Theorem idomdomd 20671

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 20641	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2847	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4159	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3902  CRingccrg 20181  Domncdomn 20637  IDomncidom 20638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-idom 20641

This theorem is referenced by:  domnprodeq0  33370  idomrcan  33373  idomrcanOLD  33376  subridom  33380  fracfld  33402  rprmasso2  33619  1arithufdlem1  33637  1arithufdlem3  33639  dfufd2lem  33642  zringfrac  33647  deg1prod  33676  ply1dg3rt0irred  33677  m1pmeq  33678  r1pid2OLD  33702  vietadeg1  33755  assafld  33815  minplyirredlem  33888  minplyirred  33889  algextdeglem7  33901  algextdeglem8  33902  deg1gprod  42510  deg1pow  42511