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Theorem fldcrngd 20370
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 20368 . . 3 (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
32simprbi 498 . 2 (𝑅 ∈ Field → 𝑅 ∈ CRing)
41, 3syl 17 1 (𝜑𝑅 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  CRingccrg 20057  DivRingcdr 20357  Fieldcfield 20358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-field 20360
This theorem is referenced by:  resrng  21174  frlmphl  21336  ply1asclunit  32654  irngnzply1lem  32754  irngnzply1  32755  ply1annig1p  32765  minplycl  32767  ply1annprmidl  32768  minplyirredlem  32769  minplyirred  32770  irngnminplynz  32771  algextdeglem1  32772  prjcrv0  41375
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