Theorem idomdomd 20772

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 20742	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2872	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4157	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2142   ∩ cin 3903  CRingccrg 20280  Domncdomn 20738  IDomncidom 20739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-idom 20742

This theorem is referenced by:  domnprodeq0  33457  idomrcan  33460  idomrcanOLD  33463  subridom  33467  fracfld  33492  rprmasso2  33719  1arithufdlem1  33737  1arithufdlem3  33739  dfufd2lem  33742  zringfrac  33747  deg1prod  33776  ply1dg3rt0irred  33777  m1pmeq  33778  r1pid2OLD  33802  mplidomlem  33821  vietadeg1  33872  assafld  33931  minplyirredlem  34004  minplyirred  34005  algextdeglem7  34017  algextdeglem8  34018  deg1gprod  42754  deg1pow  42755