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Mirrors > Home > MPE Home > Th. List > msxms | Structured version Visualization version GIF version |
Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
msxms | ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀) | |
2 | eqid 2771 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | eqid 2771 | . . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) | |
4 | 1, 2, 3 | isms 22467 | . 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀)))) |
5 | 4 | simplbi 485 | 1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 × cxp 5247 ↾ cres 5251 ‘cfv 6029 Basecbs 16057 distcds 16151 TopOpenctopn 16283 Metcme 19940 ∞MetSpcxme 22335 MetSpcmt 22336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-xp 5255 df-res 5261 df-iota 5992 df-fv 6037 df-ms 22339 |
This theorem is referenced by: mstps 22473 imasf1oms 22508 ressms 22544 prdsms 22549 ngpxms 22618 ngptgp 22653 nlmvscnlem2 22702 nlmvscn 22704 nrginvrcn 22709 nghmcn 22762 cnfldxms 22793 nmhmcn 23132 ipcnlem2 23255 ipcn 23257 nglmle 23312 cmetcusp1 23361 dya2icoseg2 30673 |
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