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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opococ | Structured version Visualization version GIF version |
Description: Double negative law for orthoposets. (ococ 30659 analog.) (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
opoccl.b | β’ π΅ = (BaseβπΎ) |
opoccl.o | β’ β₯ = (ocβπΎ) |
Ref | Expression |
---|---|
opococ | β’ ((πΎ β OP β§ π β π΅) β ( β₯ β( β₯ βπ)) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opoccl.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2733 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
3 | opoccl.o | . . . . 5 β’ β₯ = (ocβπΎ) | |
4 | eqid 2733 | . . . . 5 β’ (joinβπΎ) = (joinβπΎ) | |
5 | eqid 2733 | . . . . 5 β’ (meetβπΎ) = (meetβπΎ) | |
6 | eqid 2733 | . . . . 5 β’ (0.βπΎ) = (0.βπΎ) | |
7 | eqid 2733 | . . . . 5 β’ (1.βπΎ) = (1.βπΎ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 38052 | . . . 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 | simp2d 1144 | 1 β’ ((πΎ β OP β§ π β π΅) β ( β₯ β( β₯ βπ)) = 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 5149 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 occoc 17205 joincjn 18264 meetcmee 18265 0.cp0 18376 1.cp1 18377 OPcops 38042 |
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 ax-nul 5307 |
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-ne 2942 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-dm 5687 df-iota 6496 df-fv 6552 df-ov 7412 df-oposet 38046 |
This theorem is referenced by: opcon3b 38066 opcon2b 38067 oplecon3b 38070 oplecon1b 38071 opltcon1b 38075 opltcon2b 38076 oldmm2 38088 oldmm3N 38089 oldmm4 38090 oldmj1 38091 oldmj2 38092 oldmj3 38093 oldmj4 38094 olm11 38097 omllaw4 38116 cmt2N 38120 glbconN 38247 glbconNOLD 38248 1cvratex 38344 1cvrjat 38346 polval2N 38777 2polpmapN 38784 2polvalN 38785 2polatN 38803 lhpoc2N 38886 doch2val2 40235 dochocss 40237 dochoc 40238 |
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