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Theorem crngringd 13246
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 13245 . 2 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
31, 2syl 14 1 (𝜑𝑅 ∈ Ring)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2158  Ringcrg 13233  CRingccrg 13234
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-rab 2474  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-cring 13236
This theorem is referenced by:  crnggrpd  13247  unitmulclb  13346  rdivmuldivd  13376
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