Theorem hvadd12i

Description: Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvass.1	|- A e. ~H
	hvass.2	|- B e. ~H
	hvass.3	|- C e. ~H
Assertion	hvadd12i	|- ( A +h ( B +h C ) ) = ( B +h ( A +h C ) )

Proof

Step	Hyp	Ref	Expression
1		hvass.1	|- A e. ~H
2		hvass.2	|- B e. ~H
3		hvass.3	|- C e. ~H
4	1 2	hvcomi	|- ( A +h B ) = ( B +h A )
5	4	oveq1i	|- ( ( A +h B ) +h C ) = ( ( B +h A ) +h C )
6	1 2 3	hvassi	|- ( ( A +h B ) +h C ) = ( A +h ( B +h C ) )
7	2 1 3	hvassi	|- ( ( B +h A ) +h C ) = ( B +h ( A +h C ) )
8	5 6 7	3eqtr3i	|- ( A +h ( B +h C ) ) = ( B +h ( A +h C ) )