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Mirrors > Home > ILE Home > Th. List > xmstps | GIF version |
Description: An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
xmstps | ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2140 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2140 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2140 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isxms 12659 | . 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))))) |
5 | 4 | simplbi 272 | 1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 × cxp 4545 ↾ cres 4549 ‘cfv 5131 Basecbs 11998 distcds 12069 TopOpenctopn 12160 MetOpencmopn 12193 TopSpctps 12236 ∞MetSpcxms 12544 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-rab 2426 df-v 2691 df-un 3080 df-in 3082 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-res 4559 df-iota 5096 df-fv 5139 df-xms 12547 |
This theorem is referenced by: mstps 12667 |
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