Theorem blenn0

Description: The binary length of a "number" not being 0. (Contributed by AV, 20-May-2020)

	Ref	Expression
Assertion	blenn0	⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 0 ) → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		blenval	⊢ ( 𝑁 ∈ 𝑉 → ( #b ‘ 𝑁 ) = if ( 𝑁 = 0 , 1 , ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) ) )
2		ifnefalse	⊢ ( 𝑁 ≠ 0 → if ( 𝑁 = 0 , 1 , ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) ) = ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) )
3	1 2	sylan9eq	⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 0 ) → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) )