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Metamath Proof Explorer

Theorem pjoml

Description: Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of Kalmbach p. 22. Derived using projections; compare omlsi . (Contributed by NM, 14-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion pjoml 
|- ( ( ( A e. CH /\ B e. SH ) /\ ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) ) -> A = 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 fveq2 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( _|_ ` A ) = ( _|_ ` if ( A e. CH , A , 0H ) ) )
3 2 ineq2d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( B i^i ( _|_ ` A ) ) = ( B i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) )
4 3 eqeq1d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( B i^i ( _|_ ` A ) ) = 0H <-> ( B i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H ) )
5 1 4 anbi12d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) <-> ( if ( A e. CH , A , 0H ) C_ B /\ ( B i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H ) ) )
6 eqeq1 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( A = B <-> if ( A e. CH , A , 0H ) = B ) )
7 5 6 imbi12d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) -> A = B ) <-> ( ( if ( A e. CH , A , 0H ) C_ B /\ ( B i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H ) -> if ( A e. CH , A , 0H ) = B ) ) )
8 sseq2 
 |-  ( B = if ( B e. SH , B , 0H ) -> ( if ( A e. CH , A , 0H ) C_ B <-> if ( A e. CH , A , 0H ) C_ if ( B e. SH , B , 0H ) ) )
9 ineq1 
 |-  ( B = if ( B e. SH , B , 0H ) -> ( B i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = ( if ( B e. SH , B , 0H ) i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) )
10 9 eqeq1d 
 |-  ( B = if ( B e. SH , B , 0H ) -> ( ( B i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H <-> ( if ( B e. SH , B , 0H ) i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H ) )
11 8 10 anbi12d 
 |-  ( B = if ( B e. SH , B , 0H ) -> ( ( if ( A e. CH , A , 0H ) C_ B /\ ( B i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H ) <-> ( if ( A e. CH , A , 0H ) C_ if ( B e. SH , B , 0H ) /\ ( if ( B e. SH , B , 0H ) i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H ) ) )
12 eqeq2 
 |-  ( B = if ( B e. SH , B , 0H ) -> ( if ( A e. CH , A , 0H ) = B <-> if ( A e. CH , A , 0H ) = if ( B e. SH , B , 0H ) ) )
13 11 12 imbi12d 
 |-  ( B = if ( B e. SH , B , 0H ) -> ( ( ( if ( A e. CH , A , 0H ) C_ B /\ ( B i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H ) -> if ( A e. CH , A , 0H ) = B ) <-> ( ( if ( A e. CH , A , 0H ) C_ if ( B e. SH , B , 0H ) /\ ( if ( B e. SH , B , 0H ) i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H ) -> if ( A e. CH , A , 0H ) = if ( B e. SH , B , 0H ) ) ) )
14 h0elch 
 |-  0H e. CH
15 14 elimel 
 |-  if ( A e. CH , A , 0H ) e. CH
16 h0elsh 
 |-  0H e. SH
17 16 elimel 
 |-  if ( B e. SH , B , 0H ) e. SH
18 15 17 pjomli 
 |-  ( ( if ( A e. CH , A , 0H ) C_ if ( B e. SH , B , 0H ) /\ ( if ( B e. SH , B , 0H ) i^i ( _|_ ` if ( A e. CH , A , 0H ) ) ) = 0H ) -> if ( A e. CH , A , 0H ) = if ( B e. SH , B , 0H ) )
19 7 13 18 dedth2h 
 |-  ( ( A e. CH /\ B e. SH ) -> ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) -> A = B ) )
20 19 imp 
 |-  ( ( ( A e. CH /\ B e. SH ) /\ ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) ) -> A = B )