Theorem idomcringd 20749

Description: An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 20750. (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 20718	. . 3 ⊢ IDomn = (CRing ∩ Domn)
3	1, 2	eleqtrdi 2854	. 2 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn))
4	3	elin1d 4227	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

Syntax hints:   → wi 4   ∈ wcel 2108   ∩ cin 3975  CRingccrg 20261  Domncdomn 20714  IDomncidom 20715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-in 3983  df-idom 20718

This theorem is referenced by:  idomringd  20750  subridom  33255  fracfld  33275  idomsubr  33276  dvdsruasso2  33379  rsprprmprmidlb  33516  rprmasso  33518  rprmasso2  33519  rprmirredlem  33523  rprmirred  33524  rprmirredb  33525  1arithidomlem1  33528  1arithidom  33530  1arithufdlem1  33537  1arithufdlem3  33539  1arithufdlem4  33540  dfufd2lem  33542  zringfrac  33547  ply1dg3rt0irred  33572  assafld  33650  unitscyglem5  42156  aks5lem7  42157