Theorem drnggrp 20649

Description: A division ring is a group (closed form). (Contributed by NM, 8-Sep-2011.)

Assertion

Ref	Expression
drnggrp	⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)

Proof of Theorem drnggrp

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ DivRing)
2	1	drnggrpd 20648	1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2111  Grpcgrp 18841  DivRingcdr 20639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5239

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-ov 7344  df-ring 20148  df-drng 20641

This theorem is referenced by:  drgextlsp  33598  qqh0  33989  qqhghm  33993  dvhvaddass  41136  dvhgrp  41146  cdlemn4  41237  fldhmf1  42123