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 ∈ ℋ
	hvass.2	⊢ 𝐵 ∈ ℋ
	hvass.3	⊢ 𝐶 ∈ ℋ
Assertion	hvadd12i	⊢ ( 𝐴 +ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( 𝐵 +ℎ ( 𝐴 +ℎ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		hvass.1	⊢ 𝐴 ∈ ℋ
2		hvass.2	⊢ 𝐵 ∈ ℋ
3		hvass.3	⊢ 𝐶 ∈ 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 ) )