Metamath Proof Explorer

Theorem pjop

Description: Orthocomplement projection in terms of projection. (Contributed by NM, 5-Nov-1999) (New usage is discouraged.)

Ref Expression
Assertion pjop 
|- ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` ( _|_ ` H ) ) ` A ) = ( A -h ( ( projh ` H ) ` A ) ) )

Proof

Step Hyp Ref Expression
1 axpjpj 
 |-  ( ( H e. CH /\ A e. ~H ) -> A = ( ( ( projh ` H ) ` A ) +h ( ( projh ` ( _|_ ` H ) ) ` A ) ) )
2 1 eqcomd 
 |-  ( ( H e. CH /\ A e. ~H ) -> ( ( ( projh ` H ) ` A ) +h ( ( projh ` ( _|_ ` H ) ) ` A ) ) = A )
3 simpr 
 |-  ( ( H e. CH /\ A e. ~H ) -> A e. ~H )
4 pjhcl 
 |-  ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` H ) ` A ) e. ~H )
5 choccl 
 |-  ( H e. CH -> ( _|_ ` H ) e. CH )
6 pjhcl 
 |-  ( ( ( _|_ ` H ) e. CH /\ A e. ~H ) -> ( ( projh ` ( _|_ ` H ) ) ` A ) e. ~H )
7 5 6 sylan 
 |-  ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` ( _|_ ` H ) ) ` A ) e. ~H )
8 hvsubadd 
 |-  ( ( A e. ~H /\ ( ( projh ` H ) ` A ) e. ~H /\ ( ( projh ` ( _|_ ` H ) ) ` A ) e. ~H ) -> ( ( A -h ( ( projh ` H ) ` A ) ) = ( ( projh ` ( _|_ ` H ) ) ` A ) <-> ( ( ( projh ` H ) ` A ) +h ( ( projh ` ( _|_ ` H ) ) ` A ) ) = A ) )
9 3 4 7 8 syl3anc 
 |-  ( ( H e. CH /\ A e. ~H ) -> ( ( A -h ( ( projh ` H ) ` A ) ) = ( ( projh ` ( _|_ ` H ) ) ` A ) <-> ( ( ( projh ` H ) ` A ) +h ( ( projh ` ( _|_ ` H ) ) ` A ) ) = A ) )
10 2 9 mpbird 
 |-  ( ( H e. CH /\ A e. ~H ) -> ( A -h ( ( projh ` H ) ` A ) ) = ( ( projh ` ( _|_ ` H ) ) ` A ) )
11 10 eqcomd 
 |-  ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` ( _|_ ` H ) ) ` A ) = ( A -h ( ( projh ` H ) ` A ) ) )