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Mirrors > Home > MPE Home > Th. List > cbncms | Structured version Visualization version GIF version |
Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 24763 (or preferably bncms 24760) instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
iscbn.x | β’ π = (BaseSetβπ) |
iscbn.8 | β’ π· = (IndMetβπ) |
Ref | Expression |
---|---|
cbncms | β’ (π β CBan β π· β (CMetβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscbn.x | . . 3 β’ π = (BaseSetβπ) | |
2 | iscbn.8 | . . 3 β’ π· = (IndMetβπ) | |
3 | 1, 2 | iscbn 29903 | . 2 β’ (π β CBan β (π β NrmCVec β§ π· β (CMetβπ))) |
4 | 3 | simprbi 497 | 1 β’ (π β CBan β π· β (CMetβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 βcfv 6516 CMetccmet 24670 NrmCVeccnv 29623 BaseSetcba 29625 IndMetcims 29630 CBanccbn 29901 |
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 2702 |
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 2709 df-cleq 2723 df-clel 2809 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-br 5126 df-iota 6468 df-fv 6524 df-cbn 29902 |
This theorem is referenced by: bnsscmcl 29907 ubthlem1 29909 ubthlem2 29910 minvecolem4a 29916 hlcmet 29933 |
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