Theorem fzo0addelr

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

	Ref	Expression
Assertion	fzo0addelr	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 + 𝐷 ) ∈ ( 𝐷 ..^ ( 𝐷 + 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fzo0addel	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 + 𝐷 ) ∈ ( 𝐷 ..^ ( 𝐶 + 𝐷 ) ) )
2		zcn	⊢ ( 𝐷 ∈ ℤ → 𝐷 ∈ ℂ )
3		elfzoel2	⊢ ( 𝐴 ∈ ( 0 ..^ 𝐶 ) → 𝐶 ∈ ℤ )
4	3	zcnd	⊢ ( 𝐴 ∈ ( 0 ..^ 𝐶 ) → 𝐶 ∈ ℂ )
5		addcom	⊢ ( ( 𝐷 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐷 + 𝐶 ) = ( 𝐶 + 𝐷 ) )
6	2 4 5	syl2anr	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐷 + 𝐶 ) = ( 𝐶 + 𝐷 ) )
7	6	oveq2d	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐷 ..^ ( 𝐷 + 𝐶 ) ) = ( 𝐷 ..^ ( 𝐶 + 𝐷 ) ) )
8	1 7	eleqtrrd	⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 + 𝐷 ) ∈ ( 𝐷 ..^ ( 𝐷 + 𝐶 ) ) )