Theorem nv2

Description: A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvdi.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	nvdi.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
	nvdi.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
Assertion	nv2	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐴 ) = ( 2 𝑆 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		nvdi.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvdi.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		nvdi.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4		eqid	⊢ ( 1st ‘ 𝑈 ) = ( 1st ‘ 𝑈 )
5	4	nvvc	⊢ ( 𝑈 ∈ NrmCVec → ( 1st ‘ 𝑈 ) ∈ CVecOLD )
6	2	vafval	⊢ 𝐺 = ( 1st ‘ ( 1st ‘ 𝑈 ) )
7	3	smfval	⊢ 𝑆 = ( 2nd ‘ ( 1st ‘ 𝑈 ) )
8	1 2	bafval	⊢ 𝑋 = ran 𝐺
9	6 7 8	vc2OLD	⊢ ( ( ( 1st ‘ 𝑈 ) ∈ CVecOLD ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐴 ) = ( 2 𝑆 𝐴 ) )
10	5 9	sylan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐴 ) = ( 2 𝑆 𝐴 ) )