Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cmsms | Structured version Visualization version GIF version |
Description: A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
cmsms | ⊢ (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2736 | . . 3 ⊢ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) | |
3 | 1, 2 | iscms 24196 | . 2 ⊢ (𝐺 ∈ CMetSp ↔ (𝐺 ∈ MetSp ∧ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (CMet‘(Base‘𝐺)))) |
4 | 3 | simplbi 501 | 1 ⊢ (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 × cxp 5534 ↾ cres 5538 ‘cfv 6358 Basecbs 16666 distcds 16758 MetSpcms 23170 CMetccmet 24105 CMetSpccms 24183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-res 5548 df-iota 6316 df-fv 6366 df-cms 24186 |
This theorem is referenced by: cmsss 24202 cmetcusp1 24204 rlmbn 24212 cmscsscms 24224 rrhcn 31613 dya2icoseg2 31911 sitgclbn 31976 |
Copyright terms: Public domain | W3C validator |