Theorem drnggrpd 20657

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 20656	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20164	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2113  Grpcgrp 18850  DivRingcdr 20648

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 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

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 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6444  df-fv 6496  df-ov 7357  df-ring 20157  df-drng 20650

This theorem is referenced by:  drnggrp  20658  constrsdrg  33811