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	$$⊢\left(A\in 𝑵\wedge 1_{𝑜}\in 𝑵\right)\to A{\cdot }_{𝑵}{1}_{𝑜}=A{\cdot }_{𝑜}{1}_{𝑜}$$
3	1 2	mpan2	$$⊢A\in 𝑵\to A{\cdot }_{𝑵}{1}_{𝑜}=A{\cdot }_{𝑜}{1}_{𝑜}$$
4		pinn	$$⊢A\in 𝑵\to A\in \mathrm{\omega }$$
5		nnm1	$$⊢A\in \mathrm{\omega }\to A{\cdot }_{𝑜}{1}_{𝑜}=A$$
6	4 5	syl	$$⊢A\in 𝑵\to A{\cdot }_{𝑜}{1}_{𝑜}=A$$
7	3 6	eqtrd	$$⊢A\in 𝑵\to A{\cdot }_{𝑵}{1}_{𝑜}=A$$