Theorem hvadd32i

Description: Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvass.1	⊢ 𝐴 ∈ ℋ
	hvass.2	⊢ 𝐵 ∈ ℋ
	hvass.3	⊢ 𝐶 ∈ ℋ
Assertion	hvadd32i	⊢ ( ( 𝐴 +ℎ 𝐵 ) +ℎ 𝐶 ) = ( ( 𝐴 +ℎ 𝐶 ) +ℎ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		hvass.1	⊢ 𝐴 ∈ ℋ
2		hvass.2	⊢ 𝐵 ∈ ℋ
3		hvass.3	⊢ 𝐶 ∈ ℋ
4		hvadd32	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) +ℎ 𝐶 ) = ( ( 𝐴 +ℎ 𝐶 ) +ℎ 𝐵 ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 +ℎ 𝐵 ) +ℎ 𝐶 ) = ( ( 𝐴 +ℎ 𝐶 ) +ℎ 𝐵 )