Metamath Proof Explorer

Theorem domnlcanOLD

Description: Obsolete version of domnlcan 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 )
domnlcanOLD.r ⊢ ( 𝜑 → 𝑅 ∈ Domn )
domnlcanOLD.2 ⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) )
Assertion domnlcanOLD ( 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 domnlcanOLD.r ⊢ ( 𝜑 → 𝑅 ∈ Domn )
8 domnlcanOLD.2 ⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) )
9 oveq1 ⊢ ( 𝑎 = 𝑋 → ( 𝑎 · 𝑏 ) = ( 𝑋 · 𝑏 ) )
10 oveq1 ⊢ ( 𝑎 = 𝑋 → ( 𝑎 · 𝑐 ) = ( 𝑋 · 𝑐 ) )
11 9 10 eqeq12d ⊢ ( 𝑎 = 𝑋 → ( ( 𝑎 · 𝑏 ) = ( 𝑎 · 𝑐 ) ↔ ( 𝑋 · 𝑏 ) = ( 𝑋 · 𝑐 ) ) )
12 11 imbi1d ⊢ ( 𝑎 = 𝑋 → ( ( ( 𝑎 · 𝑏 ) = ( 𝑎 · 𝑐 ) → 𝑏 = 𝑐 ) ↔ ( ( 𝑋 · 𝑏 ) = ( 𝑋 · 𝑐 ) → 𝑏 = 𝑐 ) ) )
13 oveq2 ⊢ ( 𝑏 = 𝑌 → ( 𝑋 · 𝑏 ) = ( 𝑋 · 𝑌 ) )
14 13 eqeq1d ⊢ ( 𝑏 = 𝑌 → ( ( 𝑋 · 𝑏 ) = ( 𝑋 · 𝑐 ) ↔ ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑐 ) ) )
15 eqeq1 ⊢ ( 𝑏 = 𝑌 → ( 𝑏 = 𝑐 ↔ 𝑌 = 𝑐 ) )
16 14 15 imbi12d ⊢ ( 𝑏 = 𝑌 → ( ( ( 𝑋 · 𝑏 ) = ( 𝑋 · 𝑐 ) → 𝑏 = 𝑐 ) ↔ ( ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑐 ) → 𝑌 = 𝑐 ) ) )
17 oveq2 ⊢ ( 𝑐 = 𝑍 → ( 𝑋 · 𝑐 ) = ( 𝑋 · 𝑍 ) )
18 17 eqeq2d ⊢ ( 𝑐 = 𝑍 → ( ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑐 ) ↔ ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) ) )
19 eqeq2 ⊢ ( 𝑐 = 𝑍 → ( 𝑌 = 𝑐 ↔ 𝑌 = 𝑍 ) )
20 18 19 imbi12d ⊢ ( 𝑐 = 𝑍 → ( ( ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑐 ) → 𝑌 = 𝑐 ) ↔ ( ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) → 𝑌 = 𝑍 ) ) )
21 1 2 3 isdomn4 ⊢ ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ∀ 𝑎 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = ( 𝑎 · 𝑐 ) → 𝑏 = 𝑐 ) ) )
22 7 21 sylib ⊢ ( 𝜑 → ( 𝑅 ∈ NzRing ∧ ∀ 𝑎 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = ( 𝑎 · 𝑐 ) → 𝑏 = 𝑐 ) ) )
23 22 simprd ⊢ ( 𝜑 → ∀ 𝑎 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = ( 𝑎 · 𝑐 ) → 𝑏 = 𝑐 ) )
24 12 16 20 23 4 5 6 rspc3dv ⊢ ( 𝜑 → ( ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑍 ) → 𝑌 = 𝑍 ) )
25 8 24 mpd ⊢ ( 𝜑 → 𝑌 = 𝑍 )