Theorem idomdomd 20697

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 20667	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2847	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4146	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3889  CRingccrg 20209  Domncdomn 20663  IDomncidom 20664

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 3432  df-in 3897  df-idom 20667

This theorem is referenced by:  domnprodeq0  33355  idomrcan  33358  idomrcanOLD  33361  subridom  33365  fracfld  33387  rprmasso2  33604  1arithufdlem1  33622  1arithufdlem3  33624  dfufd2lem  33627  zringfrac  33632  deg1prod  33661  ply1dg3rt0irred  33662  m1pmeq  33663  r1pid2OLD  33687  vietadeg1  33740  assafld  33800  minplyirredlem  33873  minplyirred  33874  algextdeglem7  33886  algextdeglem8  33887  deg1gprod  42596  deg1pow  42597