Theorem idomdomd 20748

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 20718	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2854	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4228	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2108   ∩ cin 3975  CRingccrg 20261  Domncdomn 20714  IDomncidom 20715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-in 3983  df-idom 20718

This theorem is referenced by:  idomrcan  33248  idomrcanOLD  33251  subridom  33255  fracfld  33275  rprmasso2  33519  1arithufdlem1  33537  1arithufdlem3  33539  dfufd2lem  33542  zringfrac  33547  ply1dg3rt0irred  33572  m1pmeq  33573  r1pid2OLD  33594  assafld  33650  minplyirredlem  33703  minplyirred  33704  algextdeglem7  33714  algextdeglem8  33715  deg1gprod  42097  deg1pow  42098