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Metamath Proof Explorer

Theorem domnrcanb

Description: Right-cancellation law for domains, biconditional version of domnrcan . (Contributed by SN, 21-Jun-2025)

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

Proof

Step Hyp Ref Expression
1 domnrcan.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 domnrcan.0 ⊢ 0 = ( 0g ‘ 𝑅 )
3 domnrcan.m ⊢ · = ( .r ‘ 𝑅 )
4 domnrcan.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 domnrcan.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 domnrcan.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝐵 ∖ { 0 } ) )
7 domnrcan.r ⊢ ( 𝜑 → 𝑅 ∈ Domn )
8 oveq1 ⊢ ( 𝑎 = 𝑋 → ( 𝑎 · 𝑐 ) = ( 𝑋 · 𝑐 ) )
9 8 eqeq1d ⊢ ( 𝑎 = 𝑋 → ( ( 𝑎 · 𝑐 ) = ( 𝑏 · 𝑐 ) ↔ ( 𝑋 · 𝑐 ) = ( 𝑏 · 𝑐 ) ) )
10 eqeq1 ⊢ ( 𝑎 = 𝑋 → ( 𝑎 = 𝑏 ↔ 𝑋 = 𝑏 ) )
11 9 10 imbi12d ⊢ ( 𝑎 = 𝑋 → ( ( ( 𝑎 · 𝑐 ) = ( 𝑏 · 𝑐 ) → 𝑎 = 𝑏 ) ↔ ( ( 𝑋 · 𝑐 ) = ( 𝑏 · 𝑐 ) → 𝑋 = 𝑏 ) ) )
12 oveq1 ⊢ ( 𝑏 = 𝑌 → ( 𝑏 · 𝑐 ) = ( 𝑌 · 𝑐 ) )
13 12 eqeq2d ⊢ ( 𝑏 = 𝑌 → ( ( 𝑋 · 𝑐 ) = ( 𝑏 · 𝑐 ) ↔ ( 𝑋 · 𝑐 ) = ( 𝑌 · 𝑐 ) ) )
14 eqeq2 ⊢ ( 𝑏 = 𝑌 → ( 𝑋 = 𝑏 ↔ 𝑋 = 𝑌 ) )
15 13 14 imbi12d ⊢ ( 𝑏 = 𝑌 → ( ( ( 𝑋 · 𝑐 ) = ( 𝑏 · 𝑐 ) → 𝑋 = 𝑏 ) ↔ ( ( 𝑋 · 𝑐 ) = ( 𝑌 · 𝑐 ) → 𝑋 = 𝑌 ) ) )
16 oveq2 ⊢ ( 𝑐 = 𝑍 → ( 𝑋 · 𝑐 ) = ( 𝑋 · 𝑍 ) )
17 oveq2 ⊢ ( 𝑐 = 𝑍 → ( 𝑌 · 𝑐 ) = ( 𝑌 · 𝑍 ) )
18 16 17 eqeq12d ⊢ ( 𝑐 = 𝑍 → ( ( 𝑋 · 𝑐 ) = ( 𝑌 · 𝑐 ) ↔ ( 𝑋 · 𝑍 ) = ( 𝑌 · 𝑍 ) ) )
19 18 imbi1d ⊢ ( 𝑐 = 𝑍 → ( ( ( 𝑋 · 𝑐 ) = ( 𝑌 · 𝑐 ) → 𝑋 = 𝑌 ) ↔ ( ( 𝑋 · 𝑍 ) = ( 𝑌 · 𝑍 ) → 𝑋 = 𝑌 ) ) )
20 1 2 3 isdomn4r ⊢ ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ ( 𝐵 ∖ { 0 } ) ( ( 𝑎 · 𝑐 ) = ( 𝑏 · 𝑐 ) → 𝑎 = 𝑏 ) ) )
21 7 20 sylib ⊢ ( 𝜑 → ( 𝑅 ∈ NzRing ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ ( 𝐵 ∖ { 0 } ) ( ( 𝑎 · 𝑐 ) = ( 𝑏 · 𝑐 ) → 𝑎 = 𝑏 ) ) )
22 21 simprd ⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ ( 𝐵 ∖ { 0 } ) ( ( 𝑎 · 𝑐 ) = ( 𝑏 · 𝑐 ) → 𝑎 = 𝑏 ) )
23 11 15 19 22 4 5 6 rspc3dv ⊢ ( 𝜑 → ( ( 𝑋 · 𝑍 ) = ( 𝑌 · 𝑍 ) → 𝑋 = 𝑌 ) )
24 oveq1 ⊢ ( 𝑋 = 𝑌 → ( 𝑋 · 𝑍 ) = ( 𝑌 · 𝑍 ) )
25 23 24 impbid1 ⊢ ( 𝜑 → ( ( 𝑋 · 𝑍 ) = ( 𝑌 · 𝑍 ) ↔ 𝑋 = 𝑌 ) )