Theorem blennn

Description: The binary length of a positive integer. (Contributed by AV, 21-May-2020)

	Ref	Expression
Assertion	blennn	⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		nnne0	⊢ ( 𝑁 ∈ ℕ → 𝑁 ≠ 0 )
2		blenn0	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 ≠ 0 ) → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) )
3	1 2	mpdan	⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) )
4		nnre	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℝ )
5		nnnn0	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 )
6	5	nn0ge0d	⊢ ( 𝑁 ∈ ℕ → 0 ≤ 𝑁 )
7	4 6	absidd	⊢ ( 𝑁 ∈ ℕ → ( abs ‘ 𝑁 ) = 𝑁 )
8	7	oveq2d	⊢ ( 𝑁 ∈ ℕ → ( 2 logb ( abs ‘ 𝑁 ) ) = ( 2 logb 𝑁 ) )
9	8	fveq2d	⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) = ( ⌊ ‘ ( 2 logb 𝑁 ) ) )
10	9	oveq1d	⊢ ( 𝑁 ∈ ℕ → ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) )
11	3 10	eqtrd	⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) )