Theorem drnggrp 20739

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 20738	1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2108  Grpcgrp 18951  DivRingcdr 20729

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 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-ring 20232  df-drng 20731

This theorem is referenced by:  drgextlsp  33644  qqh0  33985  qqhghm  33989  dvhvaddass  41099  dvhgrp  41109  cdlemn4  41200  fldhmf1  42091