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Metamath Proof Explorer

Theorem cmj2i

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

Ref Expression
Hypotheses pjoml2.1 ⊢ 𝐴 ∈ Cℋ
pjoml2.2 ⊢ 𝐵 ∈ Cℋ
Assertion cmj2i 𝐵 𝐶ℋ ( 𝐴 ∨ℋ 𝐵 )

Proof

Step Hyp Ref Expression
1 pjoml2.1 ⊢ 𝐴 ∈ Cℋ
2 pjoml2.2 ⊢ 𝐵 ∈ Cℋ
3 1 2 chjcli ⊢ ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ
4 2 1 chub2i ⊢ 𝐵 ⊆ ( 𝐴 ∨ℋ 𝐵 )
5 2 3 4 lecmii ⊢ 𝐵 𝐶ℋ ( 𝐴 ∨ℋ 𝐵 )