Theorem crngringd 20068

Description: A commutative ring is a ring. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crngringd	⊢ (𝜑 → 𝑅 ∈ Ring)

Proof of Theorem crngringd

Step	Hyp	Ref	Expression
1		crngringd.1	. 2 ⊢ (𝜑 → 𝑅 ∈ CRing)
2		crngring 20067	. 2 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2106  Ringcrg 20055  CRingccrg 20056

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-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-cring 20058

This theorem is referenced by:  crnggrpd  20069  psrassa  21533  evlslem1  21644  mdetrsca  22104  recvs  24661  idomringd  32370  frobrhm  32377  fermltlchr  32473  znfermltl  32474  qusmul  32510  mxidlprmALT  32608  idlsrgmulrssin  32622  evls1expd  32639  evls1fpws  32641  ressply1evl  32642  ply1fermltlchr  32657  elirng  32745  0ringirng  32748  irngnzply1lem  32749  evls1maprhm  32754  ply1annidl  32758  ply1annnr  32759  algextdeglem1  32767  zarclsun  32845  zarmxt1  32855  zarcmplem  32856  fldhmf1  40950  crng12d  41090  riccrng1  41098  pwsgprod  41116  evl0  41131  evlsval3  41133  evlsvvvallem  41135  evlsvvvallem2  41136  evlsvvval  41137  evlsbagval  41140  evlsexpval  41141  evlsmaprhm  41144  evlsevl  41145  evlvvval  41147  evlvvvallem  41148  selvvvval  41159  evlselv  41161  selvmul  41163  evlsmhpvvval  41169  mhphf  41171  mhphf4  41174