Theorem idomcringd 20612

Description: An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 20613. (Proof shortened by SN, 14-May-2025.)

Hypothesis

Ref	Expression
idomringd.1	⊢ (𝜑 → 𝑅 ∈ IDomn)

Assertion

Ref	Expression
idomcringd	⊢ (𝜑 → 𝑅 ∈ CRing)

Proof of Theorem idomcringd

Step	Hyp	Ref	Expression
1		idomringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ IDomn)
2		df-idom 20581	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2838	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin1d 4163	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

Syntax hints:   → wi 4   ∈ wcel 2109   ∩ cin 3910  CRingccrg 20119  Domncdomn 20577  IDomncidom 20578

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 3446  df-in 3918  df-idom 20581

This theorem is referenced by:  idomringd  20613  subridom  33209  fracfld  33231  idomsubr  33232  dvdsruasso2  33330  rsprprmprmidlb  33467  rprmasso  33469  rprmasso2  33470  rprmirredlem  33474  rprmirred  33475  rprmirredb  33476  1arithidomlem1  33479  1arithidom  33481  1arithufdlem1  33488  1arithufdlem3  33490  1arithufdlem4  33491  dfufd2lem  33493  zringfrac  33498  ply1dg3rt0irred  33524  assafld  33606  fldextrspunfld  33644  unitscyglem5  42160  aks5lem7  42161