Theorem ringm2neg

Description: Double negation of a product in a ring. ( mul2neg analog.) (Contributed by Mario Carneiro, 4-Dec-2014) (Proof shortened by AV, 30-Mar-2025)

	Ref	Expression
Hypotheses	ringneglmul.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringneglmul.t	⊢ · = ( .r ‘ 𝑅 )
	ringneglmul.n	⊢ 𝑁 = ( invg ‘ 𝑅 )
	ringneglmul.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	ringneglmul.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ringneglmul.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	ringm2neg	⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) · ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 · 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		ringneglmul.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringneglmul.t	⊢ · = ( .r ‘ 𝑅 )
3		ringneglmul.n	⊢ 𝑁 = ( invg ‘ 𝑅 )
4		ringneglmul.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
5		ringneglmul.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		ringneglmul.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		ringrng	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Rng )
8	4 7	syl	⊢ ( 𝜑 → 𝑅 ∈ Rng )
9	1 2 3 8 5 6	rngm2neg	⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) · ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 · 𝑌 ) )