Theorem lecmi

Description: Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of Beran p. 40. (Contributed by NM, 16-Jan-2005) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	⊢ 𝐴 ∈ Cℋ
	pjoml2.2	⊢ 𝐵 ∈ Cℋ
Assertion	lecmi	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 𝐶ℋ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	⊢ 𝐵 ∈ Cℋ
3		ssinss1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) ⊆ 𝐵 )
4	1 2	cmbr4i	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) ⊆ 𝐵 )
5	3 4	sylibr	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 𝐶ℋ 𝐵 )