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Theorem idomringd 13759
Description: An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
Hypothesis
Ref Expression
idomringd.1  |-  ( ph  ->  R  e. IDomn )
Assertion
Ref Expression
idomringd  |-  ( ph  ->  R  e.  Ring )

Proof of Theorem idomringd
StepHypRef Expression
1 idomringd.1 . . 3  |-  ( ph  ->  R  e. IDomn )
21idomcringd 13758 . 2  |-  ( ph  ->  R  e.  CRing )
32crngringd 13489 1  |-  ( ph  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2164   Ringcrg 13476  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-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5207  df-fv 5254  df-cring 13479  df-idom 13740
This theorem is referenced by:  lgseisenlem3  15136
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