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| 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 2737 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
| 2 | eqid 2737 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
| 3 | eqid 2737 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
| 4 | 1, 2, 3 | isms 24459 | . 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 2108 × cxp 5683 ↾ cres 5687 ‘cfv 6561 Basecbs 17247 distcds 17306 TopOpenctopn 17466 Metcmet 21350 ∞MetSpcxms 24327 MetSpcms 24328 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 df-iota 6514 df-fv 6569 df-ms 24331 | 
| This theorem is referenced by: mstps 24465 imasf1oms 24503 ressms 24539 prdsms 24544 ngpxms 24614 ngptgp 24649 nlmvscnlem2 24706 nlmvscn 24708 nrginvrcn 24713 nghmcn 24766 cnfldxms 24797 nmhmcn 25153 ipcnlem2 25278 ipcn 25280 nglmle 25336 cmetcusp1 25387 dya2icoseg2 34280 | 
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