Theorem drngringd 20512

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 20511	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2105  Ringcrg 20131  DivRingcdr 20504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-drng 20506

This theorem is referenced by:  drnggrpd  20513  imadrhmcl  20560  frlmphl  21559  sdrgdvcl  32682  fldgensdrg  32689  primefldgen1  32696  ply1lvec  32927  ig1pnunit  32961  ig1pmindeg  32962  rlmdim  32997  ply1degltdimlem  33010  ply1degltdim  33011  irngnzply1lem  33058  minplyirredlem  33073  minplym1p  33076  algextdeglem4  33080  algextdeglem7  33083  algextdeglem8  33084  drngmulcanad  41416  drngmulcan2ad  41417  drnginvmuld  41418  prjspner1  41683