Metamath Proof Explorer

Theorem chdmm1

Description: De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chdmm1 
|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A i^i B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) ) )

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 fveq2d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( _|_ ` ( A i^i B ) ) = ( _|_ ` ( if ( A e. CH , A , ~H ) i^i B ) ) )
3 fveq2 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( _|_ ` A ) = ( _|_ ` if ( A e. CH , A , ~H ) ) )
4 3 oveq1d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( _|_ ` A ) vH ( _|_ ` B ) ) = ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH ( _|_ ` B ) ) )
5 2 4 eqeq12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( _|_ ` ( A i^i B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) ) <-> ( _|_ ` ( if ( A e. CH , A , ~H ) i^i B ) ) = ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH ( _|_ ` B ) ) ) )
6 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 ) ) )
7 6 fveq2d 
 |-  ( 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 fveq2 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( _|_ ` B ) = ( _|_ ` if ( B e. CH , B , ~H ) ) )
9 8 oveq2d 
 |-  ( 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 ) ) ) )
10 7 9 eqeq12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( _|_ ` ( if ( A e. CH , A , ~H ) i^i B ) ) = ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH ( _|_ ` B ) ) <-> ( _|_ ` ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) ) ) = ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH ( _|_ ` if ( B e. CH , B , ~H ) ) ) ) )
11 ifchhv 
 |-  if ( A e. CH , A , ~H ) e. CH
12 ifchhv 
 |-  if ( B e. CH , B , ~H ) e. CH
13 11 12 chdmm1i 
 |-  ( _|_ ` ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) ) ) = ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH ( _|_ ` if ( B e. CH , B , ~H ) ) )
14 5 10 13 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A i^i B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) ) )