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Theorem domnrng 17476
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 17475 . 2  |-  ( R  e. Domn  ->  R  e. NzRing )
2 nzrrng 17451 . 2  |-  ( R  e. NzRing  ->  R  e.  Ring )
31, 2syl 16 1  |-  ( R  e. Domn  ->  R  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   Ringcrg 16753  NzRingcnzr 17447  Domncdomn 17459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4521
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-iota 5481  df-fv 5526  df-ov 6195  df-nzr 17448  df-domn 17463
This theorem is referenced by:  domneq0  17477  abvn0b  17482  fidomndrnglem  17486  fidomndrng  17487  domnchr  18074  znidomb  18105  deg1ldgdomn  21683  ply1domn  21713  proot1mul  29704  proot1hash  29708  deg1mhm  29715  domnmsuppn0  30922
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