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Metamath Proof Explorer

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 )