Theorem drngringd 21039

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 21038	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Ringcrg 20510  DivRingcdr 21031

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-drng 21033

This theorem is used by:  drnggrpd  21040  imadrhmcl  21105  frlmphl  22142  sdrgdvcl  33965  fldgensdrg  33980  primefldgen1  33987  ply1lvec  34194  m1pmeq  34220  ig1pnunit  34236  ig1pmindeg  34237  rlmdim  34353  ply1degltdimlem  34365  ply1degltdim  34366  fedgmul  34374  fldgenfldext  34411  fldextrspunlsplem  34416  fldextrspunfld  34419  fldextrspunlem2  34420  fldextrspundgdvdslem  34423  fldextrspundgdvds  34424  irngnzply1lem  34433  minplyirredlem  34453  minplym1p  34456  minplynzm1p  34457  irredminply  34459  algextdeglem4  34463  algextdeglem7  34466  algextdeglem8  34467  constrsdrg  34518  2sqr3minply  34523  cos9thpiminplylem6  34530  cos9thpiminply  34531  drnginvmuld  43854  frlmnzcoordsca  43924  prjspnequivnorm  43926  prjspnnorm  43927