Theorem fmtno0

Description: The 0 th Fermat number, see remark in ApostolNT p. 7. (Contributed by AV, 13-Jun-2021)

	Ref	Expression
Assertion	fmtno0	$$⊢\mathrm{FermatNo}\left(0\right)=3$$

Proof

Step	Hyp	Ref	Expression
1		0nn0	$$⊢0\in {\mathbb{N}}_0$$
2		fmtno	$$⊢0\in {\mathbb{N}}_0\to \mathrm{FermatNo}\left(0\right)=2^{2^0}+1$$
3	1 2	ax-mp	$$⊢\mathrm{FermatNo}\left(0\right)=2^{2^0}+1$$
4		2cn	$$⊢2\in ℂ$$
5		exp0	$$⊢2\in ℂ\to 2^0=1$$
6	4 5	ax-mp	$$⊢2^0=1$$
7	6	oveq2i	$$⊢2^{2^0}=2^1$$
8	7	oveq1i	$$⊢2^{2^0}+1=2^1+1$$
9		exp1	$$⊢2\in ℂ\to 2^1=2$$
10	4 9	ax-mp	$$⊢2^1=2$$
11	10	oveq1i	$$⊢2^1+1=2+1$$
12		2p1e3	$$⊢2+1=3$$
13	8 11 12	3eqtri	$$⊢2^{2^0}+1=3$$
14	3 13	eqtri	$$⊢\mathrm{FermatNo}\left(0\right)=3$$