Theorem drngringd 20714

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 20713	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2114  Ringcrg 20214  DivRingcdr 20706

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-drng 20708

This theorem is referenced by:  drnggrpd  20715  imadrhmcl  20774  frlmphl  21761  sdrgdvcl  33360  fldgensdrg  33375  primefldgen1  33382  ply1lvec  33619  m1pmeq  33645  ig1pnunit  33661  ig1pmindeg  33662  rlmdim  33754  ply1degltdimlem  33766  ply1degltdim  33767  fldgenfldext  33812  fldextrspunlsplem  33817  fldextrspunfld  33820  fldextrspunlem2  33821  fldextrspundgdvdslem  33824  fldextrspundgdvds  33825  irngnzply1lem  33834  minplyirredlem  33854  minplym1p  33857  minplynzm1p  33858  irredminply  33860  algextdeglem4  33864  algextdeglem7  33867  algextdeglem8  33868  constrsdrg  33919  2sqr3minply  33924  cos9thpiminplylem6  33931  cos9thpiminply  33932  drnginvmuld  42972  prjspner1  43059