Theorem lecm

Description: Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of Beran p. 40. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	lecm	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 𝐶ℋ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sseq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( 𝐴 ⊆ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 ) )
2		breq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( 𝐴 𝐶ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ 𝐵 ) )
3	1 2	imbi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( 𝐴 ⊆ 𝐵 → 𝐴 𝐶ℋ 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 → if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ 𝐵 ) ) )
4		sseq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
5		breq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
6	4 5	imbi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 → if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) ) )
7		h0elch	⊢ 0ℋ ∈ Cℋ
8	7	elimel	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∈ Cℋ
9	7	elimel	⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ∈ Cℋ
10	8 9	lecmi	⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) 𝐶ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) )
11	3 6 10	dedth2h	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 → 𝐴 𝐶ℋ 𝐵 ) )
12	11	3impia	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 𝐶ℋ 𝐵 )