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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 25223 | . 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 25225 | . 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 5667 βΎ cres 5671 βcfv 6536 Basecbs 17151 distcds 17213 CMetccmet 25133 CMetSpccms 25211 Bancbn 25212 |
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 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-xp 5675 df-res 5681 df-iota 6488 df-fv 6544 df-cms 25214 df-bn 25215 |
This theorem is referenced by: (None) |
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