Theorem blenre

Description: The binary length of a positive real number. (Contributed by AV, 20-May-2020)

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

Proof

Step	Hyp	Ref	Expression
1		rpne0	⊢ ( 𝑁 ∈ ℝ+ → 𝑁 ≠ 0 )
2		blenn0	⊢ ( ( 𝑁 ∈ ℝ+ ∧ 𝑁 ≠ 0 ) → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) )
3	1 2	mpdan	⊢ ( 𝑁 ∈ ℝ+ → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) )
4		rpre	⊢ ( 𝑁 ∈ ℝ+ → 𝑁 ∈ ℝ )
5		rpge0	⊢ ( 𝑁 ∈ ℝ+ → 0 ≤ 𝑁 )
6	4 5	absidd	⊢ ( 𝑁 ∈ ℝ+ → ( abs ‘ 𝑁 ) = 𝑁 )
7	6	oveq2d	⊢ ( 𝑁 ∈ ℝ+ → ( 2 logb ( abs ‘ 𝑁 ) ) = ( 2 logb 𝑁 ) )
8	7	fveq2d	⊢ ( 𝑁 ∈ ℝ+ → ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) = ( ⌊ ‘ ( 2 logb 𝑁 ) ) )
9	8	oveq1d	⊢ ( 𝑁 ∈ ℝ+ → ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) )
10	3 9	eqtrd	⊢ ( 𝑁 ∈ ℝ+ → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) )