Theorem cmm1i

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	cmm1i	$$⊢A{𝐶}_{\mathscr{H}}\left(A\cap B\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	chincli	$$⊢A\cap B\in {\mathbf{C}}_{\mathscr{H}}$$
4		inss1	$$⊢A\cap B\subseteq A$$
5	3 1 4	lecmii	$$⊢\left(A\cap B\right){𝐶}_{\mathscr{H}}A$$
6	3 1 5	cmcmii	$$⊢A{𝐶}_{\mathscr{H}}\left(A\cap B\right)$$