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Metamath Proof Explorer

Theorem hvaddlid

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

		Ref	Expression
	Assertion	hvaddlid	$⊢ A \in ℋ \to 0_{ℎ} +_{ℎ} A = A$

Step	Hyp	Ref	Expression
1		ax-hv0cl	$⊢ 0_{ℎ} \in ℋ$
2		ax-hvcom	$⊢ (A \in ℋ \land 0_{ℎ} \in ℋ) \to A +_{ℎ} 0_{ℎ} = 0_{ℎ} +_{ℎ} A$
3	1 2	mpan2	$⊢ A \in ℋ \to A +_{ℎ} 0_{ℎ} = 0_{ℎ} +_{ℎ} A$
4		ax-hvaddid	$⊢ A \in ℋ \to A +_{ℎ} 0_{ℎ} = A$
5	3 4	eqtr3d	$⊢ A \in ℋ \to 0_{ℎ} +_{ℎ} A = A$