Theorem elfzoextl

Description: Membership of an integer in an extended open range of integers, extension added to the left. (Contributed by AV, 31-Aug-2025) Generalized by replacing the left border of the ranges. (Revised by SN, 18-Sep-2025)

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

Proof

Step	Hyp	Ref	Expression
1		elfzoel2	⊢ ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℤ )
2		nn0pzuz	⊢ ( ( 𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ) → ( 𝐼 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑁 ) )
3	1 2	sylan2	⊢ ( ( 𝐼 ∈ ℕ0 ∧ 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐼 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑁 ) )
4		fzoss2	⊢ ( ( 𝐼 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑁 ) → ( 𝑀 ..^ 𝑁 ) ⊆ ( 𝑀 ..^ ( 𝐼 + 𝑁 ) ) )
5	3 4	syl	⊢ ( ( 𝐼 ∈ ℕ0 ∧ 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑀 ..^ 𝑁 ) ⊆ ( 𝑀 ..^ ( 𝐼 + 𝑁 ) ) )
6	5	sseld	⊢ ( ( 𝐼 ∈ ℕ0 ∧ 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑍 ∈ ( 𝑀 ..^ ( 𝐼 + 𝑁 ) ) ) )
7	6	syldbl2	⊢ ( ( 𝐼 ∈ ℕ0 ∧ 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑍 ∈ ( 𝑀 ..^ ( 𝐼 + 𝑁 ) ) )
8	7	ancoms	⊢ ( ( 𝑍 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝐼 ∈ ℕ0 ) → 𝑍 ∈ ( 𝑀 ..^ ( 𝐼 + 𝑁 ) ) )