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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 3927 | . . 3 β’ (π β (NrmVec β© CMetSp) β (π β NrmVec β§ π β CMetSp)) | |
2 | 1 | anbi1i 625 | . 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 6843 | . . . . 5 β’ (π€ = π β (Scalarβπ€) = (Scalarβπ)) | |
4 | isbn.1 | . . . . 5 β’ πΉ = (Scalarβπ) | |
5 | 3, 4 | eqtr4di 2795 | . . . 4 β’ (π€ = π β (Scalarβπ€) = πΉ) |
6 | 5 | eleq1d 2823 | . . 3 β’ (π€ = π β ((Scalarβπ€) β CMetSp β πΉ β CMetSp)) |
7 | df-bn 24703 | . . 3 β’ Ban = {π€ β (NrmVec β© CMetSp) β£ (Scalarβπ€) β CMetSp} | |
8 | 6, 7 | elrab2 3649 | . 2 β’ (π β Ban β (π β (NrmVec β© CMetSp) β§ πΉ β CMetSp)) |
9 | df-3an 1090 | . 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β© cin 3910 βcfv 6497 Scalarcsca 17137 NrmVeccnvc 23940 CMetSpccms 24699 Bancbn 24700 |
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 2708 |
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 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-bn 24703 |
This theorem is referenced by: bnsca 24706 bnnvc 24707 bncms 24711 lssbn 24719 srabn 24727 ishl2 24737 cmslssbn 24739 |
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