Theorem drngringd 20738

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 20737	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2107  Ringcrg 20231  DivRingcdr 20730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-drng 20732

This theorem is referenced by:  drnggrpd  20739  imadrhmcl  20799  frlmphl  21802  sdrgdvcl  33302  fldgensdrg  33317  primefldgen1  33324  ply1lvec  33586  m1pmeq  33609  ig1pnunit  33622  ig1pmindeg  33623  rlmdim  33661  ply1degltdimlem  33674  ply1degltdim  33675  fldgenfldext  33719  fldextrspunlsplem  33724  fldextrspunfld  33727  fldextrspunlem2  33728  fldextrspundgdvdslem  33731  fldextrspundgdvds  33732  irngnzply1lem  33741  minplyirredlem  33754  minplym1p  33757  irredminply  33758  algextdeglem4  33762  algextdeglem7  33765  algextdeglem8  33766  2sqr3minply  33792  drnginvmuld  42542  prjspner1  42641