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 ∈ Cℋ
	pjoml2.2	⊢ 𝐵 ∈ Cℋ
Assertion	cmm1i	⊢ 𝐴 𝐶ℋ ( 𝐴 ∩ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	⊢ 𝐵 ∈ Cℋ
3	1 2	chincli	⊢ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ
4		inss1	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴
5	3 1 4	lecmii	⊢ ( 𝐴 ∩ 𝐵 ) 𝐶ℋ 𝐴
6	3 1 5	cmcmii	⊢ 𝐴 𝐶ℋ ( 𝐴 ∩ 𝐵 )