Theorem hoadd12i

Description: Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hods.1	$$⊢R:\mathscr{H}⟶\mathscr{H}$$
	hods.2	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
	hods.3	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
Assertion	hoadd12i	$$⊢R{+}_{\mathrm{op}}\left(S{+}_{\mathrm{op}}T\right)=S{+}_{\mathrm{op}}\left(R{+}_{\mathrm{op}}T\right)$$

Proof

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