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Metamath Proof Explorer

Theorem omlsi

Description: Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of Kalmbach p. 22. (Contributed by NM, 14-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses omls.1 
|- A e. CH
omls.2 
|- B e. SH
Assertion omlsi 
|- ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) -> A = B )

Proof

Step Hyp Ref Expression
1 omls.1 
 |-  A e. CH
2 omls.2 
 |-  B e. SH
3 eqeq1 
 |-  ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( A = B <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) = B ) )
4 eqeq2 
 |-  ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) = B <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) ) )
5 h0elch 
 |-  0H e. CH
6 1 5 ifcli 
 |-  if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) e. CH
7 h0elsh 
 |-  0H e. SH
8 2 7 ifcli 
 |-  if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) e. SH
9 sseq1 
 |-  ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( A C_ B <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ B ) )
10 fveq2 
 |-  ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( _|_ ` A ) = ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) )
11 10 ineq2d 
 |-  ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( B i^i ( _|_ ` A ) ) = ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) )
12 11 eqeq1d 
 |-  ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( ( B i^i ( _|_ ` A ) ) = 0H <-> ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) )
13 9 12 anbi12d 
 |-  ( A = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ B /\ ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) ) )
14 sseq2 
 |-  ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ B <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) ) )
15 ineq1 
 |-  ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) )
16 15 eqeq1d 
 |-  ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) )
17 14 16 anbi12d 
 |-  ( B = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ B /\ ( B i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) /\ ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) ) )
18 sseq1 
 |-  ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( 0H C_ 0H <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ 0H ) )
19 fveq2 
 |-  ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( _|_ ` 0H ) = ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) )
20 19 ineq2d 
 |-  ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( 0H i^i ( _|_ ` 0H ) ) = ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) )
21 20 eqeq1d 
 |-  ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( ( 0H i^i ( _|_ ` 0H ) ) = 0H <-> ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) )
22 18 21 anbi12d 
 |-  ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) -> ( ( 0H C_ 0H /\ ( 0H i^i ( _|_ ` 0H ) ) = 0H ) <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ 0H /\ ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) ) )
23 sseq2 
 |-  ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ 0H <-> if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) ) )
24 ineq1 
 |-  ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) )
25 24 eqeq1d 
 |-  ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) )
26 23 25 anbi12d 
 |-  ( 0H = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) -> ( ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ 0H /\ ( 0H i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) <-> ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) /\ ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H ) ) )
27 ssid 
 |-  0H C_ 0H
28 ocin 
 |-  ( 0H e. SH -> ( 0H i^i ( _|_ ` 0H ) ) = 0H )
29 7 28 ax-mp 
 |-  ( 0H i^i ( _|_ ` 0H ) ) = 0H
30 27 29 pm3.2i 
 |-  ( 0H C_ 0H /\ ( 0H i^i ( _|_ ` 0H ) ) = 0H )
31 13 17 22 26 30 elimhyp2v 
 |-  ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) /\ ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H )
32 31 simpli 
 |-  if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) C_ if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H )
33 31 simpri 
 |-  ( if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H ) i^i ( _|_ ` if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) ) ) = 0H
34 6 8 32 33 omlsii 
 |-  if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , A , 0H ) = if ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) , B , 0H )
35 3 4 34 dedth2v 
 |-  ( ( A C_ B /\ ( B i^i ( _|_ ` A ) ) = 0H ) -> A = B )