Theorem nvadd32

Description: Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvgcl.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	nvgcl.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
Assertion	nvadd32	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		nvgcl.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvgcl.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3	2	nvablo	⊢ ( 𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp )
4	1 2	bafval	⊢ 𝑋 = ran 𝐺
5	4	ablo32	⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐵 ) )
6	3 5	sylan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐵 ) )