Theorem fmtno2

Description: The 2 nd Fermat number, see remark in ApostolNT p. 7. (Contributed by AV, 13-Jun-2021)

	Ref	Expression
Assertion	fmtno2	$$⊢\mathrm{FermatNo}\left(2\right)=17$$

Proof

Step	Hyp	Ref	Expression
1		2nn0	$$⊢2\in {\mathbb{N}}_0$$
2		fmtno	$$⊢2\in {\mathbb{N}}_0\to \mathrm{FermatNo}\left(2\right)=2^{2^2}+1$$
3	1 2	ax-mp	$$⊢\mathrm{FermatNo}\left(2\right)=2^{2^2}+1$$
4		sq2	$$⊢2^2=4$$
5	4	oveq2i	$$⊢2^{2^2}=2^4$$
6	5	oveq1i	$$⊢2^{2^2}+1=2^4+1$$
7		2exp4	$$⊢2^4=16$$
8	7	oveq1i	$$⊢2^4+1=16+1$$
9		1nn0	$$⊢1\in {\mathbb{N}}_0$$
10		6nn0	$$⊢6\in {\mathbb{N}}_0$$
11		6p1e7	$$⊢6+1=7$$
12		eqid	$$⊢16=16$$
13	9 10 11 12	decsuc	$$⊢16+1=17$$
14	6 8 13	3eqtri	$$⊢2^{2^2}+1=17$$
15	3 14	eqtri	$$⊢\mathrm{FermatNo}\left(2\right)=17$$