Theorem idomcringd 20643

Description: An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 20644. (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 20612	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2839	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin1d 4170	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

Syntax hints:   → wi 4   ∈ wcel 2109   ∩ cin 3916  CRingccrg 20150  Domncdomn 20608  IDomncidom 20609

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3924  df-idom 20612

This theorem is referenced by:  idomringd  20644  subridom  33243  fracfld  33265  idomsubr  33266  dvdsruasso2  33364  rsprprmprmidlb  33501  rprmasso  33503  rprmasso2  33504  rprmirredlem  33508  rprmirred  33509  rprmirredb  33510  1arithidomlem1  33513  1arithidom  33515  1arithufdlem1  33522  1arithufdlem3  33524  1arithufdlem4  33525  dfufd2lem  33527  zringfrac  33532  ply1dg3rt0irred  33558  assafld  33640  fldextrspunfld  33678  unitscyglem5  42194  aks5lem7  42195