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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 2738 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2738 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2738 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isxms 23200 | . 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))))) |
5 | 4 | simplbi 501 | 1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 × cxp 5523 ↾ cres 5527 ‘cfv 6339 Basecbs 16586 distcds 16677 TopOpenctopn 16798 MetOpencmopn 20207 TopSpctps 21683 ∞MetSpcxms 23070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-rab 3062 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-xp 5531 df-res 5537 df-iota 6297 df-fv 6347 df-xms 23073 |
This theorem is referenced by: mstps 23208 ressxms 23278 prdsxmslem2 23282 tmsxpsmopn 23290 minveclem4a 24182 rrhcn 31517 rrhf 31518 rrexttps 31526 sitmcl 31888 |
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