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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 30351 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 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 | 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 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: opcon3b 37661 opcon2b 37662 oplecon3b 37665 oplecon1b 37666 opltcon1b 37670 opltcon2b 37671 oldmm2 37683 oldmm3N 37684 oldmm4 37685 oldmj1 37686 oldmj2 37687 oldmj3 37688 oldmj4 37689 olm11 37692 omllaw4 37711 cmt2N 37715 glbconN 37842 glbconNOLD 37843 1cvratex 37939 1cvrjat 37941 polval2N 38372 2polpmapN 38379 2polvalN 38380 2polatN 38398 lhpoc2N 38481 doch2val2 39830 dochocss 39832 dochoc 39833 |
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