Theorem idomdomd 21028

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

Hypothesis

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

Assertion

Ref	Expression
idomdomd	⊢ (𝜑 → 𝑅 ∈ Domn)

Proof of Theorem idomdomd

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	elin2d 4105	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

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:  domnprodeq0  33939  idomrcan  33942  subridom  33946  fracfld  33974  rprmasso2  34162  1arithufdlem1  34180  1arithufdlem3  34182  dfufd2lem  34185  zringfrac  34189  deg1prod  34218  ply1dg3rt0irred  34219  m1pmeq  34220  mplidomlem  34270  vietadeg1  34321  assafld  34380  minplyirredlem  34453  minplyirred  34454  algextdeglem7  34466  algextdeglem8  34467  deg1gprod  43456  deg1pow  43457