Theorem hocoi

Description: Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hoeq.1	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
	hoeq.2	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
Assertion	hocoi	$$⊢A\in \mathscr{H}\to \left(S\circ T\right)\left(A\right)=S\left(T\left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		hoeq.1	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
2		hoeq.2	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
3		fvco3	$$⊢\left(T:\mathscr{H}⟶\mathscr{H}\wedge A\in \mathscr{H}\right)\to \left(S\circ T\right)\left(A\right)=S\left(T\left(A\right)\right)$$
4	2 3	mpan	$$⊢A\in \mathscr{H}\to \left(S\circ T\right)\left(A\right)=S\left(T\left(A\right)\right)$$