Theorem drngmulrcan

Description: Cancellation of a nonzero factor on the right for multiplication. ( mulcan2ad analog). (Contributed by SN, 14-Aug-2024) (Proof shortened by SN, 25-Jun-2025)

	Ref	Expression
Hypotheses	drngmullcan.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	drngmullcan.0	⊢ 0 = ( 0g ‘ 𝑅 )
	drngmullcan.t	⊢ · = ( .r ‘ 𝑅 )
	drngmullcan.r	⊢ ( 𝜑 → 𝑅 ∈ DivRing )
	drngmullcan.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	drngmullcan.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	drngmullcan.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
	drngmullcan.1	⊢ ( 𝜑 → 𝑍 ≠ 0 )
	drngmulrcan.2	⊢ ( 𝜑 → ( 𝑋 · 𝑍 ) = ( 𝑌 · 𝑍 ) )
Assertion	drngmulrcan	⊢ ( 𝜑 → 𝑋 = 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		drngmullcan.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		drngmullcan.0	⊢ 0 = ( 0g ‘ 𝑅 )
3		drngmullcan.t	⊢ · = ( .r ‘ 𝑅 )
4		drngmullcan.r	⊢ ( 𝜑 → 𝑅 ∈ DivRing )
5		drngmullcan.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		drngmullcan.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		drngmullcan.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8		drngmullcan.1	⊢ ( 𝜑 → 𝑍 ≠ 0 )
9		drngmulrcan.2	⊢ ( 𝜑 → ( 𝑋 · 𝑍 ) = ( 𝑌 · 𝑍 ) )
10	7 8	eldifsnd	⊢ ( 𝜑 → 𝑍 ∈ ( 𝐵 ∖ { 0 } ) )
11		drngdomn	⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Domn )
12	4 11	syl	⊢ ( 𝜑 → 𝑅 ∈ Domn )
13	1 2 3 5 6 10 12 9	domnrcan	⊢ ( 𝜑 → 𝑋 = 𝑌 )