Theorem fldcrngd 20759

Description: A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)

Hypothesis

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

Assertion

Ref	Expression
fldcrngd	⊢ (𝜑 → 𝑅 ∈ CRing)

Proof of Theorem fldcrngd

Step	Hyp	Ref	Expression
1		fldcrngd.1	. 2 ⊢ (𝜑 → 𝑅 ∈ Field)
2		isfld 20757	. . 3 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
3	2	simprbi 496	. 2 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

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:  resrng  21657  frlmphl  21819  zringfrac  33562  ply1asclunit  33579  ply1unit  33580  ply1dg1rt  33584  ply1dg3rt0irred  33587  fldgenfldext  33693  evls1fldgencl  33695  irngnzply1lem  33705  irngnzply1  33706  ply1annig1p  33712  minplycl  33714  ply1annprmidl  33715  minplymindeg  33716  minplyann  33717  minplyirredlem  33718  minplyirred  33719  irngnminplynz  33720  minplym1p  33721  irredminply  33722  algextdeglem1  33723  algextdeglem2  33724  algextdeglem3  33725  algextdeglem4  33726  algextdeglem5  33727  algextdeglem6  33728  algextdeglem7  33729  algextdeglem8  33730  rtelextdg2lem  33732  2sqr3minply  33753  aks6d1c1p3  42092  aks6d1c1p4  42093  aks6d1c1p5  42094  aks6d1c1p7  42095  aks6d1c1p6  42096  aks6d1c1p8  42097  aks6d1c1  42098  aks6d1c2lem3  42108  aks6d1c2lem4  42109  aks6d1c5lem0  42117  aks6d1c5lem1  42118  aks6d1c5lem3  42119  aks6d1c5  42121  aks6d1c6lem1  42152  aks6d1c6lem2  42153  aks6d1c6lem3  42154  aks6d1c6lem4  42155  aks6d1c6lem5  42159  aks5lem1  42168  aks5lem2  42169  aks5lem3a  42171  aks5lem5a  42173  prjcrv0  42620