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Theorem domnrng 16037
Description: A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Assertion
Ref Expression
domnrng  |-  ( R  e. Domn  ->  R  e.  Ring )

Proof of Theorem domnrng
StepHypRef Expression
1 domnnzr 16036 . 2  |-  ( R  e. Domn  ->  R  e. NzRing )
2 nzrrng 16013 . 2  |-  ( R  e. NzRing  ->  R  e.  Ring )
31, 2syl 15 1  |-  ( R  e. Domn  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   Ringcrg 15337  NzRingcnzr 16009  Domncdomn 16021
This theorem is referenced by:  domneq0  16038  abvn0b  16043  fidomndrnglem  16047  fidomndrng  16048  domnchr  16486  znidomb  16515  deg1ldgdomn  19480  ply1domn  19509  proot1mul  27515  proot1hash  27519  deg1mhm  27526
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-nzr 16010  df-domn 16025
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