Theorem idomcringd 21029

Description: An integral domain is a commutative ring with unity. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 21030. (Proof shortened by SN, 14-May-2025.)

Hypothesis

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

Assertion

Ref	Expression
idomcringd	⊢ (𝜑 → 𝑅 ∈ CRing)

Proof of Theorem idomcringd

Step	Hyp	Ref	Expression
1		idomringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ IDomn)
2		df-idom 20999	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2874	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin1d 4104	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ∩ cin 4092  CRingccrg 20511  Domncdomn 20995  IDomncidom 20996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-idom 20999

This theorem is used by:  idomringd  21030  domnprodeq0  33939  subridom  33946  ply1edom  33959  edompidlem  33960  fracfld  33974  idomsubr  33975  dvdsruasso2  34045  mxidlirredi  34100  mxidlirred  34101  rprmasso  34161  rprmasso2  34162  rprmirredlem  34166  rprmirred  34167  rprmirredb  34168  1arithidomlem1  34171  1arithidom  34173  pidufd  34179  1arithufdlem1  34180  1arithufdlem3  34182  1arithufdlem4  34183  dfufd2lem  34185  zringfrac  34189  deg1prod  34218  ply1dg3rt0irred  34219  mplidomlem  34270  vietadeg1  34321  vietalem  34322  vieta  34323  assafld  34380  fldextrspunfld  34419  unitscyglem5  43515  aks5lem7  43516