Theorem blennn0elnn

Description: The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020)

	Ref	Expression
Assertion	blennn0elnn	⊢ ( 𝑁 ∈ ℕ0 → ( #b ‘ 𝑁 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		elnn0	⊢ ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) )
2		blennnelnn	⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) ∈ ℕ )
3		fveq2	⊢ ( 𝑁 = 0 → ( #b ‘ 𝑁 ) = ( #b ‘ 0 ) )
4		blen0	⊢ ( #b ‘ 0 ) = 1
5		1nn	⊢ 1 ∈ ℕ
6	4 5	eqeltri	⊢ ( #b ‘ 0 ) ∈ ℕ
7	3 6	eqeltrdi	⊢ ( 𝑁 = 0 → ( #b ‘ 𝑁 ) ∈ ℕ )
8	2 7	jaoi	⊢ ( ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) → ( #b ‘ 𝑁 ) ∈ ℕ )
9	1 8	sylbi	⊢ ( 𝑁 ∈ ℕ0 → ( #b ‘ 𝑁 ) ∈ ℕ )