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 : ~H --> ~H
	hods.2	|- S : ~H --> ~H
	hods.3	|- T : ~H --> ~H
Assertion	hoadd32i	|- ( ( R +op S ) +op T ) = ( ( R +op T ) +op S )

Proof

Step	Hyp	Ref	Expression
1		hods.1	|- R : ~H --> ~H
2		hods.2	|- S : ~H --> ~H
3		hods.3	|- T : ~H --> ~H
4	2 3	hoaddcomi	|- ( S +op T ) = ( T +op S )
5	4	oveq2i	|- ( R +op ( S +op T ) ) = ( R +op ( T +op S ) )
6	1 2 3	hoaddassi	|- ( ( R +op S ) +op T ) = ( R +op ( S +op T ) )
7	1 3 2	hoaddassi	|- ( ( R +op T ) +op S ) = ( R +op ( T +op S ) )
8	5 6 7	3eqtr4i	|- ( ( R +op S ) +op T ) = ( ( R +op T ) +op S )