Theorem hvnegid

Description: Addition of negative of a vector to itself. (Contributed by NM, 4-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	hvnegid	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 +ℎ ( - 1 ·ℎ 𝐴 ) ) = 0ℎ )

Proof

Step	Hyp	Ref	Expression
1		hvsubval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 −ℎ 𝐴 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐴 ) ) )
2	1	anidms	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 −ℎ 𝐴 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐴 ) ) )
3		hvsubid	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 −ℎ 𝐴 ) = 0ℎ )
4	2 3	eqtr3d	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 +ℎ ( - 1 ·ℎ 𝐴 ) ) = 0ℎ )