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Mirrors > Home > MPE Home > Th. List > metres | Structured version Visualization version GIF version |
Description: A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
metres | ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘(𝑋 ∩ 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metf 22937 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
2 | fdm 6495 | . . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ → dom 𝐷 = (𝑋 × 𝑋)) | |
3 | metreslem 22969 | . . 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 4155 | . . 3 ⊢ (𝑋 ∩ 𝑅) ⊆ 𝑋 | |
6 | metres2 22970 | . . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋 ∩ 𝑅) ⊆ 𝑋) → (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅))) ∈ (Met‘(𝑋 ∩ 𝑅))) | |
7 | 5, 6 | mpan2 690 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅))) ∈ (Met‘(𝑋 ∩ 𝑅))) |
8 | 4, 7 | eqeltrd 2890 | 1 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘(𝑋 ∩ 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 × cxp 5517 dom cdm 5519 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 ℝcr 10525 Metcmet 20077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-i2m1 10594 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-xadd 12496 df-xmet 20084 df-met 20085 |
This theorem is referenced by: ressms 23133 |
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