Theorem drnggrpd 20654

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 20653	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20161	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2111  Grpcgrp 18846  DivRingcdr 20645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-ov 7349  df-ring 20154  df-drng 20647

This theorem is referenced by:  drnggrp  20655  constrsdrg  33786