Theorem idomdomd 20698

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 20668	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2849	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4134	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2119   ∩ cin 3882  CRingccrg 20206  Domncdomn 20664  IDomncidom 20665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-in 3890  df-idom 20668

This theorem is referenced by:  domnprodeq0  33357  idomrcan  33360  idomrcanOLD  33363  subridom  33367  fracfld  33392  rprmasso2  33609  1arithufdlem1  33627  1arithufdlem3  33629  dfufd2lem  33632  zringfrac  33637  deg1prod  33666  ply1dg3rt0irred  33667  m1pmeq  33668  r1pid2OLD  33692  mplidomlem  33711  vietadeg1  33762  assafld  33821  minplyirredlem  33894  minplyirred  33895  algextdeglem7  33907  algextdeglem8  33908  deg1gprod  42625  deg1pow  42626