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Theorem domnring 19117
 Description: A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Assertion
Ref Expression
domnring (𝑅 ∈ Domn → 𝑅 ∈ Ring)

Proof of Theorem domnring
StepHypRef Expression
1 domnnzr 19116 . 2 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2 nzrring 19082 . 2 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
31, 2syl 17 1 (𝑅 ∈ Domn → 𝑅 ∈ Ring)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  Ringcrg 18370  NzRingcnzr 19078  Domncdomn 19101 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-nzr 19079  df-domn 19105 This theorem is referenced by:  domneq0  19118  abvn0b  19123  fidomndrnglem  19127  fidomndrng  19128  domnchr  19699  znidomb  19729  deg1ldgdomn  23658  ply1domn  23687  proot1mul  36796  proot1hash  36797  deg1mhm  36804  lidldomn1  41711  uzlidlring  41719  domnmsuppn0  41944
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