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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 ∈ 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 ) ) )