Metamath Proof Explorer


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 = ( +𝑣𝑈 )
nvdi.4 𝑆 = ( ·𝑠OLD𝑈 )
Assertion nv2 ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ) → ( 𝐴 𝐺 𝐴 ) = ( 2 𝑆 𝐴 ) )

Proof

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