Theorem elaz

Description: Elementhood in an adjunction of a complex number to the ring of integers. Such an element is called "extended integer" in the following. (Contributed by AV, 30-Oct-2026)

	Ref	Expression
Assertion	elaz	⊢ ( 𝑆 ∈ ℂ → ( 𝐴 ∈ ( Z[] ‘ 𝑆 ) ↔ 𝐴 ∈ ( ( RingSpan ‘ ℂfld ) ‘ ( ℤ ∪ { 𝑆 } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		azval	⊢ ( 𝑆 ∈ ℂ → ( Z[] ‘ 𝑆 ) = ( ( RingSpan ‘ ℂfld ) ‘ ( ℤ ∪ { 𝑆 } ) ) )
2	1	eleq2d	⊢ ( 𝑆 ∈ ℂ → ( 𝐴 ∈ ( Z[] ‘ 𝑆 ) ↔ 𝐴 ∈ ( ( RingSpan ‘ ℂfld ) ‘ ( ℤ ∪ { 𝑆 } ) ) ) )