Theorem isomliN 38104

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 3197	. 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 38103	. 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
10	1, 3, 9	mpbir2an 709	1 ⊢ 𝐾 ∈ OML

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   class class class wbr 5148  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  lecple 17203  occoc 17204  joincjn 18263  meetcmee 18264  OLcol 38039  OMLcoml 38040

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-oml 38044

This theorem is referenced by: (None)