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| Mirrors > Home > MPE Home > Th. List > xmetres | Structured version Visualization version GIF version | ||
| Description: A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.) |
| Ref | Expression |
|---|---|
| xmetres | ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘(𝑋 ∩ 𝑅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xmetf 24339 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
| 2 | fdm 6745 | . . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋)) | |
| 3 | metreslem 24372 | . . 3 ⊢ (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅)))) | |
| 4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅)))) |
| 5 | inss1 4237 | . . 3 ⊢ (𝑋 ∩ 𝑅) ⊆ 𝑋 | |
| 6 | xmetres2 24371 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∩ 𝑅) ⊆ 𝑋) → (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅))) ∈ (∞Met‘(𝑋 ∩ 𝑅))) | |
| 7 | 5, 6 | mpan2 691 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅))) ∈ (∞Met‘(𝑋 ∩ 𝑅))) |
| 8 | 4, 7 | eqeltrd 2841 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘(𝑋 ∩ 𝑅))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 dom cdm 5685 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 ℝ*cxr 11294 ∞Metcxmet 21349 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-xr 11299 df-xmet 21357 |
| This theorem is referenced by: blres 24441 ressxms 24538 cfilresi 25329 caussi 25331 causs 25332 minvecolem4a 30896 |
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