Theorem fldcrngd 20713

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 20711	. . 3 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
3	2	simprbi 497	. 2 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

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:  resrng  21614  frlmphl  21774  gsumind  33423  zringfrac  33632  ply1asclunit  33652  ply1unit  33653  ply1dg1rt  33658  ply1dg3rt0irred  33662  psrmonprod  33714  esplyfvaln  33736  fldgenfldext  33831  evls1fldgencl  33833  fldextrspunlsp  33837  irngnzply1lem  33853  irngnzply1  33854  extdgfialglem1  33855  extdgfialglem2  33856  extdgfialg  33857  ply1annig1p  33867  minplycl  33869  ply1annprmidl  33870  minplymindeg  33871  minplyann  33872  minplyirredlem  33873  minplyirred  33874  irngnminplynz  33875  minplym1p  33876  minplynzm1p  33877  minplyelirng  33878  irredminply  33879  algextdeglem1  33880  algextdeglem2  33881  algextdeglem3  33882  algextdeglem4  33883  algextdeglem5  33884  algextdeglem6  33885  algextdeglem7  33886  algextdeglem8  33887  rtelextdg2lem  33889  2sqr3minply  33943  cos9thpiminply  33951  aks6d1c1p3  42566  aks6d1c1p4  42567  aks6d1c1p5  42568  aks6d1c1p7  42569  aks6d1c1p6  42570  aks6d1c1p8  42571  aks6d1c1  42572  aks6d1c2lem3  42582  aks6d1c2lem4  42583  aks6d1c5lem0  42591  aks6d1c5lem1  42592  aks6d1c5lem3  42593  aks6d1c5  42595  aks6d1c6lem1  42626  aks6d1c6lem2  42627  aks6d1c6lem3  42628  aks6d1c6lem4  42629  aks6d1c6lem5  42633  aks5lem1  42642  aks5lem2  42643  aks5lem3a  42645  aks5lem5a  42647  prjcrv0  43083