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Metamath Proof Explorer

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 )