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Theorem domnring 20695
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 20694 . 2 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2 nzrring 20660 . 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 19894  NzRingcnzr 20656  Domncdomn 20679
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 2709  ax-nul 5262
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 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-ral 3064  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-iota 6444  df-fv 6500  df-ov 7353  df-nzr 20657  df-domn 20683
This theorem is referenced by:  domneq0  20696  abvn0b  20701  fidomndrnglem  20706  fidomndrng  20707  domnchr  20864  znidomb  20897  deg1ldgdomn  25387  ply1domn  25416  isdomn4  40555  proot1mul  41428  proot1hash  41429  deg1mhm  41436  lidldomn1  46010  uzlidlring  46018  domnmsuppn0  46236
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