Theorem chlejb1

Description: Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chlejb1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sseq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( 𝐴 ⊆ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 ) )
2		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ 𝐵 ) )
3	2	eqeq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ 𝐵 ) = 𝐵 ) )
4	1 3	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ 𝐵 ) = 𝐵 ) ) )
5		sseq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
6		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
7		id	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) )
8	6 7	eqeq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ 𝐵 ) = 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
9	5 8	bibi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ 𝐵 ) = 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) ) )
10		h0elch	⊢ 0ℋ ∈ Cℋ
11	10	elimel	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∈ Cℋ
12	10	elimel	⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ∈ Cℋ
13	11 12	chlejb1i	⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) )
14	4 9 13	dedth2h	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 ) )