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Mirrors > Home > MPE Home > Th. List > msxms | Structured version Visualization version GIF version |
Description: A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
msxms | ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2735 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2735 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isms 24475 | . 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀)))) |
5 | 4 | simplbi 497 | 1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 × cxp 5687 ↾ cres 5691 ‘cfv 6563 Basecbs 17245 distcds 17307 TopOpenctopn 17468 Metcmet 21368 ∞MetSpcxms 24343 MetSpcms 24344 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-res 5701 df-iota 6516 df-fv 6571 df-ms 24347 |
This theorem is referenced by: mstps 24481 imasf1oms 24519 ressms 24555 prdsms 24560 ngpxms 24630 ngptgp 24665 nlmvscnlem2 24722 nlmvscn 24724 nrginvrcn 24729 nghmcn 24782 cnfldxms 24813 nmhmcn 25167 ipcnlem2 25292 ipcn 25294 nglmle 25350 cmetcusp1 25401 dya2icoseg2 34260 |
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