Theorem hvaddlid

Description: Addition with the zero vector. (Contributed by NM, 18-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	hvaddlid	⊢ ( 𝐴 ∈ ℋ → ( 0ℎ +ℎ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ax-hv0cl	⊢ 0ℎ ∈ ℋ
2		ax-hvcom	⊢ ( ( 𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ ) → ( 𝐴 +ℎ 0ℎ ) = ( 0ℎ +ℎ 𝐴 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 +ℎ 0ℎ ) = ( 0ℎ +ℎ 𝐴 ) )
4		ax-hvaddid	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 +ℎ 0ℎ ) = 𝐴 )
5	3 4	eqtr3d	⊢ ( 𝐴 ∈ ℋ → ( 0ℎ +ℎ 𝐴 ) = 𝐴 )