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Theorem flddrngd 40645
Description: A field is a division ring. (Contributed by SN, 17-Jan-2025.)
Hypothesis
Ref Expression
flddrngd.1 (𝜑𝑅 ∈ Field)
Assertion
Ref Expression
flddrngd (𝜑𝑅 ∈ DivRing)

Proof of Theorem flddrngd
StepHypRef Expression
1 flddrngd.1 . 2 (𝜑𝑅 ∈ Field)
2 isfld 20125 . . 3 (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
32simplbi 499 . 2 (𝑅 ∈ Field → 𝑅 ∈ DivRing)
41, 3syl 17 1 (𝜑𝑅 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  CRingccrg 19895  DivRingcdr 20114  Fieldcfield 20115
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 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3446  df-in 3916  df-field 20117
This theorem is referenced by:  prjcrv0  40873
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