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Mirrors > Home > MPE Home > Th. List > bnnvc | Structured version Visualization version GIF version |
Description: A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
bnnvc | ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | 1 | isbn 24654 | . 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp)) |
3 | 2 | simp1bi 1146 | 1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ‘cfv 6494 Scalarcsca 17096 NrmVeccnvc 23889 CMetSpccms 24648 Bancbn 24649 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-iota 6446 df-fv 6502 df-bn 24652 |
This theorem is referenced by: bnnlm 24657 lssbn 24668 bncssbn 24690 cmslsschl 24693 |
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