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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 2725 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2725 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2725 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isms 24399 | . 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀)))) |
5 | 4 | simplbi 496 | 1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 × cxp 5676 ↾ cres 5680 ‘cfv 6549 Basecbs 17183 distcds 17245 TopOpenctopn 17406 Metcmet 21282 ∞MetSpcxms 24267 MetSpcms 24268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-res 5690 df-iota 6501 df-fv 6557 df-ms 24271 |
This theorem is referenced by: mstps 24405 imasf1oms 24443 ressms 24479 prdsms 24484 ngpxms 24554 ngptgp 24589 nlmvscnlem2 24646 nlmvscn 24648 nrginvrcn 24653 nghmcn 24706 cnfldxms 24737 nmhmcn 25091 ipcnlem2 25216 ipcn 25218 nglmle 25274 cmetcusp1 25325 dya2icoseg2 34029 |
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