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Theorem fldcrngd 20320
Description: A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
Hypothesis
Ref Expression
fldcrngd.1 (𝜑𝑅 ∈ Field)
Assertion
Ref Expression
fldcrngd (𝜑𝑅 ∈ CRing)

Proof of Theorem fldcrngd
StepHypRef Expression
1 fldcrngd.1 . 2 (𝜑𝑅 ∈ Field)
2 isfld 20318 . . 3 (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
32simprbi 497 . 2 (𝑅 ∈ Field → 𝑅 ∈ CRing)
41, 3syl 17 1 (𝜑𝑅 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  CRingccrg 20050  DivRingcdr 20307  Fieldcfield 20308
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-field 20310
This theorem is referenced by:  resrng  21165  frlmphl  21327  ply1asclunit  32642  irngnzply1lem  32742  irngnzply1  32743  ply1annig1p  32753  minplycl  32755  ply1annprmidl  32756  minplyirredlem  32757  minplyirred  32758  irngnminplynz  32759  algextdeglem1  32760  prjcrv0  41371
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