Theorem drngringd 20656

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 20655	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2113  Ringcrg 20155  DivRingcdr 20648

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6444  df-fv 6496  df-drng 20650

This theorem is referenced by:  drnggrpd  20657  imadrhmcl  20716  frlmphl  21722  sdrgdvcl  33274  fldgensdrg  33289  primefldgen1  33296  ply1lvec  33531  m1pmeq  33556  ig1pnunit  33570  ig1pmindeg  33571  rlmdim  33645  ply1degltdimlem  33658  ply1degltdim  33659  fldgenfldext  33704  fldextrspunlsplem  33709  fldextrspunfld  33712  fldextrspunlem2  33713  fldextrspundgdvdslem  33716  fldextrspundgdvds  33717  irngnzply1lem  33726  minplyirredlem  33746  minplym1p  33749  minplynzm1p  33750  irredminply  33752  algextdeglem4  33756  algextdeglem7  33759  algextdeglem8  33760  constrsdrg  33811  2sqr3minply  33816  cos9thpiminplylem6  33823  cos9thpiminply  33824  drnginvmuld  42648  prjspner1  42747