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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 23943 | . 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)) |
3 | 2 | simp3bi 1143 | 1 ⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Scalarcsca 16570 NrmVeccnvc 23193 CMetSpccms 23937 Bancbn 23938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-bn 23941 |
This theorem is referenced by: lssbn 23957 hlprlem 23972 bncssbn 23979 cmslsschl 23982 |
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