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Metamath Proof Explorer

Theorem pjsubii

Description: Projection of vector difference is difference of projections. (Contributed by NM, 31-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses pjidm.1 
|- H e. CH
pjidm.2 
|- A e. ~H
pjsub.3 
|- B e. ~H
Assertion pjsubii 
|- ( ( projh ` H ) ` ( A -h B ) ) = ( ( ( projh ` H ) ` A ) -h ( ( projh ` H ) ` B ) )

Proof

Step Hyp Ref Expression
1 pjidm.1 
 |-  H e. CH
2 pjidm.2 
 |-  A e. ~H
3 pjsub.3 
 |-  B e. ~H
4 neg1cn 
 |-  -u 1 e. CC
5 4 3 hvmulcli 
 |-  ( -u 1 .h B ) e. ~H
6 1 2 5 pjaddii 
 |-  ( ( projh ` H ) ` ( A +h ( -u 1 .h B ) ) ) = ( ( ( projh ` H ) ` A ) +h ( ( projh ` H ) ` ( -u 1 .h B ) ) )
7 1 3 4 pjmulii 
 |-  ( ( projh ` H ) ` ( -u 1 .h B ) ) = ( -u 1 .h ( ( projh ` H ) ` B ) )
8 7 oveq2i 
 |-  ( ( ( projh ` H ) ` A ) +h ( ( projh ` H ) ` ( -u 1 .h B ) ) ) = ( ( ( projh ` H ) ` A ) +h ( -u 1 .h ( ( projh ` H ) ` B ) ) )
9 6 8 eqtri 
 |-  ( ( projh ` H ) ` ( A +h ( -u 1 .h B ) ) ) = ( ( ( projh ` H ) ` A ) +h ( -u 1 .h ( ( projh ` H ) ` B ) ) )
10 2 3 hvsubvali 
 |-  ( A -h B ) = ( A +h ( -u 1 .h B ) )
11 10 fveq2i 
 |-  ( ( projh ` H ) ` ( A -h B ) ) = ( ( projh ` H ) ` ( A +h ( -u 1 .h B ) ) )
12 1 2 pjhclii 
 |-  ( ( projh ` H ) ` A ) e. ~H
13 1 3 pjhclii 
 |-  ( ( projh ` H ) ` B ) e. ~H
14 12 13 hvsubvali 
 |-  ( ( ( projh ` H ) ` A ) -h ( ( projh ` H ) ` B ) ) = ( ( ( projh ` H ) ` A ) +h ( -u 1 .h ( ( projh ` H ) ` B ) ) )
15 9 11 14 3eqtr4i 
 |-  ( ( projh ` H ) ` ( A -h B ) ) = ( ( ( projh ` H ) ` A ) -h ( ( projh ` H ) ` B ) )