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Theorem domnring 20782
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 20781 . 2 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2 nzrring 20747 . 2 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
31, 2syl 17 1 (𝑅 ∈ Domn → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Ringcrg 19969  NzRingcnzr 20743  Domncdomn 20766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-nzr 20744  df-domn 20770
This theorem is referenced by:  domneq0  20783  abvn0b  20788  fidomndrnglem  20793  fidomndrng  20794  domnchr  20951  znidomb  20984  deg1ldgdomn  25475  ply1domn  25504  isdomn4  40670  proot1mul  41569  proot1hash  41570  deg1mhm  41577  lidldomn1  46305  uzlidlring  46313  domnmsuppn0  46531
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