Theorem flddrngd 20686

Description: A field is a division ring. (Contributed by SN, 17-Jan-2025.)

Hypothesis

Ref	Expression
flddrngd.1	⊢ (𝜑 → 𝑅 ∈ Field)

Assertion

Ref	Expression
flddrngd	⊢ (𝜑 → 𝑅 ∈ DivRing)

Proof of Theorem flddrngd

Step	Hyp	Ref	Expression
1		flddrngd.1	. 2 ⊢ (𝜑 → 𝑅 ∈ Field)
2		isfld 20685	. . 3 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
3	2	simplbi 496	. 2 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ DivRing)

Syntax hints:   → wi 4   ∈ wcel 2114  CRingccrg 20181  DivRingcdr 20674  Fieldcfield 20675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-field 20677

This theorem is referenced by:  ply1asclunit  33667  ply1unit  33668  ply1dg1rt  33673  m1pmeq  33678  fldextsdrg  33832  fldgenfldext  33846  evls1fldgencl  33848  fldextrspunlsplem  33851  fldextrspunfld  33854  fldextrspunlem2  33855  fldextrspundgdvdslem  33858  fldextrspundgdvds  33859  extdgfialglem1  33870  minplyirred  33889  algextdeglem2  33896  algextdeglem3  33897  algextdeglem4  33898  algextdeglem5  33899  algextdeglem7  33901  algextdeglem8  33902  rtelextdg2lem  33904  rtelextdg2  33905  constrsdrg  33953  aks6d1c5lem3  42507  aks6d1c5lem2  42508  aks5lem7  42570  prjcrv0  42991