Theorem idomringd 21259

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	. . 3 ⊢ (𝜑 → 𝑅 ∈ IDomn)
2	1	idomcringd 21258	. 2 ⊢ (𝜑 → 𝑅 ∈ CRing)
3	2	crngringd 20188	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2098  Ringcrg 20175  IDomncidom 21230

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 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-iota 6493  df-fv 6549  df-cring 20178  df-idom 21234

This theorem is referenced by:  fracfld  33015  dvdsruasso  33110  unitpidl1  33160  mxidlirredi  33205  mxidlirred  33206  rsprprmprmidlb  33261  rprmasso  33263  rprmasso2  33264  rprmirred  33266  rprmirredb  33267  r1pid2  33308  algextdeglem7  33420  idomnnzpownz  41631  deg1gprod  41640  deg1pow  41641  aks6d1c6lem3  41672