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Theorem domnring 20724
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 20723 . 2 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2 nzrring 20533 . 2 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
31, 2syl 17 1 (𝑅 ∈ Domn → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Ringcrg 20251  NzRingcnzr 20529  Domncdomn 20709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-nzr 20530  df-domn 20712
This theorem is referenced by:  domneq0  20725  isdomn4  20733  domneq0r  20741  fidomndrnglem  20790  fidomndrng  20791  abvtrivg  20851  domnchr  21565  znidomb  21598  deg1ldgdomn  26148  deg1mul  26169  ply1domn  26178  r1pid2  26216  domnprodn0  33262  r1peuqusdeg1  35628  deg1pow  42123  domnexpgn0cl  42510  fidomncyc  42522  proot1mul  43183  proot1hash  43184  deg1mhm  43189  lidldomn1  48075  uzlidlring  48079  domnmsuppn0  48214
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