Theorem flddrngd 20758

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 20757	. . 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 2106  CRingccrg 20252  DivRingcdr 20746  Fieldcfield 20747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-field 20749

This theorem is referenced by:  ply1asclunit  33579  ply1unit  33580  ply1dg1rt  33584  m1pmeq  33588  fldgenfldext  33693  evls1fldgencl  33695  minplyirred  33719  algextdeglem2  33724  algextdeglem3  33725  algextdeglem4  33726  algextdeglem5  33727  algextdeglem7  33729  algextdeglem8  33730  rtelextdg2lem  33732  rtelextdg2  33733  aks6d1c5lem3  42119  aks6d1c5lem2  42120  aks5lem7  42182  prjcrv0  42620