Theorem itrere

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

	Ref	Expression
Assertion	itrere	$$⊢R\in ℝ\to \left(\mathrm{i}R\in ℝ\leftrightarrow R=0\right)$$

Proof

Step	Hyp	Ref	Expression
1		rimul	$$⊢\left(R\in ℝ\wedge \mathrm{i}R\in ℝ\right)\to R=0$$
2	1	ex	$$⊢R\in ℝ\to \left(\mathrm{i}R\in ℝ\to R=0\right)$$
3		oveq2	$$⊢R=0\to \mathrm{i}R=\mathrm{i}\cdot 0$$
4		it0e0	$$⊢\mathrm{i}\cdot 0=0$$
5		0re	$$⊢0\in ℝ$$
6	4 5	eqeltri	$$⊢\mathrm{i}\cdot 0\in ℝ$$
7	3 6	eqeltrdi	$$⊢R=0\to \mathrm{i}R\in ℝ$$
8	2 7	impbid1	$$⊢R\in ℝ\to \left(\mathrm{i}R\in ℝ\leftrightarrow R=0\right)$$