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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 2170 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2170 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2170 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isxms 13245 | . 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 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: mstps 13253 |
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