Theorem fzisfzounsn

Description: A finite interval of integers as union of a half-open integer range and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018)

	Ref	Expression
Assertion	fzisfzounsn	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐴 ... 𝐵 ) = ( ( 𝐴 ..^ 𝐵 ) ∪ { 𝐵 } ) )

Proof

Step	Hyp	Ref	Expression
1		eluzelz	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → 𝐵 ∈ ℤ )
2		fzval3	⊢ ( 𝐵 ∈ ℤ → ( 𝐴 ... 𝐵 ) = ( 𝐴 ..^ ( 𝐵 + 1 ) ) )
3	1 2	syl	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐴 ... 𝐵 ) = ( 𝐴 ..^ ( 𝐵 + 1 ) ) )
4		fzosplitsn	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐴 ..^ ( 𝐵 + 1 ) ) = ( ( 𝐴 ..^ 𝐵 ) ∪ { 𝐵 } ) )
5	3 4	eqtrd	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐴 ... 𝐵 ) = ( ( 𝐴 ..^ 𝐵 ) ∪ { 𝐵 } ) )