Theorem drngringd 20705

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

Hypothesis

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

Assertion

Ref	Expression
drngringd	⊢ (𝜑 → 𝑅 ∈ Ring)

Proof of Theorem drngringd

Step	Hyp	Ref	Expression
1		drngringd.1	. 2 ⊢ (𝜑 → 𝑅 ∈ DivRing)
2		drngring 20704	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2114  Ringcrg 20205  DivRingcdr 20697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-drng 20699

This theorem is referenced by:  drnggrpd  20706  imadrhmcl  20765  frlmphl  21771  sdrgdvcl  33375  fldgensdrg  33390  primefldgen1  33397  ply1lvec  33634  m1pmeq  33660  ig1pnunit  33676  ig1pmindeg  33677  rlmdim  33769  ply1degltdimlem  33782  ply1degltdim  33783  fldgenfldext  33828  fldextrspunlsplem  33833  fldextrspunfld  33836  fldextrspunlem2  33837  fldextrspundgdvdslem  33840  fldextrspundgdvds  33841  irngnzply1lem  33850  minplyirredlem  33870  minplym1p  33873  minplynzm1p  33874  irredminply  33876  algextdeglem4  33880  algextdeglem7  33883  algextdeglem8  33884  constrsdrg  33935  2sqr3minply  33940  cos9thpiminplylem6  33947  cos9thpiminply  33948  drnginvmuld  42986  prjspner1  43073