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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opoccl | Structured version Visualization version GIF version |
Description: Closure of orthocomplement operation. (choccl 30251 analog.) (Contributed by NM, 20-Oct-2011.) |
Ref | Expression |
---|---|
opoccl.b | β’ π΅ = (BaseβπΎ) |
opoccl.o | β’ β₯ = (ocβπΎ) |
Ref | Expression |
---|---|
opoccl | β’ ((πΎ β OP β§ π β π΅) β ( β₯ βπ) β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opoccl.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2737 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
3 | opoccl.o | . . . . 5 β’ β₯ = (ocβπΎ) | |
4 | eqid 2737 | . . . . 5 β’ (joinβπΎ) = (joinβπΎ) | |
5 | eqid 2737 | . . . . 5 β’ (meetβπΎ) = (meetβπΎ) | |
6 | eqid 2737 | . . . . 5 β’ (0.βπΎ) = (0.βπΎ) | |
7 | eqid 2737 | . . . . 5 β’ (1.βπΎ) = (1.βπΎ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 37647 | . . . 4 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β ((( β₯ βπ) β π΅ β§ ( β₯ β( β₯ βπ)) = π β§ (π(leβπΎ)π β ( β₯ βπ)(leβπΎ)( β₯ βπ))) β§ (π(joinβπΎ)( β₯ βπ)) = (1.βπΎ) β§ (π(meetβπΎ)( β₯ βπ)) = (0.βπΎ))) |
9 | 8 | 3anidm23 1422 | . . 3 β’ ((πΎ β OP β§ π β π΅) β ((( β₯ βπ) β π΅ β§ ( β₯ β( β₯ βπ)) = π β§ (π(leβπΎ)π β ( β₯ βπ)(leβπΎ)( β₯ βπ))) β§ (π(joinβπΎ)( β₯ βπ)) = (1.βπΎ) β§ (π(meetβπΎ)( β₯ βπ)) = (0.βπΎ))) |
10 | 9 | simp1d 1143 | . 2 β’ ((πΎ β OP β§ π β π΅) β (( β₯ βπ) β π΅ β§ ( β₯ β( β₯ βπ)) = π β§ (π(leβπΎ)π β ( β₯ βπ)(leβπΎ)( β₯ βπ)))) |
11 | 10 | simp1d 1143 | 1 β’ ((πΎ β OP β§ π β π΅) β ( β₯ βπ) β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17084 lecple 17141 occoc 17142 joincjn 18201 meetcmee 18202 0.cp0 18313 1.cp1 18314 OPcops 37637 |
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-ral 3066 df-rab 3409 df-v 3448 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-dm 5644 df-iota 6449 df-fv 6505 df-ov 7361 df-oposet 37641 |
This theorem is referenced by: opcon2b 37662 oplecon3b 37665 oplecon1b 37666 opoc1 37667 opltcon3b 37669 opltcon1b 37670 opltcon2b 37671 riotaocN 37674 oldmm1 37682 oldmm2 37683 oldmm3N 37684 oldmm4 37685 oldmj1 37686 oldmj2 37687 oldmj3 37688 oldmj4 37689 olm11 37692 latmassOLD 37694 omllaw2N 37709 omllaw4 37711 cmtcomlemN 37713 cmt2N 37715 cmt3N 37716 cmt4N 37717 cmtbr2N 37718 cmtbr3N 37719 cmtbr4N 37720 lecmtN 37721 omlfh1N 37723 omlfh3N 37724 omlspjN 37726 cvrcon3b 37742 cvrcmp2 37749 atlatmstc 37784 glbconN 37842 glbconNOLD 37843 glbconxN 37844 cvrexch 37886 1cvrco 37938 1cvratex 37939 1cvrjat 37941 polval2N 38372 polsubN 38373 2polpmapN 38379 2polvalN 38380 poldmj1N 38394 pmapj2N 38395 polatN 38397 2polatN 38398 pnonsingN 38399 ispsubcl2N 38413 polsubclN 38418 poml4N 38419 pmapojoinN 38434 pl42lem1N 38445 lhpoc2N 38481 lhpocnle 38482 lhpmod2i2 38504 lhpmod6i1 38505 lhprelat3N 38506 trlcl 38630 trlle 38650 docaclN 39590 doca2N 39592 djajN 39603 dih1 39752 dih1dimatlem 39795 dochcl 39819 dochvalr3 39829 doch2val2 39830 dochss 39831 dochocss 39832 dochoc 39833 dochnoncon 39857 djhlj 39867 |
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