Theorem lnopmulsubi

Description: Product/subtraction 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	lnopmulsubi	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( ( A .h B ) -h C ) ) = ( ( A .h ( T ` B ) ) -h ( T ` C ) ) )

Proof

Step	Hyp	Ref	Expression
1		lnopl.1	|- T e. LinOp
2		hvmulcl	|- ( ( A e. CC /\ B e. ~H ) -> ( A .h B ) e. ~H )
3	1	lnopsubi	|- ( ( ( A .h B ) e. ~H /\ C e. ~H ) -> ( T ` ( ( A .h B ) -h C ) ) = ( ( T ` ( A .h B ) ) -h ( T ` C ) ) )
4	2 3	stoic3	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( ( A .h B ) -h C ) ) = ( ( T ` ( A .h B ) ) -h ( T ` C ) ) )
5	1	lnopmuli	|- ( ( A e. CC /\ B e. ~H ) -> ( T ` ( A .h B ) ) = ( A .h ( T ` B ) ) )
6	5	3adant3	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( A .h B ) ) = ( A .h ( T ` B ) ) )
7	6	oveq1d	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( T ` ( A .h B ) ) -h ( T ` C ) ) = ( ( A .h ( T ` B ) ) -h ( T ` C ) ) )
8	4 7	eqtrd	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( ( A .h B ) -h C ) ) = ( ( A .h ( T ` B ) ) -h ( T ` C ) ) )