Theorem drngringd 20640

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 20639	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2109  Ringcrg 20136  DivRingcdr 20632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-drng 20634

This theorem is referenced by:  drnggrpd  20641  imadrhmcl  20700  frlmphl  21706  sdrgdvcl  33248  fldgensdrg  33263  primefldgen1  33270  ply1lvec  33504  m1pmeq  33528  ig1pnunit  33542  ig1pmindeg  33543  rlmdim  33581  ply1degltdimlem  33594  ply1degltdim  33595  fldgenfldext  33639  fldextrspunlsplem  33644  fldextrspunfld  33647  fldextrspunlem2  33648  fldextrspundgdvdslem  33651  fldextrspundgdvds  33652  irngnzply1lem  33661  minplyirredlem  33676  minplym1p  33679  minplynzm1p  33680  irredminply  33682  algextdeglem4  33686  algextdeglem7  33689  algextdeglem8  33690  constrsdrg  33741  2sqr3minply  33746  cos9thpiminplylem6  33753  cos9thpiminply  33754  drnginvmuld  42500  prjspner1  42599