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Metamath Proof Explorer

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 )