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 ∈ ℝ → ( ( i · 𝑅 ) ∈ ℝ ↔ 𝑅 = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		rimul	⊢ ( ( 𝑅 ∈ ℝ ∧ ( i · 𝑅 ) ∈ ℝ ) → 𝑅 = 0 )
2	1	ex	⊢ ( 𝑅 ∈ ℝ → ( ( i · 𝑅 ) ∈ ℝ → 𝑅 = 0 ) )
3		oveq2	⊢ ( 𝑅 = 0 → ( i · 𝑅 ) = ( i · 0 ) )
4		it0e0	⊢ ( i · 0 ) = 0
5		0re	⊢ 0 ∈ ℝ
6	4 5	eqeltri	⊢ ( i · 0 ) ∈ ℝ
7	3 6	eqeltrdi	⊢ ( 𝑅 = 0 → ( i · 𝑅 ) ∈ ℝ )
8	2 7	impbid1	⊢ ( 𝑅 ∈ ℝ → ( ( i · 𝑅 ) ∈ ℝ ↔ 𝑅 = 0 ) )