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Theorem isomliN 38620
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 3191 . 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 38619 . 2 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))))
101, 3, 9mpbir2an 708 1 𝐾 ∈ OML
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   class class class wbr 5141  β€˜cfv 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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