Theorem domnlcan

Description: Left-cancellation law for domains. (Contributed by Thierry Arnoux, 22-Mar-2025) (Proof shortened by SN, 21-Jun-2025)

	Ref	Expression
Hypotheses	domncan.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	domncan.0	⊢ 0 = ( 0g ‘ 𝑅 )
	domncan.m	⊢ · = ( .r ‘ 𝑅 )
	domncan.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ∖ { 0 } ) )
	domncan.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	domncan.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
	domncan.r	⊢ ( 𝜑 → 𝑅 ∈ Domn )
	domnlcan.1	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) )
Assertion	domnlcan	⊢ ( 𝜑 → 𝑌 = 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		domncan.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		domncan.0	⊢ 0 = ( 0g ‘ 𝑅 )
3		domncan.m	⊢ · = ( .r ‘ 𝑅 )
4		domncan.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ∖ { 0 } ) )
5		domncan.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		domncan.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7		domncan.r	⊢ ( 𝜑 → 𝑅 ∈ Domn )
8		domnlcan.1	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) )
9	1 2 3 4 5 6 7	domnlcanb	⊢ ( 𝜑 → ( ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ↔ 𝑌 = 𝑍 ) )
10	8 9	mpbid	⊢ ( 𝜑 → 𝑌 = 𝑍 )