Theorem cmj1i

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	cmj1i	⊢ 𝐴 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	1 2	chub1i	⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 )
5	1 3 4	lecmii	⊢ 𝐴 𝐶ℋ ( 𝐴 ∨ℋ 𝐵 )