Theorem oesuc

Description: Ordinal exponentiation with a successor exponent. Definition 8.30 of TakeutiZaring p. 67. Definition 2.6 of Schloeder p. 4. (Contributed by NM, 31-Dec-2004) (Revised by Mario Carneiro, 8-Sep-2013)

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

Proof

Step	Hyp	Ref	Expression
1		limon	$$⊢\mathrm{Lim}\mathrm{On}$$
2		rdgsuc	$$⊢B\in \mathrm{On}\to \mathrm{rec}\left(\left(x\in \mathrm{V}⟼x{\cdot }_{𝑜}A\right),1_{𝑜}\right)\left(\mathrm{suc}B\right)=\left(x\in \mathrm{V}⟼x{\cdot }_{𝑜}A\right)\left(\mathrm{rec}\left(\left(x\in \mathrm{V}⟼x{\cdot }_{𝑜}A\right),1_{𝑜}\right)\left(B\right)\right)$$
3	1 2	oesuclem	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A{\uparrow }_{𝑜}\mathrm{suc}B=\left(A{\uparrow }_{𝑜}B\right){\cdot }_{𝑜}A$$