Theorem idomringd 21218

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 21195	. . . 4 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2837	. . 3 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin1d 4193	. 2 ⊢ (𝜑 → 𝑅 ∈ CRing)
5	4	crngringd 20151	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2098   ∩ cin 3942  Ringcrg 20138  CRingccrg 20139  Domncdomn 21190  IDomncidom 21191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-cring 20141  df-idom 21195

This theorem is referenced by:  dvdsruasso  32996  unitpidl1  33048  mxidlirredi  33093  mxidlirred  33094  r1pid2  33184  algextdeglem7  33300  idomnnzpownz  41508  deg1gprod  41517  deg1pow  41518