Theorem drnggrpd 20905

Description: A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
drngringd.1	⊢ (𝜑 → 𝑅 ∈ DivRing)

Assertion

Ref	Expression
drnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem drnggrpd

Step	Hyp	Ref	Expression
1		drngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ DivRing)
2	1	drngringd 20904	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20387	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Grpcgrp 19063  DivRingcdr 20896

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 2734  ax-nul 5267

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 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420  df-ring 20380  df-drng 20898

This theorem is used by:  drnggrp  20906  drnglring  33910  constrsdrg  34293