Theorem idomringd 32370

Description: An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)

Hypothesis

Ref	Expression
idomringd.1	⊢ (𝜑 → 𝑅 ∈ IDomn)

Assertion

Ref	Expression
idomringd	⊢ (𝜑 → 𝑅 ∈ Ring)

Proof of Theorem idomringd

Step	Hyp	Ref	Expression
1		idomringd.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ IDomn)
2		df-idom 20900	. . . 4 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2843	. . 3 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin1d 4198	. 2 ⊢ (𝜑 → 𝑅 ∈ CRing)
5	4	crngringd 20068	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2106   ∩ cin 3947  Ringcrg 20055  CRingccrg 20056  Domncdomn 20895  IDomncidom 20896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-cring 20058  df-idom 20900

This theorem is referenced by:  dvdsruasso  32485  unitpidl1  32537  mxidlirredi  32582  mxidlirred  32583