Theorem fmtno1

Description: The 1 st Fermat number, see remark in ApostolNT p. 7. (Contributed by AV, 13-Jun-2021)

	Ref	Expression
Assertion	fmtno1	$$⊢\mathrm{FermatNo}\left(1\right)=5$$

Proof

Step	Hyp	Ref	Expression
1		1nn0	$$⊢1\in {\mathbb{N}}_0$$
2		fmtno	$$⊢1\in {\mathbb{N}}_0\to \mathrm{FermatNo}\left(1\right)=2^{2^1}+1$$
3	1 2	ax-mp	$$⊢\mathrm{FermatNo}\left(1\right)=2^{2^1}+1$$
4		2cn	$$⊢2\in ℂ$$
5		exp1	$$⊢2\in ℂ\to 2^1=2$$
6	4 5	ax-mp	$$⊢2^1=2$$
7	6	oveq2i	$$⊢2^{2^1}=2^2$$
8	7	oveq1i	$$⊢2^{2^1}+1=2^2+1$$
9		sq2	$$⊢2^2=4$$
10	9	oveq1i	$$⊢2^2+1=4+1$$
11		4p1e5	$$⊢4+1=5$$
12	8 10 11	3eqtri	$$⊢2^{2^1}+1=5$$
13	3 12	eqtri	$$⊢\mathrm{FermatNo}\left(1\right)=5$$