Theorem nnmsuc

Description: Multiplication with successor. Theorem 4J(A2) of Enderton p. 80. (Contributed by NM, 20-Sep-1995) (Revised by Mario Carneiro, 14-Nov-2014)

	Ref	Expression
Assertion	nnmsuc	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to A{\cdot }_{𝑜}\mathrm{suc}B=\left(A{\cdot }_{𝑜}B\right){+}_{𝑜}A$$

Proof

Step	Hyp	Ref	Expression
1		nnon	$$⊢A\in \mathrm{\omega }\to A\in \mathrm{On}$$
2		onmsuc	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{\omega }\right)\to A{\cdot }_{𝑜}\mathrm{suc}B=\left(A{\cdot }_{𝑜}B\right){+}_{𝑜}A$$
3	1 2	sylan	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to A{\cdot }_{𝑜}\mathrm{suc}B=\left(A{\cdot }_{𝑜}B\right){+}_{𝑜}A$$