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Theorem drngringd 39472
 Description: A division ring is a ring. (Contributed by SN, 16-May-2024.)
Hypothesis
Ref Expression
drngringd.1 (𝜑𝑅 ∈ DivRing)
Assertion
Ref Expression
drngringd (𝜑𝑅 ∈ Ring)

Proof of Theorem drngringd
StepHypRef Expression
1 drngringd.1 . 2 (𝜑𝑅 ∈ DivRing)
2 drngring 19506 . 2 (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
31, 2syl 17 1 (𝜑𝑅 ∈ Ring)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  Ringcrg 19294  DivRingcdr 19499 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 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-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-drng 19501 This theorem is referenced by:  drnggrpd  39473
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