Theorem idomcringd 14110

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

Hypothesis

Ref	Expression
idomringd.1	|- ( ph -> R e. IDomn )

Assertion

Ref	Expression
idomcringd	|- ( ph -> R e. CRing )

Proof of Theorem idomcringd

Step	Hyp	Ref	Expression
1		idomringd.1	. . 3 |- ( ph -> R e. IDomn )
2		df-idom 14092	. . 3 |- IDomn = ( CRing i^i Domn )
3	1, 2	eleqtrdi 2299	. 2 |- ( ph -> R e. ( CRing i^i Domn ) )
4	3	elin1d 3366	1 |- ( ph -> R e. CRing )

Syntax hints:   -> wi 4   e. wcel 2177   i^i cin 3169   CRingccrg 13829  Domncdomn 14088  IDomncidom 14089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-idom 14092

This theorem is referenced by:  idomringd  14111  lgseisenlem3  15619