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 4164	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2109   ∩ cin 3910  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 3446  df-in 3918  df-idom 20581

This theorem is referenced by:  idomrcan  33202  idomrcanOLD  33205  subridom  33209  fracfld  33231  rprmasso2  33470  1arithufdlem1  33488  1arithufdlem3  33490  dfufd2lem  33493  zringfrac  33498  ply1dg3rt0irred  33524  m1pmeq  33525  r1pid2OLD  33547  assafld  33606  minplyirredlem  33673  minplyirred  33674  algextdeglem7  33686  algextdeglem8  33687  deg1gprod  42101  deg1pow  42102