Theorem domnlcan

Description: Left-cancellation law for domains. (Contributed by Thierry Arnoux, 22-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		domncan.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		domncan.1	⊢ 0 = ( 0g ‘ 𝑅 )
3		domncan.m	⊢ · = ( .r ‘ 𝑅 )
4		domncan.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ∖ { 0 } ) )
5		domncan.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		domncan.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7		domnlcan.r	⊢ ( 𝜑 → 𝑅 ∈ Domn )
8		domnlcan.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 )