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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 2821 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2821 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2821 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isxms 23051 | . 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))))) |
5 | 4 | simplbi 500 | 1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 × cxp 5547 ↾ cres 5551 ‘cfv 6349 Basecbs 16477 distcds 16568 TopOpenctopn 16689 MetOpencmopn 20529 TopSpctps 21534 ∞MetSpcxms 22921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-res 5561 df-iota 6308 df-fv 6357 df-xms 22924 |
This theorem is referenced by: mstps 23059 ressxms 23129 prdsxmslem2 23133 tmsxpsmopn 23141 minveclem4a 24027 rrhcn 31233 rrhf 31234 rrexttps 31242 sitmcl 31604 |
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