Theorem idomdomd 20635

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 20605	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2838	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4168	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2109   ∩ cin 3913  CRingccrg 20143  Domncdomn 20601  IDomncidom 20602

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 3449  df-in 3921  df-idom 20605

This theorem is referenced by:  idomrcan  33229  idomrcanOLD  33232  subridom  33236  fracfld  33258  rprmasso2  33497  1arithufdlem1  33515  1arithufdlem3  33517  dfufd2lem  33520  zringfrac  33525  ply1dg3rt0irred  33551  m1pmeq  33552  r1pid2OLD  33574  assafld  33633  minplyirredlem  33700  minplyirred  33701  algextdeglem7  33713  algextdeglem8  33714  deg1gprod  42128  deg1pow  42129