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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 2818 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2818 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2818 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isms 22986 | . 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀)))) |
5 | 4 | simplbi 498 | 1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 × cxp 5546 ↾ cres 5550 ‘cfv 6348 Basecbs 16471 distcds 16562 TopOpenctopn 16683 Metcmet 20459 ∞MetSpcxms 22854 MetSpcms 22855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-res 5560 df-iota 6307 df-fv 6356 df-ms 22858 |
This theorem is referenced by: mstps 22992 imasf1oms 23027 ressms 23063 prdsms 23068 ngpxms 23137 ngptgp 23172 nlmvscnlem2 23221 nlmvscn 23223 nrginvrcn 23228 nghmcn 23281 cnfldxms 23312 nmhmcn 23651 ipcnlem2 23774 ipcn 23776 nglmle 23832 cmetcusp1 23883 dya2icoseg2 31435 |
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