Theorem rngm2neg

Description: Double negation of a product in a non-unital ring ( mul2neg analog). (Contributed by Mario Carneiro, 4-Dec-2014) Generalization of ringm2neg . (Revised by AV, 17-Feb-2025)

	Ref	Expression
Hypotheses	rngneglmul.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rngneglmul.t	⊢ · = ( .r ‘ 𝑅 )
	rngneglmul.n	⊢ 𝑁 = ( invg ‘ 𝑅 )
	rngneglmul.r	⊢ ( 𝜑 → 𝑅 ∈ Rng )
	rngneglmul.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	rngneglmul.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	rngm2neg	⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) · ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 · 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		rngneglmul.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rngneglmul.t	⊢ · = ( .r ‘ 𝑅 )
3		rngneglmul.n	⊢ 𝑁 = ( invg ‘ 𝑅 )
4		rngneglmul.r	⊢ ( 𝜑 → 𝑅 ∈ Rng )
5		rngneglmul.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		rngneglmul.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		rnggrp	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Grp )
8	4 7	syl	⊢ ( 𝜑 → 𝑅 ∈ Grp )
9	1 3 8 6	grpinvcld	⊢ ( 𝜑 → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 )
10	1 2 3 4 5 9	rngmneg1	⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) · ( 𝑁 ‘ 𝑌 ) ) = ( 𝑁 ‘ ( 𝑋 · ( 𝑁 ‘ 𝑌 ) ) ) )
11	1 2 3 4 5 6	rngmneg2	⊢ ( 𝜑 → ( 𝑋 · ( 𝑁 ‘ 𝑌 ) ) = ( 𝑁 ‘ ( 𝑋 · 𝑌 ) ) )
12	11	fveq2d	⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑋 · ( 𝑁 ‘ 𝑌 ) ) ) = ( 𝑁 ‘ ( 𝑁 ‘ ( 𝑋 · 𝑌 ) ) ) )
13	1 2	rngcl	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )
14	4 5 6 13	syl3anc	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ 𝐵 )
15	1 3	grpinvinv	⊢ ( ( 𝑅 ∈ Grp ∧ ( 𝑋 · 𝑌 ) ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ ( 𝑋 · 𝑌 ) ) ) = ( 𝑋 · 𝑌 ) )
16	8 14 15	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ ( 𝑋 · 𝑌 ) ) ) = ( 𝑋 · 𝑌 ) )
17	10 12 16	3eqtrd	⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) · ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 · 𝑌 ) )