Theorem chj0

Description: Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chj0	⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∨ℋ 0ℋ ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( 𝐴 ∨ℋ 0ℋ ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ 0ℋ ) )
2		id	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) )
3	1 2	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( 𝐴 ∨ℋ 0ℋ ) = 𝐴 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ 0ℋ ) = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) )
4		h0elch	⊢ 0ℋ ∈ Cℋ
5	4	elimel	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∈ Cℋ
6	5	chj0i	⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ 0ℋ ) = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ )
7	3 6	dedth	⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∨ℋ 0ℋ ) = 𝐴 )