Theorem isdomn2

Description: A ring is a domain iff all nonzero elements are regular elements. (Contributed by Mario Carneiro, 28-Mar-2015) (Proof shortened by SN, 21-Jun-2025)

	Ref	Expression
Hypotheses	isdomn2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	isdomn2.t	⊢ 𝐸 = ( RLReg ‘ 𝑅 )
	isdomn2.z	⊢ 0 = ( 0g ‘ 𝑅 )
Assertion	isdomn2	⊢ ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		isdomn2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		isdomn2.t	⊢ 𝐸 = ( RLReg ‘ 𝑅 )
3		isdomn2.z	⊢ 0 = ( 0g ‘ 𝑅 )
4		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
5	1 4 3	isdomn	⊢ ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) )
6		eldifi	⊢ ( 𝑥 ∈ ( 𝐵 ∖ { 0 } ) → 𝑥 ∈ 𝐵 )
7	2 1 4 3	isrrg	⊢ ( 𝑥 ∈ 𝐸 ↔ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
8	7	baib	⊢ ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐸 ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
9	6 8	syl	⊢ ( 𝑥 ∈ ( 𝐵 ∖ { 0 } ) → ( 𝑥 ∈ 𝐸 ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
10	9	ralbiia	⊢ ( ∀ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) 𝑥 ∈ 𝐸 ↔ ∀ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) )
11		dfss3	⊢ ( ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 ↔ ∀ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) 𝑥 ∈ 𝐸 )
12		isdomn5	⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ↔ ∀ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) )
13	10 11 12	3bitr4ri	⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ↔ ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 )
14	13	anbi2i	⊢ ( ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) ↔ ( 𝑅 ∈ NzRing ∧ ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 ) )
15	5 14	bitri	⊢ ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 ) )