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Mirrors > Home > MPE Home > Th. List > xmstps | Structured version Visualization version 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 2736 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2736 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2736 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isxms 23746 | . 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))))) |
5 | 4 | simplbi 498 | 1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 × cxp 5629 ↾ cres 5633 ‘cfv 6493 Basecbs 17037 distcds 17096 TopOpenctopn 17257 MetOpencmopn 20733 TopSpctps 22227 ∞MetSpcxms 23616 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-res 5643 df-iota 6445 df-fv 6501 df-xms 23619 |
This theorem is referenced by: mstps 23754 ressxms 23827 prdsxmslem2 23831 tmsxpsmopn 23839 minveclem4a 24740 rrhcn 32406 rrhf 32407 rrexttps 32415 sitmcl 32779 |
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