Theorem fldidom

Description: A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015) (Proof shortened by SN, 11-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		drngdomn	⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Domn )
2	1	anim1ci	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing ) → ( 𝑅 ∈ CRing ∧ 𝑅 ∈ Domn ) )
3		isfld	⊢ ( 𝑅 ∈ Field ↔ ( 𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing ) )
4		isidom	⊢ ( 𝑅 ∈ IDomn ↔ ( 𝑅 ∈ CRing ∧ 𝑅 ∈ Domn ) )
5	2 3 4	3imtr4i	⊢ ( 𝑅 ∈ Field → 𝑅 ∈ IDomn )