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Theorem domnring 20060
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 20059 . 2 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2 nzrring 20025 . 2 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
31, 2syl 17 1 (𝑅 ∈ Domn → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  Ringcrg 19288  NzRingcnzr 20021  Domncdomn 20044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-nzr 20022  df-domn 20048
This theorem is referenced by:  domneq0  20061  abvn0b  20066  fidomndrnglem  20070  fidomndrng  20071  domnchr  20222  znidomb  20251  deg1ldgdomn  24693  ply1domn  24722  proot1mul  40073  proot1hash  40074  deg1mhm  40081  lidldomn1  44484  uzlidlring  44492  domnmsuppn0  44710
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