Theorem fmtno0

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

	Ref	Expression
Assertion	fmtno0	⊢ ( FermatNo ‘ 0 ) = 3

Proof

Step	Hyp	Ref	Expression
1		0nn0	⊢ 0 ∈ ℕ0
2		fmtno	⊢ ( 0 ∈ ℕ0 → ( FermatNo ‘ 0 ) = ( ( 2 ↑ ( 2 ↑ 0 ) ) + 1 ) )
3	1 2	ax-mp	⊢ ( FermatNo ‘ 0 ) = ( ( 2 ↑ ( 2 ↑ 0 ) ) + 1 )
4		2cn	⊢ 2 ∈ ℂ
5		exp0	⊢ ( 2 ∈ ℂ → ( 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 ∈ ℂ → ( 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	⊢ ( FermatNo ‘ 0 ) = 3