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Metamath Proof Explorer

Theorem pjoml2i

Description: Variation of orthomodular law. Definition in Kalmbach p. 22. (Contributed by NM, 31-Oct-2000) (New usage is discouraged.)

Ref Expression
Hypotheses pjoml2.1 
|- A e. CH
pjoml2.2 
|- B e. CH
Assertion pjoml2i 
|- ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )

Proof

Step Hyp Ref Expression
1 pjoml2.1 
 |-  A e. CH
2 pjoml2.2 
 |-  B e. CH
3 inss2 
 |-  ( ( _|_ ` A ) i^i B ) C_ B
4 1 choccli 
 |-  ( _|_ ` A ) e. CH
5 4 2 chincli 
 |-  ( ( _|_ ` A ) i^i B ) e. CH
6 1 5 2 chlubii 
 |-  ( ( A C_ B /\ ( ( _|_ ` A ) i^i B ) C_ B ) -> ( A vH ( ( _|_ ` A ) i^i B ) ) C_ B )
7 3 6 mpan2 
 |-  ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) C_ B )
8 1 5 chdmj1i 
 |-  ( _|_ ` ( A vH ( ( _|_ ` A ) i^i B ) ) ) = ( ( _|_ ` A ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) )
9 8 ineq2i 
 |-  ( B i^i ( _|_ ` ( A vH ( ( _|_ ` A ) i^i B ) ) ) ) = ( B i^i ( ( _|_ ` A ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) )
10 incom 
 |-  ( B i^i ( _|_ ` A ) ) = ( ( _|_ ` A ) i^i B )
11 10 ineq1i 
 |-  ( ( B i^i ( _|_ ` A ) ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) = ( ( ( _|_ ` A ) i^i B ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) )
12 inass 
 |-  ( ( B i^i ( _|_ ` A ) ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) = ( B i^i ( ( _|_ ` A ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) )
13 5 chocini 
 |-  ( ( ( _|_ ` A ) i^i B ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) = 0H
14 11 12 13 3eqtr3i 
 |-  ( B i^i ( ( _|_ ` A ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) ) = 0H
15 9 14 eqtri 
 |-  ( B i^i ( _|_ ` ( A vH ( ( _|_ ` A ) i^i B ) ) ) ) = 0H
16 1 5 chjcli 
 |-  ( A vH ( ( _|_ ` A ) i^i B ) ) e. CH
17 2 chshii 
 |-  B e. SH
18 16 17 pjomli 
 |-  ( ( ( A vH ( ( _|_ ` A ) i^i B ) ) C_ B /\ ( B i^i ( _|_ ` ( A vH ( ( _|_ ` A ) i^i B ) ) ) ) = 0H ) -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )
19 7 15 18 sylancl 
 |-  ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )