Theorem drngmullcan

Description: Cancellation of a nonzero factor on the left for multiplication. ( mulcanad 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 )
	drngmullcan.2	⊢ ( 𝜑 → ( 𝑍 · 𝑋 ) = ( 𝑍 · 𝑌 ) )
Assertion	drngmullcan	⊢ ( 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		drngmullcan.2	⊢ ( 𝜑 → ( 𝑍 · 𝑋 ) = ( 𝑍 · 𝑌 ) )
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 10 5 6 12 9	domnlcan	⊢ ( 𝜑 → 𝑋 = 𝑌 )