Theorem drngringd 20759

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 20758	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2108  Ringcrg 20260  DivRingcdr 20751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-drng 20753

This theorem is referenced by:  drnggrpd  20760  imadrhmcl  20820  frlmphl  21824  sdrgdvcl  33266  fldgensdrg  33281  primefldgen1  33288  ply1lvec  33550  m1pmeq  33573  ig1pnunit  33586  ig1pmindeg  33587  rlmdim  33622  ply1degltdimlem  33635  ply1degltdim  33636  fldgenfldext  33678  irngnzply1lem  33690  minplyirredlem  33703  minplym1p  33706  irredminply  33707  algextdeglem4  33711  algextdeglem7  33714  algextdeglem8  33715  2sqr3minply  33738  drnginvmuld  42482  prjspner1  42581