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 e. CH /\ B e. CH ) -> ( ( A i^i B )  A

Proof

Step Hyp Ref Expression
1 ineq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A i^i B ) = ( if ( A e. CH , A , ~H ) i^i B ) )
2 1 breq1d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( A i^i B )  ( if ( A e. CH , A , ~H ) i^i B )
3 id 
 |-  ( A = if ( A e. CH , A , ~H ) -> A = if ( A e. CH , A , ~H ) )
4 oveq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A vH B ) = ( if ( A e. CH , A , ~H ) vH B ) )
5 3 4 breq12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A  if ( A e. CH , A , ~H )
6 2 5 bibi12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( ( A i^i B )  A  ( ( if ( A e. CH , A , ~H ) i^i B )  if ( A e. CH , A , ~H )
7 ineq2 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( if ( A e. CH , A , ~H ) i^i B ) = ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) ) )
8 id 
 |-  ( B = if ( B e. CH , B , ~H ) -> B = if ( B e. CH , B , ~H ) )
9 7 8 breq12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( if ( A e. CH , A , ~H ) i^i B )  ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) )
10 oveq2 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( if ( A e. CH , A , ~H ) vH B ) = ( if ( A e. CH , A , ~H ) vH if ( B e. CH , B , ~H ) ) )
11 10 breq2d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( if ( A e. CH , A , ~H )  if ( A e. CH , A , ~H )
12 9 11 bibi12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( ( if ( A e. CH , A , ~H ) i^i B )  if ( A e. CH , A , ~H )  ( ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) )  if ( A e. CH , A , ~H )
13 ifchhv 
 |-  if ( A e. CH , A , ~H ) e. CH
14 ifchhv 
 |-  if ( B e. CH , B , ~H ) e. CH
15 13 14 cvexchi 
 |-  ( ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) )  if ( A e. CH , A , ~H )
16 6 12 15 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( ( A i^i B )  A