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Mirrors > Home > MPE Home > Th. List > bncmet | Structured version Visualization version GIF version |
Description: The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
iscms.1 | β’ π = (Baseβπ) |
iscms.2 | β’ π· = ((distβπ) βΎ (π Γ π)) |
Ref | Expression |
---|---|
bncmet | β’ (π β Ban β π· β (CMetβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bncms 25292 | . 2 β’ (π β Ban β π β CMetSp) | |
2 | iscms.1 | . . 3 β’ π = (Baseβπ) | |
3 | iscms.2 | . . 3 β’ π· = ((distβπ) βΎ (π Γ π)) | |
4 | 2, 3 | cmscmet 25294 | . 2 β’ (π β CMetSp β π· β (CMetβπ)) |
5 | 1, 4 | syl 17 | 1 β’ (π β Ban β π· β (CMetβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Γ cxp 5680 βΎ cres 5684 βcfv 6553 Basecbs 17187 distcds 17249 CMetccmet 25202 CMetSpccms 25280 Bancbn 25281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-xp 5688 df-res 5694 df-iota 6505 df-fv 6561 df-cms 25283 df-bn 25284 |
This theorem is referenced by: (None) |
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