Theorem pjomli

Description: Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of Kalmbach p. 22. Derived using projections; compare omlsi . (Contributed by NM, 6-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	pjoml.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
Assertion	pjomli	$$⊢\left(A\subseteq B\wedge B\cap \perp \left(A\right)=0_{\mathscr{H}}\right)\to A=B$$

Proof

Step	Hyp	Ref	Expression
1		pjoml.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
3	1 2	omlsi	$$⊢\left(A\subseteq B\wedge B\cap \perp \left(A\right)=0_{\mathscr{H}}\right)\to A=B$$