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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isomliN | Structured version Visualization version GIF version |
Description: Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isomli.0 | β’ πΎ β OL |
isomli.b | β’ π΅ = (BaseβπΎ) |
isomli.l | β’ β€ = (leβπΎ) |
isomli.j | β’ β¨ = (joinβπΎ) |
isomli.m | β’ β§ = (meetβπΎ) |
isomli.o | β’ β₯ = (ocβπΎ) |
isomli.7 | β’ ((π₯ β π΅ β§ π¦ β π΅) β (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯))))) |
Ref | Expression |
---|---|
isomliN | β’ πΎ β OML |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isomli.0 | . 2 β’ πΎ β OL | |
2 | isomli.7 | . . 3 β’ ((π₯ β π΅ β§ π¦ β π΅) β (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯))))) | |
3 | 2 | rgen2 3191 | . 2 β’ βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯)))) |
4 | isomli.b | . . 3 β’ π΅ = (BaseβπΎ) | |
5 | isomli.l | . . 3 β’ β€ = (leβπΎ) | |
6 | isomli.j | . . 3 β’ β¨ = (joinβπΎ) | |
7 | isomli.m | . . 3 β’ β§ = (meetβπΎ) | |
8 | isomli.o | . . 3 β’ β₯ = (ocβπΎ) | |
9 | 4, 5, 6, 7, 8 | isoml 38619 | . 2 β’ (πΎ β OML β (πΎ β OL β§ βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯)))))) |
10 | 1, 3, 9 | mpbir2an 708 | 1 β’ πΎ β OML |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 class class class wbr 5141 βcfv 6536 (class class class)co 7404 Basecbs 17151 lecple 17211 occoc 17212 joincjn 18274 meetcmee 18275 OLcol 38555 OMLcoml 38556 |
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 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6488 df-fv 6544 df-ov 7407 df-oml 38560 |
This theorem is referenced by: (None) |
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