Theorem drngringd 20774

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 20773	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2141  Ringcrg 20270  DivRingcdr 20766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-drng 20768

This theorem is referenced by:  drnggrpd  20775  imadrhmcl  20834  frlmphl  21821  sdrgdvcl  33447  fldgensdrg  33462  primefldgen1  33469  ply1lvec  33716  m1pmeq  33742  ig1pnunit  33758  ig1pmindeg  33759  rlmdim  33868  ply1degltdimlem  33880  ply1degltdim  33881  fldgenfldext  33926  fldextrspunlsplem  33931  fldextrspunfld  33934  fldextrspunlem2  33935  fldextrspundgdvdslem  33938  fldextrspundgdvds  33939  irngnzply1lem  33948  minplyirredlem  33968  minplym1p  33971  minplynzm1p  33972  irredminply  33974  algextdeglem4  33978  algextdeglem7  33981  algextdeglem8  33982  constrsdrg  34033  2sqr3minply  34038  cos9thpiminplylem6  34045  cos9thpiminply  34046  drnginvmuld  43106  prjspner1  43169