Theorem drnggrpd 40173

Description: A division ring is a group. (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 40172	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 19707	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2108  Grpcgrp 18492  DivRingcdr 19906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-ring 19700  df-drng 19908

This theorem is referenced by: (None)