Theorem fldidom

Description: A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015)

	Ref	Expression
Assertion	fldidom	⊢ ( 𝑅 ∈ Field → 𝑅 ∈ IDomn )

Proof

Step	Hyp	Ref	Expression
1		isfld	⊢ ( 𝑅 ∈ Field ↔ ( 𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing ) )
2	1	simprbi	⊢ ( 𝑅 ∈ Field → 𝑅 ∈ CRing )
3	1	simplbi	⊢ ( 𝑅 ∈ Field → 𝑅 ∈ DivRing )
4		drngdomn	⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Domn )
5	3 4	syl	⊢ ( 𝑅 ∈ Field → 𝑅 ∈ Domn )
6		isidom	⊢ ( 𝑅 ∈ IDomn ↔ ( 𝑅 ∈ CRing ∧ 𝑅 ∈ Domn ) )
7	2 5 6	sylanbrc	⊢ ( 𝑅 ∈ Field → 𝑅 ∈ IDomn )