Theorem isomliN 35314

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	rgen2a 3186	. 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 35313	. 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
10	1, 3, 9	mpbir2an 704	1 ⊢ 𝐾 ∈ OML

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117   class class class wbr 4873  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  occoc 16313  joincjn 17297  meetcmee 17298  OLcol 35249  OMLcoml 35250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-oml 35254

This theorem is referenced by: (None)