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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 30716 | . 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 30693 | . 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 1533 β wcel 2098 βcfv 6551 CMetccmet 25200 BaseSetcba 30414 IndMetcims 30419 CBanccbn 30690 CHilOLDchlo 30713 |
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 2698 |
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 2705 df-cleq 2719 df-clel 2805 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-iota 6503 df-fv 6559 df-cbn 30691 df-hlo 30714 |
This theorem is referenced by: hlmet 30723 hlcompl 30743 |
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