Theorem idomdomd 20611

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 20581	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2838	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4156	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2109   ∩ cin 3902  CRingccrg 20119  Domncdomn 20577  IDomncidom 20578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-in 3910  df-idom 20581

This theorem is referenced by:  idomrcan  33218  idomrcanOLD  33221  subridom  33225  fracfld  33247  rprmasso2  33463  1arithufdlem1  33481  1arithufdlem3  33483  dfufd2lem  33486  zringfrac  33491  ply1dg3rt0irred  33518  m1pmeq  33519  r1pid2OLD  33541  assafld  33604  minplyirredlem  33677  minplyirred  33678  algextdeglem7  33690  algextdeglem8  33691  deg1gprod  42113  deg1pow  42114