Theorem cvexch

Description: The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of Kalmbach p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvexch	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ineq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∩ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) )
2	1	breq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ⋖ℋ 𝐵 ) )
3		id	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) )
4		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) )
5	3 4	breq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⋖ℋ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ) )
6	2 5	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⋖ℋ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ) ) )
7		ineq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
8		id	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) )
9	7 8	breq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ⋖ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
10		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
11	10	breq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⋖ℋ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⋖ℋ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
12	9 11	bibi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⋖ℋ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ⋖ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⋖ℋ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) ) )
13		ifchhv	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
14		ifchhv	⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
15	13 14	cvexchi	⊢ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ⋖ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⋖ℋ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
16	6 12 15	dedth2h	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )