Metamath Proof Explorer


Theorem normneg

Description: The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005) (New usage is discouraged.)

Ref Expression
Assertion normneg ( 𝐴 ∈ ℋ → ( norm ‘ ( - 1 · 𝐴 ) ) = ( norm𝐴 ) )

Proof

Step Hyp Ref Expression
1 ax-hv0cl 0 ∈ ℋ
2 normsub ( ( 0 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( norm ‘ ( 0 𝐴 ) ) = ( norm ‘ ( 𝐴 0 ) ) )
3 1 2 mpan ( 𝐴 ∈ ℋ → ( norm ‘ ( 0 𝐴 ) ) = ( norm ‘ ( 𝐴 0 ) ) )
4 hv2neg ( 𝐴 ∈ ℋ → ( 0 𝐴 ) = ( - 1 · 𝐴 ) )
5 4 fveq2d ( 𝐴 ∈ ℋ → ( norm ‘ ( 0 𝐴 ) ) = ( norm ‘ ( - 1 · 𝐴 ) ) )
6 hvsub0 ( 𝐴 ∈ ℋ → ( 𝐴 0 ) = 𝐴 )
7 6 fveq2d ( 𝐴 ∈ ℋ → ( norm ‘ ( 𝐴 0 ) ) = ( norm𝐴 ) )
8 3 5 7 3eqtr3d ( 𝐴 ∈ ℋ → ( norm ‘ ( - 1 · 𝐴 ) ) = ( norm𝐴 ) )