Metamath Proof Explorer

Theorem pjssmi

Description: Projection meet property. Remark in Kalmbach p. 66. Also Theorem 4.5(i)->(iv) of Beran p. 112. (Contributed by NM, 26-Sep-2001) (New usage is discouraged.)

Ref Expression
Hypotheses pjco.1 
|- G e. CH
pjco.2 
|- H e. CH
Assertion pjssmi 
|- ( A e. ~H -> ( H C_ G -> ( ( ( projh ` G ) ` A ) -h ( ( projh ` H ) ` A ) ) = ( ( projh ` ( G i^i ( _|_ ` H ) ) ) ` A ) ) )

Proof

Step Hyp Ref Expression
1 pjco.1 
 |-  G e. CH
2 pjco.2 
 |-  H e. CH
3 fveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( projh ` G ) ` A ) = ( ( projh ` G ) ` if ( A e. ~H , A , 0h ) ) )
4 fveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( projh ` H ) ` A ) = ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) )
5 3 4 oveq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( projh ` G ) ` A ) -h ( ( projh ` H ) ` A ) ) = ( ( ( projh ` G ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) ) )
6 fveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( projh ` ( G i^i ( _|_ ` H ) ) ) ` A ) = ( ( projh ` ( G i^i ( _|_ ` H ) ) ) ` if ( A e. ~H , A , 0h ) ) )
7 5 6 eqeq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( ( projh ` G ) ` A ) -h ( ( projh ` H ) ` A ) ) = ( ( projh ` ( G i^i ( _|_ ` H ) ) ) ` A ) <-> ( ( ( projh ` G ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) ) = ( ( projh ` ( G i^i ( _|_ ` H ) ) ) ` if ( A e. ~H , A , 0h ) ) ) )
8 7 imbi2d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( H C_ G -> ( ( ( projh ` G ) ` A ) -h ( ( projh ` H ) ` A ) ) = ( ( projh ` ( G i^i ( _|_ ` H ) ) ) ` A ) ) <-> ( H C_ G -> ( ( ( projh ` G ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) ) = ( ( projh ` ( G i^i ( _|_ ` H ) ) ) ` if ( A e. ~H , A , 0h ) ) ) ) )
9 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
10 2 9 1 pjssmii 
 |-  ( H C_ G -> ( ( ( projh ` G ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) ) = ( ( projh ` ( G i^i ( _|_ ` H ) ) ) ` if ( A e. ~H , A , 0h ) ) )
11 8 10 dedth 
 |-  ( A e. ~H -> ( H C_ G -> ( ( ( projh ` G ) ` A ) -h ( ( projh ` H ) ` A ) ) = ( ( projh ` ( G i^i ( _|_ ` H ) ) ) ` A ) ) )