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Mirrors > Home > MPE Home > Th. List > Mathboxes > omlol | Structured version Visualization version GIF version |
Description: An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.) |
Ref | Expression |
---|---|
omlol | โข (๐พ โ OML โ ๐พ โ OL) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 โข (Baseโ๐พ) = (Baseโ๐พ) | |
2 | eqid 2737 | . . 3 โข (leโ๐พ) = (leโ๐พ) | |
3 | eqid 2737 | . . 3 โข (joinโ๐พ) = (joinโ๐พ) | |
4 | eqid 2737 | . . 3 โข (meetโ๐พ) = (meetโ๐พ) | |
5 | eqid 2737 | . . 3 โข (ocโ๐พ) = (ocโ๐พ) | |
6 | 1, 2, 3, 4, 5 | isoml 37703 | . 2 โข (๐พ โ OML โ (๐พ โ OL โง โ๐ฅ โ (Baseโ๐พ)โ๐ฆ โ (Baseโ๐พ)(๐ฅ(leโ๐พ)๐ฆ โ ๐ฆ = (๐ฅ(joinโ๐พ)(๐ฆ(meetโ๐พ)((ocโ๐พ)โ๐ฅ)))))) |
7 | 6 | simplbi 499 | 1 โข (๐พ โ OML โ ๐พ โ OL) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โwral 3065 class class class wbr 5106 โcfv 6497 (class class class)co 7358 Basecbs 17084 lecple 17141 occoc 17142 joincjn 18201 meetcmee 18202 OLcol 37639 OMLcoml 37640 |
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 |
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-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-iota 6449 df-fv 6505 df-ov 7361 df-oml 37644 |
This theorem is referenced by: omlop 37706 omllat 37707 omllaw3 37710 omllaw4 37711 cmtcomlemN 37713 cmtbr2N 37718 cmtbr3N 37719 omlfh1N 37723 omlfh3N 37724 omlspjN 37726 hlol 37826 |
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