Theorem idomdomd 20887

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 20858	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2870	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4151	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ∩ cin 3898  CRingccrg 20373  Domncdomn 20854  IDomncidom 20855

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 3906  df-idom 20858

This theorem is used by:  domnprodeq0  33719  idomrcan  33722  subridom  33726  fracfld  33749  rprmasso2  33936  1arithufdlem1  33954  1arithufdlem3  33956  dfufd2lem  33959  zringfrac  33964  deg1prod  33993  ply1dg3rt0irred  33994  m1pmeq  33995  mplidomlem  34037  vietadeg1  34088  assafld  34147  minplyirredlem  34220  minplyirred  34221  algextdeglem7  34233  algextdeglem8  34234  deg1gprod  43006  deg1pow  43007