Theorem nn2m

Description: Multiply an element of _om by 2o . (Contributed by Scott Fenton, 16-Apr-2012) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	nn2m	$$⊢A\in \mathrm{\omega }\to 2_{𝑜}{\cdot }_{𝑜}A=A{+}_{𝑜}A$$

Proof

Step	Hyp	Ref	Expression
1		2onn	$$⊢2_{𝑜}\in \mathrm{\omega }$$
2		nnmcom	$$⊢\left(2_{𝑜}\in \mathrm{\omega }\wedge A\in \mathrm{\omega }\right)\to 2_{𝑜}{\cdot }_{𝑜}A=A{\cdot }_{𝑜}{2}_{𝑜}$$
3	1 2	mpan	$$⊢A\in \mathrm{\omega }\to 2_{𝑜}{\cdot }_{𝑜}A=A{\cdot }_{𝑜}{2}_{𝑜}$$
4		nnm2	$$⊢A\in \mathrm{\omega }\to A{\cdot }_{𝑜}{2}_{𝑜}=A{+}_{𝑜}A$$
5	3 4	eqtrd	$$⊢A\in \mathrm{\omega }\to 2_{𝑜}{\cdot }_{𝑜}A=A{+}_{𝑜}A$$