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Metamath Proof Explorer

Theorem chlejb1

Description: Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chlejb1 
|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( A vH 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 oveq1 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( A vH B ) = ( if ( A e. CH , A , 0H ) vH B ) )
3 2 eqeq1d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( A vH B ) = B <-> ( if ( A e. CH , A , 0H ) vH B ) = B ) )
4 1 3 bibi12d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( A C_ B <-> ( A vH B ) = B ) <-> ( if ( A e. CH , A , 0H ) C_ B <-> ( if ( A e. CH , A , 0H ) vH B ) = B ) ) )
5 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 ) ) )
6 oveq2 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( if ( A e. CH , A , 0H ) vH B ) = ( if ( A e. CH , A , 0H ) vH if ( B e. CH , B , 0H ) ) )
7 id 
 |-  ( B = if ( B e. CH , B , 0H ) -> B = if ( B e. CH , B , 0H ) )
8 6 7 eqeq12d 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( ( if ( A e. CH , A , 0H ) vH B ) = B <-> ( if ( A e. CH , A , 0H ) vH if ( B e. CH , B , 0H ) ) = if ( B e. CH , B , 0H ) ) )
9 5 8 bibi12d 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( ( if ( A e. CH , A , 0H ) C_ B <-> ( if ( A e. CH , A , 0H ) vH B ) = B ) <-> ( if ( A e. CH , A , 0H ) C_ if ( B e. CH , B , 0H ) <-> ( if ( A e. CH , A , 0H ) vH if ( B e. CH , B , 0H ) ) = if ( B e. CH , B , 0H ) ) ) )
10 h0elch 
 |-  0H e. CH
11 10 elimel 
 |-  if ( A e. CH , A , 0H ) e. CH
12 10 elimel 
 |-  if ( B e. CH , B , 0H ) e. CH
13 11 12 chlejb1i 
 |-  ( if ( A e. CH , A , 0H ) C_ if ( B e. CH , B , 0H ) <-> ( if ( A e. CH , A , 0H ) vH if ( B e. CH , B , 0H ) ) = if ( B e. CH , B , 0H ) )
14 4 9 13 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( A vH B ) = B ) )