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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlomcmcv | Structured version Visualization version GIF version |
Description: A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
hlomcmcv | β’ (πΎ β HL β (πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | eqid 2733 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
3 | eqid 2733 | . . 3 β’ (ltβπΎ) = (ltβπΎ) | |
4 | eqid 2733 | . . 3 β’ (joinβπΎ) = (joinβπΎ) | |
5 | eqid 2733 | . . 3 β’ (0.βπΎ) = (0.βπΎ) | |
6 | eqid 2733 | . . 3 β’ (1.βπΎ) = (1.βπΎ) | |
7 | eqid 2733 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ishlat1 37860 | . 2 β’ (πΎ β HL β ((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ (βπ₯ β (AtomsβπΎ)βπ¦ β (AtomsβπΎ)(π₯ β π¦ β βπ§ β (AtomsβπΎ)(π§ β π₯ β§ π§ β π¦ β§ π§(leβπΎ)(π₯(joinβπΎ)π¦))) β§ βπ₯ β (BaseβπΎ)βπ¦ β (BaseβπΎ)βπ§ β (BaseβπΎ)(((0.βπΎ)(ltβπΎ)π₯ β§ π₯(ltβπΎ)π¦) β§ (π¦(ltβπΎ)π§ β§ π§(ltβπΎ)(1.βπΎ)))))) |
9 | 8 | simplbi 499 | 1 β’ (πΎ β HL β (πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 β wcel 2107 β wne 2940 βwral 3061 βwrex 3070 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17088 lecple 17145 ltcplt 18202 joincjn 18205 0.cp0 18317 1.cp1 18318 CLatccla 18392 OMLcoml 37683 Atomscatm 37771 CvLatclc 37773 HLchlt 37858 |
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-ral 3062 df-rex 3071 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-ov 7361 df-hlat 37859 |
This theorem is referenced by: hloml 37865 hlclat 37866 hlcvl 37867 cvr1 37919 cvrp 37925 atcvr1 37926 atcvr2 37927 |
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