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Metamath Proof Explorer

Theorem chjass

Description: Associative law for Hilbert lattice join. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 10-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chjass 
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( A vH B ) vH C ) = ( A vH ( B vH C ) ) )

Proof

Step Hyp Ref Expression
1 oveq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A vH B ) = ( if ( A e. CH , A , ~H ) vH B ) )
2 1 oveq1d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( A vH B ) vH C ) = ( ( if ( A e. CH , A , ~H ) vH B ) vH C ) )
3 oveq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A vH ( B vH C ) ) = ( if ( A e. CH , A , ~H ) vH ( B vH C ) ) )
4 2 3 eqeq12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( ( A vH B ) vH C ) = ( A vH ( B vH C ) ) <-> ( ( if ( A e. CH , A , ~H ) vH B ) vH C ) = ( if ( A e. CH , A , ~H ) vH ( B vH C ) ) ) )
5 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 ) ) )
6 5 oveq1d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( if ( A e. CH , A , ~H ) vH B ) vH C ) = ( ( if ( A e. CH , A , ~H ) vH if ( B e. CH , B , ~H ) ) vH C ) )
7 oveq1 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( B vH C ) = ( if ( B e. CH , B , ~H ) vH C ) )
8 7 oveq2d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( if ( A e. CH , A , ~H ) vH ( B vH C ) ) = ( if ( A e. CH , A , ~H ) vH ( if ( B e. CH , B , ~H ) vH C ) ) )
9 6 8 eqeq12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( ( if ( A e. CH , A , ~H ) vH B ) vH C ) = ( if ( A e. CH , A , ~H ) vH ( B vH C ) ) <-> ( ( if ( A e. CH , A , ~H ) vH if ( B e. CH , B , ~H ) ) vH C ) = ( if ( A e. CH , A , ~H ) vH ( if ( B e. CH , B , ~H ) vH C ) ) ) )
10 oveq2 
 |-  ( C = if ( C e. CH , C , ~H ) -> ( ( if ( A e. CH , A , ~H ) vH if ( B e. CH , B , ~H ) ) vH C ) = ( ( if ( A e. CH , A , ~H ) vH if ( B e. CH , B , ~H ) ) vH if ( C e. CH , C , ~H ) ) )
11 oveq2 
 |-  ( C = if ( C e. CH , C , ~H ) -> ( if ( B e. CH , B , ~H ) vH C ) = ( if ( B e. CH , B , ~H ) vH if ( C e. CH , C , ~H ) ) )
12 11 oveq2d 
 |-  ( C = if ( C e. CH , C , ~H ) -> ( if ( A e. CH , A , ~H ) vH ( if ( B e. CH , B , ~H ) vH C ) ) = ( if ( A e. CH , A , ~H ) vH ( if ( B e. CH , B , ~H ) vH if ( C e. CH , C , ~H ) ) ) )
13 10 12 eqeq12d 
 |-  ( C = if ( C e. CH , C , ~H ) -> ( ( ( if ( A e. CH , A , ~H ) vH if ( B e. CH , B , ~H ) ) vH C ) = ( if ( A e. CH , A , ~H ) vH ( if ( B e. CH , B , ~H ) vH C ) ) <-> ( ( if ( A e. CH , A , ~H ) vH if ( B e. CH , B , ~H ) ) vH if ( C e. CH , C , ~H ) ) = ( if ( A e. CH , A , ~H ) vH ( if ( B e. CH , B , ~H ) vH if ( C e. CH , C , ~H ) ) ) ) )
14 ifchhv 
 |-  if ( A e. CH , A , ~H ) e. CH
15 ifchhv 
 |-  if ( B e. CH , B , ~H ) e. CH
16 ifchhv 
 |-  if ( C e. CH , C , ~H ) e. CH
17 14 15 16 chjassi 
 |-  ( ( if ( A e. CH , A , ~H ) vH if ( B e. CH , B , ~H ) ) vH if ( C e. CH , C , ~H ) ) = ( if ( A e. CH , A , ~H ) vH ( if ( B e. CH , B , ~H ) vH if ( C e. CH , C , ~H ) ) )
18 4 9 13 17 dedth3h 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( A vH B ) vH C ) = ( A vH ( B vH C ) ) )