Theorem flddrngd 20712

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 20711	. . 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 20209  DivRingcdr 20700  Fieldcfield 20701

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 3432  df-in 3897  df-field 20703

This theorem is referenced by:  ply1asclunit  33652  ply1unit  33653  ply1dg1rt  33658  m1pmeq  33663  fldextsdrg  33817  fldgenfldext  33831  evls1fldgencl  33833  fldextrspunlsplem  33836  fldextrspunfld  33839  fldextrspunlem2  33840  fldextrspundgdvdslem  33843  fldextrspundgdvds  33844  extdgfialglem1  33855  minplyirred  33874  algextdeglem2  33881  algextdeglem3  33882  algextdeglem4  33883  algextdeglem5  33884  algextdeglem7  33886  algextdeglem8  33887  rtelextdg2lem  33889  rtelextdg2  33890  constrsdrg  33938  aks6d1c5lem3  42593  aks6d1c5lem2  42594  aks5lem7  42656  prjcrv0  43083