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Mirrors > Home > MPE Home > Th. List > bnsca | Structured version Visualization version GIF version |
Description: The scalar field of a Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
isbn.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
bnsca | ⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isbn.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | isbn 24654 | . 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)) |
3 | 2 | simp3bi 1147 | 1 ⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 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 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-bn 24652 |
This theorem is referenced by: lssbn 24668 hlprlem 24683 bncssbn 24690 cmslsschl 24693 |
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