Theorem hoadd32i

Description: Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006) (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	hoadd32i	$$⊢\left(R{+}_{\mathrm{op}}S\right){+}_{\mathrm{op}}T=\left(R{+}_{\mathrm{op}}T\right){+}_{\mathrm{op}}S$$

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	2 3	hoaddcomi	$$⊢S{+}_{\mathrm{op}}T=T{+}_{\mathrm{op}}S$$
5	4	oveq2i	$$⊢R{+}_{\mathrm{op}}\left(S{+}_{\mathrm{op}}T\right)=R{+}_{\mathrm{op}}\left(T{+}_{\mathrm{op}}S\right)$$
6	1 2 3	hoaddassi	$$⊢\left(R{+}_{\mathrm{op}}S\right){+}_{\mathrm{op}}T=R{+}_{\mathrm{op}}\left(S{+}_{\mathrm{op}}T\right)$$
7	1 3 2	hoaddassi	$$⊢\left(R{+}_{\mathrm{op}}T\right){+}_{\mathrm{op}}S=R{+}_{\mathrm{op}}\left(T{+}_{\mathrm{op}}S\right)$$
8	5 6 7	3eqtr4i	$$⊢\left(R{+}_{\mathrm{op}}S\right){+}_{\mathrm{op}}T=\left(R{+}_{\mathrm{op}}T\right){+}_{\mathrm{op}}S$$