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Mirrors > Home > MPE Home > Th. List > hlcmet | Structured version Visualization version GIF version |
Description: The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlcmet.x | β’ π = (BaseSetβπ) |
hlcmet.8 | β’ π· = (IndMetβπ) |
Ref | Expression |
---|---|
hlcmet | β’ (π β CHilOLD β π· β (CMetβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlobn 29872 | . 2 β’ (π β CHilOLD β π β CBan) | |
2 | hlcmet.x | . . 3 β’ π = (BaseSetβπ) | |
3 | hlcmet.8 | . . 3 β’ π· = (IndMetβπ) | |
4 | 2, 3 | cbncms 29849 | . 2 β’ (π β CBan β π· β (CMetβπ)) |
5 | 1, 4 | syl 17 | 1 β’ (π β CHilOLD β π· β (CMetβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6497 CMetccmet 24634 BaseSetcba 29570 IndMetcims 29575 CBanccbn 29846 CHilOLDchlo 29869 |
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 2704 |
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 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 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-cbn 29847 df-hlo 29870 |
This theorem is referenced by: hlmet 29879 hlcompl 29899 |
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