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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 5496 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
2 | isms.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | eqtr4di 2221 | . . 3 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) |
4 | fveq2 5496 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾)) | |
5 | fveq2 5496 | . . . . . . . 8 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
6 | isms.x | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐾) | |
7 | 5, 6 | eqtr4di 2221 | . . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋) |
8 | 7 | sqxpeqd 4637 | . . . . . 6 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋)) |
9 | 4, 8 | reseq12d 4892 | . . . . 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 | eqtr4di 2221 | . . . 4 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷) |
12 | 11 | fveq2d 5500 | . . 3 ⊢ (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷)) |
13 | 3, 12 | eqeq12d 2185 | . 2 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷))) |
14 | df-xms 13133 | . 2 ⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} | |
15 | 13, 14 | elrab2 2889 | 1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 × cxp 4609 ↾ cres 4613 ‘cfv 5198 Basecbs 12416 distcds 12489 TopOpenctopn 12580 MetOpencmopn 12779 TopSpctps 12822 ∞MetSpcxms 13130 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-res 4623 df-iota 5160 df-fv 5206 df-xms 13133 |
This theorem is referenced by: isxms2 13246 xmstopn 13249 xmstps 13251 xmspropd 13271 |
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