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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 2778 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2778 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2778 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isms 22666 | . 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀)))) |
5 | 4 | simplbi 493 | 1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 × cxp 5355 ↾ cres 5359 ‘cfv 6137 Basecbs 16259 distcds 16351 TopOpenctopn 16472 Metcmet 20132 ∞MetSpcxms 22534 MetSpcms 22535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-xp 5363 df-res 5369 df-iota 6101 df-fv 6145 df-ms 22538 |
This theorem is referenced by: mstps 22672 imasf1oms 22707 ressms 22743 prdsms 22748 ngpxms 22817 ngptgp 22852 nlmvscnlem2 22901 nlmvscn 22903 nrginvrcn 22908 nghmcn 22961 cnfldxms 22992 nmhmcn 23331 ipcnlem2 23454 ipcn 23456 nglmle 23512 cmetcusp1 23563 dya2icoseg2 30942 |
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