Theorem idomcringd 20672

Description: An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 20673. (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 20641	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2847	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin1d 4158	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3902  CRingccrg 20181  Domncdomn 20637  IDomncidom 20638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-idom 20641

This theorem is referenced by:  idomringd  20673  domnprodeq0  33370  subridom  33380  fracfld  33402  idomsubr  33403  dvdsruasso2  33479  rsprprmprmidlb  33616  rprmasso  33618  rprmasso2  33619  rprmirredlem  33623  rprmirred  33624  rprmirredb  33625  1arithidomlem1  33628  1arithidom  33630  1arithufdlem1  33637  1arithufdlem3  33639  1arithufdlem4  33640  dfufd2lem  33642  zringfrac  33647  deg1prod  33676  ply1dg3rt0irred  33677  vietadeg1  33755  vietalem  33756  vieta  33757  assafld  33815  fldextrspunfld  33854  unitscyglem5  42569  aks5lem7  42570