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Metamath Proof Explorer

Theorem pjoml2

Description: Variation of orthomodular law. Definition in Kalmbach p. 22. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion pjoml2 
|- ( ( A e. CH /\ B e. CH /\ A C_ B ) -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )

Proof

Step Hyp Ref Expression
1 sseq1 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( A C_ B <-> if ( A e. CH , A , 0H ) C_ B ) )
2 id 
 |-  ( A = if ( A e. CH , A , 0H ) -> A = if ( A e. CH , A , 0H ) )
3 fveq2 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( _|_ ` A ) = ( _|_ ` if ( A e. CH , A , 0H ) ) )
4 3 ineq1d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( _|_ ` A ) i^i B ) = ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i B ) )
5 2 4 oveq12d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( A vH ( ( _|_ ` A ) i^i B ) ) = ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i B ) ) )
6 5 eqeq1d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( A vH ( ( _|_ ` A ) i^i B ) ) = B <-> ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i B ) ) = B ) )
7 1 6 imbi12d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) <-> ( if ( A e. CH , A , 0H ) C_ B -> ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i B ) ) = B ) ) )
8 sseq2 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( if ( A e. CH , A , 0H ) C_ B <-> if ( A e. CH , A , 0H ) C_ if ( B e. CH , B , 0H ) ) )
9 ineq2 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i B ) = ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i if ( B e. CH , B , 0H ) ) )
10 9 oveq2d 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i B ) ) = ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i if ( B e. CH , B , 0H ) ) ) )
11 id 
 |-  ( B = if ( B e. CH , B , 0H ) -> B = if ( B e. CH , B , 0H ) )
12 10 11 eqeq12d 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i B ) ) = B <-> ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i if ( B e. CH , B , 0H ) ) ) = if ( B e. CH , B , 0H ) ) )
13 8 12 imbi12d 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( ( if ( A e. CH , A , 0H ) C_ B -> ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i B ) ) = B ) <-> ( if ( A e. CH , A , 0H ) C_ if ( B e. CH , B , 0H ) -> ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i if ( B e. CH , B , 0H ) ) ) = if ( B e. CH , B , 0H ) ) ) )
14 h0elch 
 |-  0H e. CH
15 14 elimel 
 |-  if ( A e. CH , A , 0H ) e. CH
16 14 elimel 
 |-  if ( B e. CH , B , 0H ) e. CH
17 15 16 pjoml2i 
 |-  ( if ( A e. CH , A , 0H ) C_ if ( B e. CH , B , 0H ) -> ( if ( A e. CH , A , 0H ) vH ( ( _|_ ` if ( A e. CH , A , 0H ) ) i^i if ( B e. CH , B , 0H ) ) ) = if ( B e. CH , B , 0H ) )
18 7 13 17 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) )
19 18 3impia 
 |-  ( ( A e. CH /\ B e. CH /\ A C_ B ) -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )