Theorem fz0sn0fz1

Description: A finite set of sequential nonnegative integers is the union of the singleton containing 0 and a finite set of sequential positive integers. (Contributed by AV, 20-Mar-2021)

	Ref	Expression
Assertion	fz0sn0fz1	⊢ ( 𝑁 ∈ ℕ0 → ( 0 ... 𝑁 ) = ( { 0 } ∪ ( 1 ... 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		0elfz	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑁 ) )
2		fzsplit	⊢ ( 0 ∈ ( 0 ... 𝑁 ) → ( 0 ... 𝑁 ) = ( ( 0 ... 0 ) ∪ ( ( 0 + 1 ) ... 𝑁 ) ) )
3		0p1e1	⊢ ( 0 + 1 ) = 1
4	3	oveq1i	⊢ ( ( 0 + 1 ) ... 𝑁 ) = ( 1 ... 𝑁 )
5	4	uneq2i	⊢ ( ( 0 ... 0 ) ∪ ( ( 0 + 1 ) ... 𝑁 ) ) = ( ( 0 ... 0 ) ∪ ( 1 ... 𝑁 ) )
6	2 5	eqtrdi	⊢ ( 0 ∈ ( 0 ... 𝑁 ) → ( 0 ... 𝑁 ) = ( ( 0 ... 0 ) ∪ ( 1 ... 𝑁 ) ) )
7	1 6	syl	⊢ ( 𝑁 ∈ ℕ0 → ( 0 ... 𝑁 ) = ( ( 0 ... 0 ) ∪ ( 1 ... 𝑁 ) ) )
8		0z	⊢ 0 ∈ ℤ
9		fzsn	⊢ ( 0 ∈ ℤ → ( 0 ... 0 ) = { 0 } )
10	8 9	mp1i	⊢ ( 𝑁 ∈ ℕ0 → ( 0 ... 0 ) = { 0 } )
11	10	uneq1d	⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ... 0 ) ∪ ( 1 ... 𝑁 ) ) = ( { 0 } ∪ ( 1 ... 𝑁 ) ) )
12	7 11	eqtrd	⊢ ( 𝑁 ∈ ℕ0 → ( 0 ... 𝑁 ) = ( { 0 } ∪ ( 1 ... 𝑁 ) ) )