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Mirrors > Home > MPE Home > Th. List > bnnv | Structured version Visualization version GIF version |
Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 25388 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
bnnv | ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
2 | eqid 2735 | . . 3 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
3 | 1, 2 | iscbn 30893 | . 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 2106 ‘cfv 6563 CMetccmet 25302 NrmCVeccnv 30613 BaseSetcba 30615 IndMetcims 30620 CBanccbn 30891 |
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 |
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-rab 3434 df-v 3480 df-dif 3966 df-un 3968 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-iota 6516 df-fv 6571 df-cbn 30892 |
This theorem is referenced by: bnrel 30896 bnsscmcl 30897 ubthlem1 30899 ubthlem2 30900 ubthlem3 30901 minvecolem1 30903 hlnv 30920 |
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