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Theorem drnggrp 18524
Description: A division ring is a group. (Contributed by NM, 8-Sep-2011.)
Assertion
Ref Expression
drnggrp (𝑅 ∈ DivRing → 𝑅 ∈ Grp)

Proof of Theorem drnggrp
StepHypRef Expression
1 drngring 18523 . 2 (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
2 ringgrp 18321 . 2 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
31, 2syl 17 1 (𝑅 ∈ DivRing → 𝑅 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  Grpcgrp 17191  Ringcrg 18316  DivRingcdr 18516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-ring 18318  df-drng 18518
This theorem is referenced by:  qqh0  29162  qqhghm  29166  dvhvaddass  35200  dvhgrp  35210  cdlemn4  35301
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