Theorem drngringd 20682

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 20681	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2114  Ringcrg 20180  DivRingcdr 20674

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-drng 20676

This theorem is referenced by:  drnggrpd  20683  imadrhmcl  20742  frlmphl  21748  sdrgdvcl  33392  fldgensdrg  33407  primefldgen1  33414  ply1lvec  33651  m1pmeq  33677  ig1pnunit  33693  ig1pmindeg  33694  rlmdim  33786  ply1degltdimlem  33799  ply1degltdim  33800  fldgenfldext  33845  fldextrspunlsplem  33850  fldextrspunfld  33853  fldextrspunlem2  33854  fldextrspundgdvdslem  33857  fldextrspundgdvds  33858  irngnzply1lem  33867  minplyirredlem  33887  minplym1p  33890  minplynzm1p  33891  irredminply  33893  algextdeglem4  33897  algextdeglem7  33900  algextdeglem8  33901  constrsdrg  33952  2sqr3minply  33957  cos9thpiminplylem6  33964  cos9thpiminply  33965  drnginvmuld  42894  prjspner1  42981