Theorem omlol

Description: An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011)

	Ref	Expression
Assertion	omlol	⊢ ( 𝐾 ∈ OML → 𝐾 ∈ OL )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
2		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
3		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
4		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
5		eqid	⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
6	1 2 3 4 5	isoml	⊢ ( 𝐾 ∈ OML ↔ ( 𝐾 ∈ OL ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑦 → 𝑦 = ( 𝑥 ( join ‘ 𝐾 ) ( 𝑦 ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) ) ) ) )
7	6	simplbi	⊢ ( 𝐾 ∈ OML → 𝐾 ∈ OL )