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Theorem omlol 37705
Description: An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
Assertion
Ref Expression
omlol (๐พ โˆˆ OML โ†’ ๐พ โˆˆ OL)

Proof of Theorem omlol
Dummy variables ๐‘ฅ ๐‘ฆ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 (Baseโ€˜๐พ) = (Baseโ€˜๐พ)
2 eqid 2737 . . 3 (leโ€˜๐พ) = (leโ€˜๐พ)
3 eqid 2737 . . 3 (joinโ€˜๐พ) = (joinโ€˜๐พ)
4 eqid 2737 . . 3 (meetโ€˜๐พ) = (meetโ€˜๐พ)
5 eqid 2737 . . 3 (ocโ€˜๐พ) = (ocโ€˜๐พ)
61, 2, 3, 4, 5isoml 37703 . 2 (๐พ โˆˆ OML โ†” (๐พ โˆˆ OL โˆง โˆ€๐‘ฅ โˆˆ (Baseโ€˜๐พ)โˆ€๐‘ฆ โˆˆ (Baseโ€˜๐พ)(๐‘ฅ(leโ€˜๐พ)๐‘ฆ โ†’ ๐‘ฆ = (๐‘ฅ(joinโ€˜๐พ)(๐‘ฆ(meetโ€˜๐พ)((ocโ€˜๐พ)โ€˜๐‘ฅ))))))
76simplbi 499 1 (๐พ โˆˆ OML โ†’ ๐พ โˆˆ OL)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   = wceq 1542   โˆˆ wcel 2107  โˆ€wral 3065   class class class wbr 5106  โ€˜cfv 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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