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Theorem drnggrpd 20647
Description: A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)
Hypothesis
Ref Expression
drngringd.1 (𝜑𝑅 ∈ DivRing)
Assertion
Ref Expression
drnggrpd (𝜑𝑅 ∈ Grp)

Proof of Theorem drnggrpd
StepHypRef Expression
1 drngringd.1 . . 3 (𝜑𝑅 ∈ DivRing)
21drngringd 20646 . 2 (𝜑𝑅 ∈ Ring)
32ringgrpd 20196 1 (𝜑𝑅 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Grpcgrp 18904  DivRingcdr 20638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-ring 20189  df-drng 20640
This theorem is referenced by:  drnggrp  20648
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