Theorem isomliN 40054

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 3208	. 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 40053	. 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
10	1, 3, 9	mpbir2an 724	1 ⊢ 𝐾 ∈ OML

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  lecple 17342  occoc 17343  joincjn 18392  meetcmee 18393  OLcol 39989  OMLcoml 39990

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-oml 39994

This theorem is used by: (None)