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Metamath Proof Explorer

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 ∈ Cℋ
pjoml2.2 ⊢ 𝐵 ∈ Cℋ
Assertion cmm2i 𝐵 𝐶ℋ ( 𝐴 ∩ 𝐵 )

Proof

Step Hyp Ref Expression
1 pjoml2.1 ⊢ 𝐴 ∈ Cℋ
2 pjoml2.2 ⊢ 𝐵 ∈ Cℋ
3 2 1 cmm1i ⊢ 𝐵 𝐶ℋ ( 𝐵 ∩ 𝐴 )
4 incom ⊢ ( 𝐵 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐵 )
5 3 4 breqtri ⊢ 𝐵 𝐶ℋ ( 𝐴 ∩ 𝐵 )