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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 23948 | . 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 1537 ∈ wcel 2114 × cxp 5553 ↾ cres 5557 ‘cfv 6355 Basecbs 16483 distcds 16574 MetSpcms 22928 CMetccmet 23857 CMetSpccms 23935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-res 5567 df-iota 6314 df-fv 6363 df-cms 23938 |
This theorem is referenced by: bncmet 23950 cmsss 23954 cmetcusp1 23956 cmscsscms 23976 minveclem3a 24030 |
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