Theorem drnggrp 20597

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 20596	1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2098  Grpcgrp 18863  DivRingcdr 20587

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 2697  ax-nul 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-ring 20140  df-drng 20589

This theorem is referenced by:  drgextlsp  33198  qqh0  33494  qqhghm  33498  dvhvaddass  40481  dvhgrp  40491  cdlemn4  40582  fldhmf1  41472