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Metamath Proof Explorer

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 ∈ C_ℋ
	pjoml2.2	⊢ 𝐵 ∈ C_ℋ
Assertion	lecmi	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 𝐶_ℋ 𝐵 )

Proof

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