Metamath Proof Explorer


Theorem domnnzr

Description: A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015)

Ref Expression
Assertion domnnzr ( 𝑅 ∈ Domn → 𝑅 ∈ NzRing )

Proof

Step Hyp Ref Expression
1 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2 eqid ( .r𝑅 ) = ( .r𝑅 )
3 eqid ( 0g𝑅 ) = ( 0g𝑅 )
4 1 2 3 isdomn ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r𝑅 ) 𝑦 ) = ( 0g𝑅 ) → ( 𝑥 = ( 0g𝑅 ) ∨ 𝑦 = ( 0g𝑅 ) ) ) ) )
5 4 simplbi ( 𝑅 ∈ Domn → 𝑅 ∈ NzRing )