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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 23946 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
bnnv | ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
2 | eqid 2824 | . . 3 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
3 | 1, 2 | iscbn 28644 | . 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘𝑈) ∈ (CMet‘(BaseSet‘𝑈)))) |
4 | 3 | simplbi 500 | 1 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ‘cfv 6358 CMetccmet 23860 NrmCVeccnv 28364 BaseSetcba 28366 IndMetcims 28371 CBanccbn 28642 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-cbn 28643 |
This theorem is referenced by: bnrel 28647 bnsscmcl 28648 ubthlem1 28650 ubthlem2 28651 ubthlem3 28652 minvecolem1 28654 hlnv 28671 |
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