Theorem fzo0sn0fzo1

Description: A half-open range of nonnegative integers is the union of the singleton set containing 0 and a half-open range of positive integers. (Contributed by Alexander van der Vekens, 18-May-2018)

	Ref	Expression
Assertion	fzo0sn0fzo1	⊢ ( 𝑁 ∈ ℕ → ( 0 ..^ 𝑁 ) = ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		1nn0	⊢ 1 ∈ ℕ0
2	1	a1i	⊢ ( 𝑁 ∈ ℕ → 1 ∈ ℕ0 )
3		nnnn0	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 )
4		nnge1	⊢ ( 𝑁 ∈ ℕ → 1 ≤ 𝑁 )
5		elfz2nn0	⊢ ( 1 ∈ ( 0 ... 𝑁 ) ↔ ( 1 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ) )
6	2 3 4 5	syl3anbrc	⊢ ( 𝑁 ∈ ℕ → 1 ∈ ( 0 ... 𝑁 ) )
7		fzosplit	⊢ ( 1 ∈ ( 0 ... 𝑁 ) → ( 0 ..^ 𝑁 ) = ( ( 0 ..^ 1 ) ∪ ( 1 ..^ 𝑁 ) ) )
8	6 7	syl	⊢ ( 𝑁 ∈ ℕ → ( 0 ..^ 𝑁 ) = ( ( 0 ..^ 1 ) ∪ ( 1 ..^ 𝑁 ) ) )
9		fzo01	⊢ ( 0 ..^ 1 ) = { 0 }
10	9	a1i	⊢ ( 𝑁 ∈ ℕ → ( 0 ..^ 1 ) = { 0 } )
11	10	uneq1d	⊢ ( 𝑁 ∈ ℕ → ( ( 0 ..^ 1 ) ∪ ( 1 ..^ 𝑁 ) ) = ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) )
12	8 11	eqtrd	⊢ ( 𝑁 ∈ ℕ → ( 0 ..^ 𝑁 ) = ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) )