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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 24711 | . 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 24713 | . 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 1542 β wcel 2107 Γ cxp 5632 βΎ cres 5636 βcfv 6497 Basecbs 17084 distcds 17143 CMetccmet 24621 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 ax-nul 5264 |
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-ne 2945 df-rab 3409 df-v 3448 df-sbc 3741 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-opab 5169 df-xp 5640 df-res 5646 df-iota 6449 df-fv 6505 df-cms 24702 df-bn 24703 |
This theorem is referenced by: (None) |
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