Theorem flddrngd 20599

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 20598	. . 3 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
3	2	simplbi 497	. 2 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ DivRing)

Syntax hints:   → wi 4   ∈ wcel 2098  CRingccrg 20139  DivRingcdr 20587  Fieldcfield 20588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-field 20590

This theorem is referenced by:  ply1asclunit  33163  ply1unit  33164  m1pmeq  33165  evls1fldgencl  33263  minplyirred  33290  algextdeglem2  33295  algextdeglem3  33296  algextdeglem4  33297  algextdeglem5  33298  algextdeglem7  33300  algextdeglem8  33301  aks6d1c5lem3  41513  aks6d1c5lem2  41514  prjcrv0  41953