This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )