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Metamath Proof Explorer

Theorem pjorthi

Description: Projection components on orthocomplemented subspaces are orthogonal. (Contributed by NM, 26-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses pjorth.1 
|- A e. ~H
pjorth.2 
|- B e. ~H
Assertion pjorthi 
|- ( H e. CH -> ( ( ( projh ` H ) ` A ) .ih ( ( projh ` ( _|_ ` H ) ) ` B ) ) = 0 )

Proof

Step Hyp Ref Expression
1 pjorth.1 
 |-  A e. ~H
2 pjorth.2 
 |-  B e. ~H
3 chsh 
 |-  ( H e. CH -> H e. SH )
4 axpjcl 
 |-  ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` H ) ` A ) e. H )
5 1 4 mpan2 
 |-  ( H e. CH -> ( ( projh ` H ) ` A ) e. H )
6 choccl 
 |-  ( H e. CH -> ( _|_ ` H ) e. CH )
7 axpjcl 
 |-  ( ( ( _|_ ` H ) e. CH /\ B e. ~H ) -> ( ( projh ` ( _|_ ` H ) ) ` B ) e. ( _|_ ` H ) )
8 6 2 7 sylancl 
 |-  ( H e. CH -> ( ( projh ` ( _|_ ` H ) ) ` B ) e. ( _|_ ` H ) )
9 5 8 jca 
 |-  ( H e. CH -> ( ( ( projh ` H ) ` A ) e. H /\ ( ( projh ` ( _|_ ` H ) ) ` B ) e. ( _|_ ` H ) ) )
10 shocorth 
 |-  ( H e. SH -> ( ( ( ( projh ` H ) ` A ) e. H /\ ( ( projh ` ( _|_ ` H ) ) ` B ) e. ( _|_ ` H ) ) -> ( ( ( projh ` H ) ` A ) .ih ( ( projh ` ( _|_ ` H ) ) ` B ) ) = 0 ) )
11 3 9 10 sylc 
 |-  ( H e. CH -> ( ( ( projh ` H ) ` A ) .ih ( ( projh ` ( _|_ ` H ) ) ` B ) ) = 0 )