Theorem ho0subi

Description: Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hodseq.2	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
	hodseq.3	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
Assertion	ho0subi	$$⊢S{-}_{\mathrm{op}}T=S{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)$$

Proof

Step	Hyp	Ref	Expression
1		hodseq.2	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
2		hodseq.3	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
3		ho0f	$$⊢0_{\mathrm{hop}}:\mathscr{H}⟶\mathscr{H}$$
4	3 2	hosubcli	$$⊢\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right):\mathscr{H}⟶\mathscr{H}$$
5	2 1 4	hoadd12i	$$⊢T{+}_{\mathrm{op}}\left(S{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)\right)=S{+}_{\mathrm{op}}\left(T{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)\right)$$
6	2 3	hodseqi	$$⊢T{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)=0_{\mathrm{hop}}$$
7	6	oveq2i	$$⊢S{+}_{\mathrm{op}}\left(T{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)\right)=S{+}_{\mathrm{op}}{0}_{\mathrm{hop}}$$
8	1	hoaddid1i	$$⊢S{+}_{\mathrm{op}}{0}_{\mathrm{hop}}=S$$
9	5 7 8	3eqtri	$$⊢T{+}_{\mathrm{op}}\left(S{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)\right)=S$$
10	1 4	hoaddcli	$$⊢\left(S{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)\right):\mathscr{H}⟶\mathscr{H}$$
11	1 2 10	hodsi	$$⊢S{-}_{\mathrm{op}}T=S{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)\leftrightarrow T{+}_{\mathrm{op}}\left(S{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)\right)=S$$
12	9 11	mpbir	$$⊢S{-}_{\mathrm{op}}T=S{+}_{\mathrm{op}}\left(0_{\mathrm{hop}}{-}_{\mathrm{op}}T\right)$$