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Metamath Proof Explorer

Theorem mulidpi

Description: 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996) (Revised by Mario Carneiro, 17-Nov-2014) (New usage is discouraged.)

		Ref	Expression
	Assertion	mulidpi	$⊢ A \in 𝑵 \to A \cdot_{𝑵} 1_{𝑜} = A$

Proof

Step	Hyp	Ref	Expression
1		1pi	$⊢ 1_{𝑜} \in 𝑵$
2		mulpiord	$⊢ (A \in 𝑵 \land 1_{𝑜} \in 𝑵) \to A \cdot_{𝑵} 1_{𝑜} = A \cdot_{𝑜} 1_{𝑜}$
3	1 2	mpan2	$⊢ A \in 𝑵 \to A \cdot_{𝑵} 1_{𝑜} = A \cdot_{𝑜} 1_{𝑜}$
4		pinn	$⊢ A \in 𝑵 \to A \in ω$
5		nnm1	$⊢ A \in ω \to A \cdot_{𝑜} 1_{𝑜} = A$
6	4 5	syl	$⊢ A \in 𝑵 \to A \cdot_{𝑜} 1_{𝑜} = A$
7	3 6	eqtrd	$⊢ A \in 𝑵 \to A \cdot_{𝑵} 1_{𝑜} = A$