Theorem fzo0addel

Description: Translate membership in a 0-based half-open integer range. (Contributed by AV, 30-Apr-2020)

	Ref	Expression
Assertion	fzo0addel	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 + 𝐷 ) ∈ ( 𝐷 ..^ ( 𝐶 + 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fzoaddel	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 + 𝐷 ) ∈ ( ( 0 + 𝐷 ) ..^ ( 𝐶 + 𝐷 ) ) )
2		zcn	⊢ ( 𝐷 ∈ ℤ → 𝐷 ∈ ℂ )
3		addlid	⊢ ( 𝐷 ∈ ℂ → ( 0 + 𝐷 ) = 𝐷 )
4	3	eqcomd	⊢ ( 𝐷 ∈ ℂ → 𝐷 = ( 0 + 𝐷 ) )
5	2 4	syl	⊢ ( 𝐷 ∈ ℤ → 𝐷 = ( 0 + 𝐷 ) )
6	5	adantl	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → 𝐷 = ( 0 + 𝐷 ) )
7	6	oveq1d	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐷 ..^ ( 𝐶 + 𝐷 ) ) = ( ( 0 + 𝐷 ) ..^ ( 𝐶 + 𝐷 ) ) )
8	1 7	eleqtrrd	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 + 𝐷 ) ∈ ( 𝐷 ..^ ( 𝐶 + 𝐷 ) ) )