Theorem drnggrp 20697

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 20696	1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2108  Grpcgrp 18914  DivRingcdr 20687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6483  df-fv 6538  df-ov 7406  df-ring 20193  df-drng 20689

This theorem is referenced by:  drgextlsp  33579  qqh0  33961  qqhghm  33965  dvhvaddass  41062  dvhgrp  41072  cdlemn4  41163  fldhmf1  42049