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| Description: A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| bncms | ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 2 | 1 | isbn 25373 | . 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp)) | 
| 3 | 2 | simp2bi 1146 | 1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ‘cfv 6560 Scalarcsca 17301 NrmVeccnvc 24595 CMetSpccms 25367 Bancbn 25368 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-bn 25371 | 
| This theorem is referenced by: bncmet 25382 lssbn 25387 hlcms 25401 bncssbn 25409 sitgclbn 34346 | 
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