Theorem nnesuc

Description: Exponentiation with a successor exponent. Definition 8.30 of TakeutiZaring p. 67. (Contributed by Mario Carneiro, 14-Nov-2014)

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

Proof

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