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Metamath Proof Explorer

Theorem idomrcan

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

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

Proof

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