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Metamath Proof Explorer

Theorem chincl

Description: Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chincl ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∩ 𝐵 ) ∈ Cℋ )

Proof

Step Hyp Ref Expression
1 ineq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∩ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) )
2 1 eleq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ∩ 𝐵 ) ∈ Cℋ ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ∈ Cℋ ) )
3 ineq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
4 3 eleq1d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ∈ Cℋ ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ∈ Cℋ ) )
5 ifchhv ⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
6 ifchhv ⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
7 5 6 chincli ⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ∈ Cℋ
8 2 4 7 dedth2h ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∩ 𝐵 ) ∈ Cℋ )