Theorem idomdomd 21262

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 21239	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2839	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4201	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2098   ∩ cin 3948  CRingccrg 20181  Domncdomn 21234  IDomncidom 21235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-idom 21239

This theorem is referenced by:  idomrcan  32972  subridom  32978  fracfld  33019  rprmasso2  33268  zringfrac  33277  m1pmeq  33294  r1pid2  33312  minplyirredlem  33413  minplyirred  33414  deg1gprod  41644  deg1pow  41645