Theorem fz0dif1

Description: Split the first element of a finite set of sequential nonnegative integers. (Contributed by AV, 12-Sep-2025)

	Ref	Expression
Assertion	fz0dif1	⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ... 𝑁 ) ∖ { 0 } ) = ( 1 ... 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		elnn0uz	⊢ ( 𝑁 ∈ ℕ0 ↔ 𝑁 ∈ ( ℤ≥ ‘ 0 ) )
2		fzdif1	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( ( 0 ... 𝑁 ) ∖ { 0 } ) = ( ( 0 + 1 ) ... 𝑁 ) )
3	1 2	sylbi	⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ... 𝑁 ) ∖ { 0 } ) = ( ( 0 + 1 ) ... 𝑁 ) )
4		0p1e1	⊢ ( 0 + 1 ) = 1
5	4	oveq1i	⊢ ( ( 0 + 1 ) ... 𝑁 ) = ( 1 ... 𝑁 )
6	3 5	eqtrdi	⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ... 𝑁 ) ∖ { 0 } ) = ( 1 ... 𝑁 ) )