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Theorem isomliN 37704
Description: Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
isomli.0 𝐾 ∈ OL
isomli.b 𝐡 = (Baseβ€˜πΎ)
isomli.l ≀ = (leβ€˜πΎ)
isomli.j ∨ = (joinβ€˜πΎ)
isomli.m ∧ = (meetβ€˜πΎ)
isomli.o βŠ₯ = (ocβ€˜πΎ)
isomli.7 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)))))
Assertion
Ref Expression
isomliN 𝐾 ∈ OML
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐾,𝑦
Allowed substitution hints:   ∨ (π‘₯,𝑦)   ≀ (π‘₯,𝑦)   ∧ (π‘₯,𝑦)   βŠ₯ (π‘₯,𝑦)

Proof of Theorem isomliN
StepHypRef Expression
1 isomli.0 . 2 𝐾 ∈ OL
2 isomli.7 . . 3 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)))))
32rgen2 3195 . 2 βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))
4 isomli.b . . 3 𝐡 = (Baseβ€˜πΎ)
5 isomli.l . . 3 ≀ = (leβ€˜πΎ)
6 isomli.j . . 3 ∨ = (joinβ€˜πΎ)
7 isomli.m . . 3 ∧ = (meetβ€˜πΎ)
8 isomli.o . . 3 βŠ₯ = (ocβ€˜πΎ)
94, 5, 6, 7, 8isoml 37703 . 2 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))))
101, 3, 9mpbir2an 710 1 𝐾 ∈ OML
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   class class class wbr 5106  β€˜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: (None)
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