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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 24514 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
bnnv | ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
2 | eqid 2738 | . . 3 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
3 | 1, 2 | iscbn 29234 | . 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘𝑈) ∈ (CMet‘(BaseSet‘𝑈)))) |
4 | 3 | simplbi 498 | 1 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ‘cfv 6426 CMetccmet 24428 NrmCVeccnv 28954 BaseSetcba 28956 IndMetcims 28961 CBanccbn 29232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-iota 6384 df-fv 6434 df-cbn 29233 |
This theorem is referenced by: bnrel 29237 bnsscmcl 29238 ubthlem1 29240 ubthlem2 29241 ubthlem3 29242 minvecolem1 29244 hlnv 29261 |
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