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Mirrors > Home > MPE Home > Th. List > cmscmet | Structured version Visualization version GIF version |
Description: The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
iscms.1 | ⊢ 𝑋 = (Base‘𝑀) |
iscms.2 | ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
cmscmet | ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscms.1 | . . 3 ⊢ 𝑋 = (Base‘𝑀) | |
2 | iscms.2 | . . 3 ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)) | |
3 | 1, 2 | iscms 23942 | . 2 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋))) |
4 | 3 | simprbi 499 | 1 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 × cxp 5548 ↾ cres 5552 ‘cfv 6350 Basecbs 16477 distcds 16568 MetSpcms 22922 CMetccmet 23851 CMetSpccms 23929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-res 5562 df-iota 6309 df-fv 6358 df-cms 23932 |
This theorem is referenced by: bncmet 23944 cmsss 23948 cmetcusp1 23950 cmscsscms 23970 minveclem3a 24024 |
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