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Mirrors > Home > ILE Home > Th. List > isxms | GIF version |
Description: Express the predicate "〈𝑋, 𝐷〉 is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
isms.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
isms.x | ⊢ 𝑋 = (Base‘𝐾) |
isms.d | ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
isxms | ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5421 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
2 | isms.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | syl6eqr 2190 | . . 3 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) |
4 | fveq2 5421 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾)) | |
5 | fveq2 5421 | . . . . . . . 8 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
6 | isms.x | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐾) | |
7 | 5, 6 | syl6eqr 2190 | . . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋) |
8 | 7 | sqxpeqd 4565 | . . . . . 6 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋)) |
9 | 4, 8 | reseq12d 4820 | . . . . 5 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋))) |
10 | isms.d | . . . . 5 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | |
11 | 9, 10 | syl6eqr 2190 | . . . 4 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷) |
12 | 11 | fveq2d 5425 | . . 3 ⊢ (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷)) |
13 | 3, 12 | eqeq12d 2154 | . 2 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷))) |
14 | df-xms 12508 | . 2 ⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} | |
15 | 13, 14 | elrab2 2843 | 1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 × cxp 4537 ↾ cres 4541 ‘cfv 5123 Basecbs 11959 distcds 12030 TopOpenctopn 12121 MetOpencmopn 12154 TopSpctps 12197 ∞MetSpcxms 12505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-res 4551 df-iota 5088 df-fv 5131 df-xms 12508 |
This theorem is referenced by: isxms2 12621 xmstopn 12624 xmstps 12626 xmspropd 12646 |
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