This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem domnlcanbOLD

Description: Obsolete version of domnlcanb as of 21-Jun-2025. (Contributed by Thierry Arnoux, 8-Jun-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 )
domnlcanbOLD.r ⊢ ( 𝜑 → 𝑅 ∈ Domn )
Assertion domnlcanbOLD ( 𝜑 → ( ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ↔ 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 domnlcanbOLD.r ⊢ ( 𝜑 → 𝑅 ∈ Domn )
8 4 adantr ⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ) → 𝑋 ∈ ( 𝐵 ∖ { 0 } ) )
9 5 adantr ⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ) → 𝑌 ∈ 𝐵 )
10 6 adantr ⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ) → 𝑍 ∈ 𝐵 )
11 7 adantr ⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ) → 𝑅 ∈ Domn )
12 simpr ⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ) → ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) )
13 1 2 3 8 9 10 11 12 domnlcan ⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ) → 𝑌 = 𝑍 )
14 simpr ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑌 = 𝑍 )
15 14 oveq2d ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) )
16 13 15 impbida ⊢ ( 𝜑 → ( ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ↔ 𝑌 = 𝑍 ) )