Theorem cmm2i

Description: A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	cmm2i	$$⊢B{𝐶}_{\mathscr{H}}\left(A\cap B\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	2 1	cmm1i	$$⊢B{𝐶}_{\mathscr{H}}\left(B\cap A\right)$$
4		incom	$$⊢B\cap A=A\cap B$$
5	3 4	breqtri	$$⊢B{𝐶}_{\mathscr{H}}\left(A\cap B\right)$$