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Metamath Proof Explorer

Theorem pjoml6i

Description: An equivalent of the orthomodular law. Theorem 29.13(e) of MaedaMaeda p. 132. (Contributed by NM, 30-May-2004) (New usage is discouraged.)

Ref Expression
Hypotheses pjoml2.1 
|- A e. CH
pjoml2.2 
|- B e. CH
Assertion pjoml6i 
|- ( A C_ B -> E. x e. CH ( A C_ ( _|_ ` x ) /\ ( A vH x ) = B ) )

Proof

Step Hyp Ref Expression
1 pjoml2.1 
 |-  A e. CH
2 pjoml2.2 
 |-  B e. CH
3 1 choccli 
 |-  ( _|_ ` A ) e. CH
4 3 2 chincli 
 |-  ( ( _|_ ` A ) i^i B ) e. CH
5 1 2 pjoml2i 
 |-  ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )
6 2 choccli 
 |-  ( _|_ ` B ) e. CH
7 1 6 chub1i 
 |-  A C_ ( A vH ( _|_ ` B ) )
8 1 2 chdmm2i 
 |-  ( _|_ ` ( ( _|_ ` A ) i^i B ) ) = ( A vH ( _|_ ` B ) )
9 7 8 sseqtrri 
 |-  A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) )
10 5 9 jctil 
 |-  ( A C_ B -> ( A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) ) /\ ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) )
11 fveq2 
 |-  ( x = ( ( _|_ ` A ) i^i B ) -> ( _|_ ` x ) = ( _|_ ` ( ( _|_ ` A ) i^i B ) ) )
12 11 sseq2d 
 |-  ( x = ( ( _|_ ` A ) i^i B ) -> ( A C_ ( _|_ ` x ) <-> A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) )
13 oveq2 
 |-  ( x = ( ( _|_ ` A ) i^i B ) -> ( A vH x ) = ( A vH ( ( _|_ ` A ) i^i B ) ) )
14 13 eqeq1d 
 |-  ( x = ( ( _|_ ` A ) i^i B ) -> ( ( A vH x ) = B <-> ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) )
15 12 14 anbi12d 
 |-  ( x = ( ( _|_ ` A ) i^i B ) -> ( ( A C_ ( _|_ ` x ) /\ ( A vH x ) = B ) <-> ( A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) ) /\ ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) ) )
16 15 rspcev 
 |-  ( ( ( ( _|_ ` A ) i^i B ) e. CH /\ ( A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) ) /\ ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) ) -> E. x e. CH ( A C_ ( _|_ ` x ) /\ ( A vH x ) = B ) )
17 4 10 16 sylancr 
 |-  ( A C_ B -> E. x e. CH ( A C_ ( _|_ ` x ) /\ ( A vH x ) = B ) )