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 : ~H --> ~H
	hods.2	|- S : ~H --> ~H
	hods.3	|- T : ~H --> ~H
Assertion	hoadd12i	|- ( R +op ( S +op T ) ) = ( S +op ( R +op T ) )

Proof

Step	Hyp	Ref	Expression
1		hods.1	|- R : ~H --> ~H
2		hods.2	|- S : ~H --> ~H
3		hods.3	|- T : ~H --> ~H
4	1 2	hoaddcomi	|- ( R +op S ) = ( S +op R )
5	4	oveq1i	|- ( ( R +op S ) +op T ) = ( ( S +op R ) +op T )
6	1 2 3	hoaddassi	|- ( ( R +op S ) +op T ) = ( R +op ( S +op T ) )
7	2 1 3	hoaddassi	|- ( ( S +op R ) +op T ) = ( S +op ( R +op T ) )
8	5 6 7	3eqtr3i	|- ( R +op ( S +op T ) ) = ( S +op ( R +op T ) )