Theorem hmopre

Description: The inner product of the value and argument of a Hermitian operator is real. (Contributed by NM, 23-Jul-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	hmopre	|- ( ( T e. HrmOp /\ A e. ~H ) -> ( ( T ` A ) .ih A ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		hmop	|- ( ( T e. HrmOp /\ A e. ~H /\ A e. ~H ) -> ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) )
2	1	3anidm23	|- ( ( T e. HrmOp /\ A e. ~H ) -> ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) )
3	2	eqcomd	|- ( ( T e. HrmOp /\ A e. ~H ) -> ( ( T ` A ) .ih A ) = ( A .ih ( T ` A ) ) )
4		hmopf	|- ( T e. HrmOp -> T : ~H --> ~H )
5	4	ffvelcdmda	|- ( ( T e. HrmOp /\ A e. ~H ) -> ( T ` A ) e. ~H )
6		hire	|- ( ( ( T ` A ) e. ~H /\ A e. ~H ) -> ( ( ( T ` A ) .ih A ) e. RR <-> ( ( T ` A ) .ih A ) = ( A .ih ( T ` A ) ) ) )
7	5 6	sylancom	|- ( ( T e. HrmOp /\ A e. ~H ) -> ( ( ( T ` A ) .ih A ) e. RR <-> ( ( T ` A ) .ih A ) = ( A .ih ( T ` A ) ) ) )
8	3 7	mpbird	|- ( ( T e. HrmOp /\ A e. ~H ) -> ( ( T ` A ) .ih A ) e. RR )