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Theorem domnrng 17716
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 17715 . 2  |-  ( R  e. Domn  ->  R  e. NzRing )
2 nzrrng 17691 . 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 1767   Ringcrg 16986  NzRingcnzr 17687  Domncdomn 17699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-nzr 17688  df-domn 17703
This theorem is referenced by:  domneq0  17717  abvn0b  17722  fidomndrnglem  17726  fidomndrng  17727  domnchr  18336  znidomb  18367  deg1ldgdomn  22229  ply1domn  22259  proot1mul  30761  proot1hash  30765  deg1mhm  30772  domnmsuppn0  32035
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