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 ) )