Theorem elfzoext

Description: Membership of an integer in an extended open range of integers, extension added to the right. (Contributed by AV, 30-Apr-2020) (Proof shortened by AV, 23-Sep-2025)

	Ref	Expression
Assertion	elfzoext	⊢ ( ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝐼 ∈ ℕ0 ) → 𝑍 ∈ ( 𝑀 ..^ ( 𝑁 + 𝐼 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elfzoextl	⊢ ( ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝐼 ∈ ℕ0 ) → 𝑍 ∈ ( 𝑀 ..^ ( 𝐼 + 𝑁 ) ) )
2		elfzoel2	⊢ ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℤ )
3	2	zcnd	⊢ ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℂ )
4	3	adantr	⊢ ( ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝐼 ∈ ℕ0 ) → 𝑁 ∈ ℂ )
5		nn0cn	⊢ ( 𝐼 ∈ ℕ0 → 𝐼 ∈ ℂ )
6	5	adantl	⊢ ( ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝐼 ∈ ℕ0 ) → 𝐼 ∈ ℂ )
7	4 6	addcomd	⊢ ( ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝐼 ∈ ℕ0 ) → ( 𝑁 + 𝐼 ) = ( 𝐼 + 𝑁 ) )
8	7	oveq2d	⊢ ( ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝐼 ∈ ℕ0 ) → ( 𝑀 ..^ ( 𝑁 + 𝐼 ) ) = ( 𝑀 ..^ ( 𝐼 + 𝑁 ) ) )
9	1 8	eleqtrrd	⊢ ( ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝐼 ∈ ℕ0 ) → 𝑍 ∈ ( 𝑀 ..^ ( 𝑁 + 𝐼 ) ) )