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Metamath Proof Explorer

Theorem pjoml3

Description: Variation of orthomodular law. (Contributed by NM, 24-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion pjoml3 
|- ( ( A e. CH /\ B e. CH ) -> ( B C_ A -> ( A i^i ( ( _|_ ` A ) vH B ) ) = B ) )

Proof

Step Hyp Ref Expression
1 sseq2 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( B C_ A <-> B C_ if ( A e. CH , A , ~H ) ) )
2 id 
 |-  ( A = if ( A e. CH , A , ~H ) -> A = if ( A e. CH , A , ~H ) )
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 ineq12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A i^i ( ( _|_ ` A ) vH B ) ) = ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH B ) ) )
6 5 eqeq1d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( A i^i ( ( _|_ ` A ) vH B ) ) = B <-> ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH B ) ) = B ) )
7 1 6 imbi12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( B C_ A -> ( A i^i ( ( _|_ ` A ) vH B ) ) = B ) <-> ( B C_ if ( A e. CH , A , ~H ) -> ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH B ) ) = B ) ) )
8 sseq1 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( B C_ if ( A e. CH , A , ~H ) <-> if ( B e. CH , B , ~H ) C_ if ( A e. CH , A , ~H ) ) )
9 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 ) ) )
10 9 ineq2d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH B ) ) = ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH if ( B e. CH , B , ~H ) ) ) )
11 id 
 |-  ( B = if ( B e. CH , B , ~H ) -> B = if ( B e. CH , B , ~H ) )
12 10 11 eqeq12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH B ) ) = B <-> ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH if ( B e. CH , B , ~H ) ) ) = if ( B e. CH , B , ~H ) ) )
13 8 12 imbi12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( B C_ if ( A e. CH , A , ~H ) -> ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH B ) ) = B ) <-> ( if ( B e. CH , B , ~H ) C_ if ( A e. CH , A , ~H ) -> ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH if ( B e. CH , B , ~H ) ) ) = if ( B e. CH , B , ~H ) ) ) )
14 ifchhv 
 |-  if ( A e. CH , A , ~H ) e. CH
15 ifchhv 
 |-  if ( B e. CH , B , ~H ) e. CH
16 14 15 pjoml3i 
 |-  ( if ( B e. CH , B , ~H ) C_ if ( A e. CH , A , ~H ) -> ( if ( A e. CH , A , ~H ) i^i ( ( _|_ ` if ( A e. CH , A , ~H ) ) vH if ( B e. CH , B , ~H ) ) ) = if ( B e. CH , B , ~H ) )
17 7 13 16 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( B C_ A -> ( A i^i ( ( _|_ ` A ) vH B ) ) = B ) )