Theorem drnggrpd 20873

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 20872	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20355	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Grpcgrp 19031  DivRingcdr 20864

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 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

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 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-ring 20348  df-drng 20866

This theorem is used by:  drnggrp  20874  drnglring  33813  constrsdrg  34196