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Mirrors > Home > MPE Home > Th. List > isbn | Structured version Visualization version GIF version |
Description: A Banach space is a normed vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
isbn.1 | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
isbn | β’ (π β Ban β (π β NrmVec β§ π β CMetSp β§ πΉ β CMetSp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3961 | . . 3 β’ (π β (NrmVec β© CMetSp) β (π β NrmVec β§ π β CMetSp)) | |
2 | 1 | anbi1i 623 | . 2 β’ ((π β (NrmVec β© CMetSp) β§ πΉ β CMetSp) β ((π β NrmVec β§ π β CMetSp) β§ πΉ β CMetSp)) |
3 | fveq2 6891 | . . . . 5 β’ (π€ = π β (Scalarβπ€) = (Scalarβπ)) | |
4 | isbn.1 | . . . . 5 β’ πΉ = (Scalarβπ) | |
5 | 3, 4 | eqtr4di 2786 | . . . 4 β’ (π€ = π β (Scalarβπ€) = πΉ) |
6 | 5 | eleq1d 2814 | . . 3 β’ (π€ = π β ((Scalarβπ€) β CMetSp β πΉ β CMetSp)) |
7 | df-bn 25257 | . . 3 β’ Ban = {π€ β (NrmVec β© CMetSp) β£ (Scalarβπ€) β CMetSp} | |
8 | 6, 7 | elrab2 3684 | . 2 β’ (π β Ban β (π β (NrmVec β© CMetSp) β§ πΉ β CMetSp)) |
9 | df-3an 1087 | . 2 β’ ((π β NrmVec β§ π β CMetSp β§ πΉ β CMetSp) β ((π β NrmVec β§ π β CMetSp) β§ πΉ β CMetSp)) | |
10 | 2, 8, 9 | 3bitr4i 303 | 1 β’ (π β Ban β (π β NrmVec β§ π β CMetSp β§ πΉ β CMetSp)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β© cin 3944 βcfv 6542 Scalarcsca 17229 NrmVeccnvc 24483 CMetSpccms 25253 Bancbn 25254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-bn 25257 |
This theorem is referenced by: bnsca 25260 bnnvc 25261 bncms 25265 lssbn 25273 srabn 25281 ishl2 25291 cmslssbn 25293 |
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