Metamath Proof Explorer

Theorem lnopsubmuli

Description: Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005) (New usage is discouraged.)

Ref Expression
Hypothesis lnopl.1 
|- T e. LinOp
Assertion lnopsubmuli 
|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( B -h ( A .h C ) ) ) = ( ( T ` B ) -h ( A .h ( T ` C ) ) ) )

Proof

Step Hyp Ref Expression
1 lnopl.1 
 |-  T e. LinOp
2 hvmulcl 
 |-  ( ( A e. CC /\ C e. ~H ) -> ( A .h C ) e. ~H )
3 1 lnopsubi 
 |-  ( ( B e. ~H /\ ( A .h C ) e. ~H ) -> ( T ` ( B -h ( A .h C ) ) ) = ( ( T ` B ) -h ( T ` ( A .h C ) ) ) )
4 2 3 sylan2 
 |-  ( ( B e. ~H /\ ( A e. CC /\ C e. ~H ) ) -> ( T ` ( B -h ( A .h C ) ) ) = ( ( T ` B ) -h ( T ` ( A .h C ) ) ) )
5 4 3impb 
 |-  ( ( B e. ~H /\ A e. CC /\ C e. ~H ) -> ( T ` ( B -h ( A .h C ) ) ) = ( ( T ` B ) -h ( T ` ( A .h C ) ) ) )
6 5 3com12 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( B -h ( A .h C ) ) ) = ( ( T ` B ) -h ( T ` ( A .h C ) ) ) )
7 1 lnopmuli 
 |-  ( ( A e. CC /\ C e. ~H ) -> ( T ` ( A .h C ) ) = ( A .h ( T ` C ) ) )
8 7 oveq2d 
 |-  ( ( A e. CC /\ C e. ~H ) -> ( ( T ` B ) -h ( T ` ( A .h C ) ) ) = ( ( T ` B ) -h ( A .h ( T ` C ) ) ) )
9 8 3adant2 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( T ` B ) -h ( T ` ( A .h C ) ) ) = ( ( T ` B ) -h ( A .h ( T ` C ) ) ) )
10 6 9 eqtrd 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( B -h ( A .h C ) ) ) = ( ( T ` B ) -h ( A .h ( T ` C ) ) ) )