Theorem dig2nn0ld

Description: The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020)

	Ref	Expression
Assertion	dig2nn0ld	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐾 ∈ ( ℤ≥ ‘ ( #b ‘ 𝑁 ) ) ) → ( 𝐾 ( digit ‘ 2 ) 𝑁 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		2z	⊢ 2 ∈ ℤ
2		uzid	⊢ ( 2 ∈ ℤ → 2 ∈ ( ℤ≥ ‘ 2 ) )
3	1 2	mp1i	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐾 ∈ ( ℤ≥ ‘ ( #b ‘ 𝑁 ) ) ) → 2 ∈ ( ℤ≥ ‘ 2 ) )
4		simpl	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐾 ∈ ( ℤ≥ ‘ ( #b ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ )
5		blennn	⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) )
6	5	fveq2d	⊢ ( 𝑁 ∈ ℕ → ( ℤ≥ ‘ ( #b ‘ 𝑁 ) ) = ( ℤ≥ ‘ ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) ) )
7	6	eleq2d	⊢ ( 𝑁 ∈ ℕ → ( 𝐾 ∈ ( ℤ≥ ‘ ( #b ‘ 𝑁 ) ) ↔ 𝐾 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) ) ) )
8	7	biimpa	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐾 ∈ ( ℤ≥ ‘ ( #b ‘ 𝑁 ) ) ) → 𝐾 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) ) )
9		dignnld	⊢ ( ( 2 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) ) ) → ( 𝐾 ( digit ‘ 2 ) 𝑁 ) = 0 )
10	3 4 8 9	syl3anc	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐾 ∈ ( ℤ≥ ‘ ( #b ‘ 𝑁 ) ) ) → ( 𝐾 ( digit ‘ 2 ) 𝑁 ) = 0 )