | sselda 3983 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌)) → (𝑓‘𝑧) ∈ 𝑌) |
| 30 | | simplr 769 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌)) → (𝑓‘𝑦) ∈ 𝑌) |
| 31 | 29, 30 | ovresd 7600 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌)) → ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) = ((𝑓‘𝑧)𝐷(𝑓‘𝑦))) |
| 32 | 31 | breq1d 5153 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌)) → (((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥 ↔ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥)) |
| 33 | 32 | biimpd 229 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌)) → (((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥 → ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥)) |
| 34 | 33 | imdistanda 571 |
. . . . . . . . . . . . . 14
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) → (((𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → ((𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 35 | 1 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) → (𝑋 ∩ 𝑌) ⊆ 𝑋) |
| 36 | 35 | sseld 3982 |
. . . . . . . . . . . . . . 15
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) → ((𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) → (𝑓‘𝑧) ∈ 𝑋)) |
| 37 | 36 | anim1d 611 |
. . . . . . . . . . . . . 14
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) → (((𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥) → ((𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 38 | 34, 37 | syld 47 |
. . . . . . . . . . . . 13
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ 𝑌) → (((𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → ((𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 39 | 26, 38 | syldan 591 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌)) → (((𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → ((𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 40 | 39 | anim2d 612 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌)) → ((𝑧 ∈ dom 𝑓 ∧ ((𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥)) → (𝑧 ∈ dom 𝑓 ∧ ((𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥)))) |
| 41 | | 3anass 1095 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) ↔ (𝑧 ∈ dom 𝑓 ∧ ((𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥))) |
| 42 | | 3anass 1095 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥) ↔ (𝑧 ∈ dom 𝑓 ∧ ((𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 43 | 40, 41, 42 | 3imtr4g 296 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌)) → ((𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → (𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 44 | 43 | ralimdv 3169 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌)) → (∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 45 | 44 | impancom 451 |
. . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ ∀𝑧 ∈
(ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥)) → ((𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌) → ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 46 | 24, 45 | mpd 15 |
. . . . . . 7
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ ∀𝑧 ∈
(ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥)) → ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥)) |
| 47 | 46 | ex 412 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) → (∀𝑧 ∈
(ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 48 | 47 | reximdva 3168 |
. . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → (∃𝑦 ∈ ℤ ∀𝑧 ∈
(ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → ∃𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 49 | 48 | ralimdv 3169 |
. . . 4
⊢ (𝐷 ∈ (∞Met‘𝑋) → (∀𝑥 ∈ ℝ+
∃𝑦 ∈ ℤ
∀𝑧 ∈
(ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℤ ∀𝑧 ∈
(ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥))) |
| 50 | 16, 49 | anim12d 609 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝑓 ∈ ((𝑋 ∩ 𝑌) ↑pm ℂ) ∧
∀𝑥 ∈
ℝ+ ∃𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥)) → (𝑓 ∈ (𝑋 ↑pm ℂ) ∧
∀𝑥 ∈
ℝ+ ∃𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥)))) |
| 51 | | xmetres 24374 |
. . . 4
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌))) |
| 52 | | iscau2 25311 |
. . . 4
⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌)) → (𝑓 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ↔ (𝑓 ∈ ((𝑋 ∩ 𝑌) ↑pm ℂ) ∧
∀𝑥 ∈
ℝ+ ∃𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥)))) |
| 53 | 51, 52 | syl 17 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ↔ (𝑓 ∈ ((𝑋 ∩ 𝑌) ↑pm ℂ) ∧
∀𝑥 ∈
ℝ+ ∃𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ (𝑋 ∩ 𝑌) ∧ ((𝑓‘𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓‘𝑦)) < 𝑥)))) |
| 54 | | iscau2 25311 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ (Cau‘𝐷) ↔ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧
∀𝑥 ∈
ℝ+ ∃𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ≥‘𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓‘𝑧) ∈ 𝑋 ∧ ((𝑓‘𝑧)𝐷(𝑓‘𝑦)) < 𝑥)))) |
| 55 | 50, 53, 54 | 3imtr4d 294 |
. 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) → 𝑓 ∈ (Cau‘𝐷))) |
| 56 | 55 | ssrdv 3989 |
1
⊢ (𝐷 ∈ (∞Met‘𝑋) → (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ⊆ (Cau‘𝐷)) |