Theorem isomliN 39578

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	⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → 𝑦 = (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

Step	Hyp	Ref	Expression
1		isomli.0	. 2 ⊢ 𝐾 ∈ OL
2		isomli.7	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
3	2	rgen2 3177	. 2 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (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)
9	4, 5, 6, 7, 8	isoml 39577	. 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
10	1, 3, 9	mpbir2an 712	1 ⊢ 𝐾 ∈ OML

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5099  ‘cfv 6493  (class class class)co 7361  Basecbs 17141  lecple 17189  occoc 17190  joincjn 18239  meetcmee 18240  OLcol 39513  OMLcoml 39514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-ov 7364  df-oml 39518

This theorem is referenced by: (None)