Theorem drnggrp 20990

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 23	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ DivRing)
2	1	drnggrpd 20989	1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Grpcgrp 19144  DivRingcdr 20980

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-ring 20461  df-drng 20982

This theorem is used by:  drgextlsp  34226  qqh0  34616  qqhghm  34620  dvhvaddass  42154  dvhgrp  42164  cdlemn4  42255  fldhmf1  43140