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	$$⊢A\in \mathscr{H}\to A{+}_{ℎ}\left(-1{\cdot }_{ℎ}A\right)=0_{ℎ}$$

Proof

Step	Hyp	Ref	Expression
1		hvsubval	$$⊢\left(A\in \mathscr{H}\wedge A\in \mathscr{H}\right)\to A{-}_{ℎ}A=A{+}_{ℎ}\left(-1{\cdot }_{ℎ}A\right)$$
2	1	anidms	$$⊢A\in \mathscr{H}\to A{-}_{ℎ}A=A{+}_{ℎ}\left(-1{\cdot }_{ℎ}A\right)$$
3		hvsubid	$$⊢A\in \mathscr{H}\to A{-}_{ℎ}A=0_{ℎ}$$
4	2 3	eqtr3d	$$⊢A\in \mathscr{H}\to A{+}_{ℎ}\left(-1{\cdot }_{ℎ}A\right)=0_{ℎ}$$