Metamath Proof Explorer


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 ∈ ℕ0𝐼 ∈ 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 ) ) )