Theorem omlol

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

	Ref	Expression
Assertion	omlol	$$⊢K\in \mathrm{OML}\to K\in \mathrm{OL}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
2		eqid	$$⊢{\le }_K={\le }_K$$
3		eqid	$$⊢\mathrm{join}\left(K\right)=\mathrm{join}\left(K\right)$$
4		eqid	$$⊢\mathrm{meet}\left(K\right)=\mathrm{meet}\left(K\right)$$
5		eqid	$$⊢\mathrm{oc}\left(K\right)=\mathrm{oc}\left(K\right)$$
6	1 2 3 4 5	isoml	$$⊢K\in \mathrm{OML}\leftrightarrow \left(K\in \mathrm{OL}\wedge \forall x\in {\mathrm{Base}}_K\forall y\in {\mathrm{Base}}_K\left(x{\le }_{K}y\to y=x\mathrm{join}\left(K\right)\left(y\mathrm{meet}\left(K\right)\mathrm{oc}\left(K\right)\left(x\right)\right)\right)\right)$$
7	6	simplbi	$$⊢K\in \mathrm{OML}\to K\in \mathrm{OL}$$