Theorem idomdomd 20657

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 20627	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2844	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin2d 4155	1 ⊢ (𝜑 → 𝑅 ∈ Domn)

Syntax hints:   → wi 4   ∈ wcel 2113   ∩ cin 3898  CRingccrg 20167  Domncdomn 20623  IDomncidom 20624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-in 3906  df-idom 20627

This theorem is referenced by:  domnprodeq0  33307  idomrcan  33310  idomrcanOLD  33313  subridom  33317  fracfld  33339  rprmasso2  33556  1arithufdlem1  33574  1arithufdlem3  33576  dfufd2lem  33579  zringfrac  33584  deg1prod  33613  ply1dg3rt0irred  33614  m1pmeq  33615  r1pid2OLD  33639  vietadeg1  33683  assafld  33743  minplyirredlem  33816  minplyirred  33817  algextdeglem7  33829  algextdeglem8  33830  deg1gprod  42333  deg1pow  42334