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| Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 25374 instead. (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bnnv | ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
| 2 | eqid 2737 | . . 3 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
| 3 | 1, 2 | iscbn 30883 | . 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘𝑈) ∈ (CMet‘(BaseSet‘𝑈)))) | 
| 4 | 3 | simplbi 497 | 1 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ‘cfv 6561 CMetccmet 25288 NrmCVeccnv 30603 BaseSetcba 30605 IndMetcims 30610 CBanccbn 30881 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-cbn 30882 | 
| This theorem is referenced by: bnrel 30886 bnsscmcl 30887 ubthlem1 30889 ubthlem2 30890 ubthlem3 30891 minvecolem1 30893 hlnv 30910 | 
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