Theorem fmtno1

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

	Ref	Expression
Assertion	fmtno1	⊢ ( FermatNo ‘ 1 ) = 5

Proof

Step	Hyp	Ref	Expression
1		1nn0	⊢ 1 ∈ ℕ0
2		fmtno	⊢ ( 1 ∈ ℕ0 → ( FermatNo ‘ 1 ) = ( ( 2 ↑ ( 2 ↑ 1 ) ) + 1 ) )
3	1 2	ax-mp	⊢ ( FermatNo ‘ 1 ) = ( ( 2 ↑ ( 2 ↑ 1 ) ) + 1 )
4		2cn	⊢ 2 ∈ ℂ
5		exp1	⊢ ( 2 ∈ ℂ → ( 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	⊢ ( FermatNo ‘ 1 ) = 5