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Theorem isomliN 35029
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 𝐵 = (Base‘𝐾)
isomli.l = (le‘𝐾)
isomli.j = (join‘𝐾)
isomli.m = (meet‘𝐾)
isomli.o = (oc‘𝐾)
isomli.7 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))))
Assertion
Ref Expression
isomliN 𝐾 ∈ OML
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦
Allowed substitution hints:   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem isomliN
StepHypRef Expression
1 isomli.0 . 2 𝐾 ∈ OL
2 isomli.7 . . 3 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))))
32rgen2a 3115 . 2 𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))
4 isomli.b . . 3 𝐵 = (Base‘𝐾)
5 isomli.l . . 3 = (le‘𝐾)
6 isomli.j . . 3 = (join‘𝐾)
7 isomli.m . . 3 = (meet‘𝐾)
8 isomli.o . . 3 = (oc‘𝐾)
94, 5, 6, 7, 8isoml 35028 . 2 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
101, 3, 9mpbir2an 993 1 𝐾 ∈ OML
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  occoc 16151  joincjn 17145  meetcmee 17146  OLcol 34964  OMLcoml 34965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-oml 34969
This theorem is referenced by: (None)
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