Theorem honpncani

Description: Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	honpncan.1	$$⊢R:\mathscr{H}⟶\mathscr{H}$$
	honpncan.2	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
	honpncan.3	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
Assertion	honpncani	$$⊢\left(R{-}_{\mathrm{op}}S\right){+}_{\mathrm{op}}\left(S{-}_{\mathrm{op}}T\right)=R{-}_{\mathrm{op}}T$$

Proof

Step	Hyp	Ref	Expression
1		honpncan.1	$$⊢R:\mathscr{H}⟶\mathscr{H}$$
2		honpncan.2	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
3		honpncan.3	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
4	1 2	hosubcli	$$⊢\left(R{-}_{\mathrm{op}}S\right):\mathscr{H}⟶\mathscr{H}$$
5	4 2 3	hoaddsubassi	$$⊢\left(\left(R{-}_{\mathrm{op}}S\right){+}_{\mathrm{op}}S\right){-}_{\mathrm{op}}T=\left(R{-}_{\mathrm{op}}S\right){+}_{\mathrm{op}}\left(S{-}_{\mathrm{op}}T\right)$$
6	1 2	honpcani	$$⊢\left(R{-}_{\mathrm{op}}S\right){+}_{\mathrm{op}}S=R$$
7	6	oveq1i	$$⊢\left(\left(R{-}_{\mathrm{op}}S\right){+}_{\mathrm{op}}S\right){-}_{\mathrm{op}}T=R{-}_{\mathrm{op}}T$$
8	5 7	eqtr3i	$$⊢\left(R{-}_{\mathrm{op}}S\right){+}_{\mathrm{op}}\left(S{-}_{\mathrm{op}}T\right)=R{-}_{\mathrm{op}}T$$