Theorem retire

Description: A real times _i is real iff the real is zero. (Contributed by SN, 25-Apr-2025)

	Ref	Expression
Assertion	retire	⊢ ( 𝑅 ∈ ℝ → ( ( 𝑅 · i ) ∈ ℝ ↔ 𝑅 = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝑅 ∈ ℝ → 𝑅 ∈ ℂ )
2		ax-icn	⊢ i ∈ ℂ
3	2	a1i	⊢ ( 𝑅 ∈ ℝ → i ∈ ℂ )
4	1 3	mulcomd	⊢ ( 𝑅 ∈ ℝ → ( 𝑅 · i ) = ( i · 𝑅 ) )
5	4	eleq1d	⊢ ( 𝑅 ∈ ℝ → ( ( 𝑅 · i ) ∈ ℝ ↔ ( i · 𝑅 ) ∈ ℝ ) )
6		itrere	⊢ ( 𝑅 ∈ ℝ → ( ( i · 𝑅 ) ∈ ℝ ↔ 𝑅 = 0 ) )
7	5 6	bitrd	⊢ ( 𝑅 ∈ ℝ → ( ( 𝑅 · i ) ∈ ℝ ↔ 𝑅 = 0 ) )