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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 25393 | . 2 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋))) |
4 | 3 | simprbi 496 | 1 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 × cxp 5687 ↾ cres 5691 ‘cfv 6563 Basecbs 17245 distcds 17307 MetSpcms 24344 CMetccmet 25302 CMetSpccms 25380 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-res 5701 df-iota 6516 df-fv 6571 df-cms 25383 |
This theorem is referenced by: bncmet 25395 cmsss 25399 cmetcusp1 25401 cmscsscms 25421 minveclem3a 25475 |
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