Theorem fmtnole4prm

Description: The first five Fermat numbers are prime. (Contributed by AV, 28-Jul-2021)

	Ref	Expression
Assertion	fmtnole4prm	$$⊢\left(N\in {\mathbb{N}}_0\wedge N\le 4\right)\to \mathrm{FermatNo}\left(N\right)\in \mathbb{P}$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(N\in {\mathbb{N}}_0\wedge N\le 4\right)\to N\in {\mathbb{N}}_0$$
2		4nn0	$$⊢4\in {\mathbb{N}}_0$$
3	2	a1i	$$⊢\left(N\in {\mathbb{N}}_0\wedge N\le 4\right)\to 4\in {\mathbb{N}}_0$$
4		simpr	$$⊢\left(N\in {\mathbb{N}}_0\wedge N\le 4\right)\to N\le 4$$
5		elfz2nn0	$$⊢N\in \left(0\dots 4\right)\leftrightarrow \left(N\in {\mathbb{N}}_0\wedge 4\in {\mathbb{N}}_0\wedge N\le 4\right)$$
6	1 3 4 5	syl3anbrc	$$⊢\left(N\in {\mathbb{N}}_0\wedge N\le 4\right)\to N\in \left(0\dots 4\right)$$
7		fmtnofz04prm	$$⊢N\in \left(0\dots 4\right)\to \mathrm{FermatNo}\left(N\right)\in \mathbb{P}$$
8	6 7	syl	$$⊢\left(N\in {\mathbb{N}}_0\wedge N\le 4\right)\to \mathrm{FermatNo}\left(N\right)\in \mathbb{P}$$