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Theorem domnrng 16357
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 16356 . 2  |-  ( R  e. Domn  ->  R  e. NzRing )
2 nzrrng 16333 . 2  |-  ( R  e. NzRing  ->  R  e.  Ring )
31, 2syl 16 1  |-  ( R  e. Domn  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   Ringcrg 15661  NzRingcnzr 16329  Domncdomn 16341
This theorem is referenced by:  domneq0  16358  abvn0b  16363  fidomndrnglem  16367  fidomndrng  16368  domnchr  16814  znidomb  16843  deg1ldgdomn  20018  ply1domn  20047  proot1mul  27493  proot1hash  27497  deg1mhm  27504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-nul 4339
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085  df-nzr 16330  df-domn 16345
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