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Theorem idomcringd 13758
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
StepHypRef Expression
1 idomringd.1 . . 3  |-  ( ph  ->  R  e. IDomn )
2 df-idom 13740 . . 3  |- IDomn  =  (
CRing  i^i Domn )
31, 2eleqtrdi 2286 . 2  |-  ( ph  ->  R  e.  ( CRing  i^i Domn
) )
43elin1d 3348 1  |-  ( ph  ->  R  e.  CRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2164    i^i cin 3152   CRingccrg 13477  Domncdomn 13736  IDomncidom 13737
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-idom 13740
This theorem is referenced by:  idomringd  13759  lgseisenlem3  15136
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