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Theorem crngringd 13330
Description: A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
Hypothesis
Ref Expression
crngringd.1 (𝜑𝑅 ∈ CRing)
Assertion
Ref Expression
crngringd (𝜑𝑅 ∈ Ring)

Proof of Theorem crngringd
StepHypRef Expression
1 crngringd.1 . 2 (𝜑𝑅 ∈ CRing)
2 crngring 13329 . 2 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
31, 2syl 14 1 (𝜑𝑅 ∈ Ring)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  Ringcrg 13317  CRingccrg 13318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5193  df-fv 5239  df-cring 13320
This theorem is referenced by:  crnggrpd  13331  unitmulclb  13431  rdivmuldivd  13461
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