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Metamath Proof Explorer

Theorem idomrcanOLD

Description: Obsolete version of idomrcan as of 21-Jun-2025. (Contributed by Thierry Arnoux, 22-Mar-2025) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses domncanOLD.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
domncanOLD.1 ⊢ 0 = ( 0g ‘ 𝑅 )
domncanOLD.m ⊢ · = ( .r ‘ 𝑅 )
domncanOLD.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ∖ { 0 } ) )
domncanOLD.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
domncanOLD.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
idomrcanOLD.r ⊢ ( 𝜑 → 𝑅 ∈ IDomn )
idomrcanOLD.2 ⊢ ( 𝜑 → ( 𝑌 · 𝑋 ) = ( 𝑍 · 𝑋 ) )
Assertion idomrcanOLD ( 𝜑 → 𝑌 = 𝑍 )

Proof

Step Hyp Ref Expression
1 domncanOLD.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 domncanOLD.1 ⊢ 0 = ( 0g ‘ 𝑅 )
3 domncanOLD.m ⊢ · = ( .r ‘ 𝑅 )
4 domncanOLD.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ∖ { 0 } ) )
5 domncanOLD.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 domncanOLD.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7 idomrcanOLD.r ⊢ ( 𝜑 → 𝑅 ∈ IDomn )
8 idomrcanOLD.2 ⊢ ( 𝜑 → ( 𝑌 · 𝑋 ) = ( 𝑍 · 𝑋 ) )
9 7 idomdomd ⊢ ( 𝜑 → 𝑅 ∈ Domn )
10 df-idom ⊢ IDomn = ( CRing ∩ Domn )
11 7 10 eleqtrdi ⊢ ( 𝜑 → 𝑅 ∈ ( CRing ∩ Domn ) )
12 11 elin1d ⊢ ( 𝜑 → 𝑅 ∈ CRing )
13 4 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
14 1 3 crngcom ⊢ ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) = ( 𝑌 · 𝑋 ) )
15 12 13 5 14 syl3anc ⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑌 · 𝑋 ) )
16 1 3 crngcom ⊢ ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑋 · 𝑍 ) = ( 𝑍 · 𝑋 ) )
17 12 13 6 16 syl3anc ⊢ ( 𝜑 → ( 𝑋 · 𝑍 ) = ( 𝑍 · 𝑋 ) )
18 8 15 17 3eqtr4d ⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) )
19 1 2 3 4 5 6 9 18 domnlcan ⊢ ( 𝜑 → 𝑌 = 𝑍 )