Theorem ellazim

Description: Representation of an "extended integer". (Contributed by AV, 25-Oct-2026)

	Ref	Expression
Assertion	ellazim	⊢ ( ( 𝑆 ∈ ℂ ∧ 𝐴 ∈ ( Z[]l ‘ 𝑆 ) ) → ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ 𝐴 = ( 𝑎 + ( 𝑏 · 𝑆 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ellaz	⊢ ( 𝑆 ∈ ℂ → ( 𝐴 ∈ ( Z[]l ‘ 𝑆 ) ↔ ( 𝐴 ∈ ℂ ∧ ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ 𝐴 = ( 𝑎 + ( 𝑏 · 𝑆 ) ) ) ) )
2	1	simplbda	⊢ ( ( 𝑆 ∈ ℂ ∧ 𝐴 ∈ ( Z[]l ‘ 𝑆 ) ) → ∃ 𝑎 ∈ ℤ ∃ 𝑏 ∈ ℤ 𝐴 = ( 𝑎 + ( 𝑏 · 𝑆 ) ) )