Theorem crngringd 20071

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 20070	. 2 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2106  Ringcrg 20058  CRingccrg 20059

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 20061

This theorem is referenced by:  crnggrpd  20072  psrassa  21540  evlslem1  21651  mdetrsca  22112  recvs  24669  idomringd  32416  frobrhm  32423  fermltlchr  32523  znfermltl  32524  qusmul  32560  mxidlprmALT  32658  idlsrgmulrssin  32672  evls1expd  32689  evls1fpws  32691  ressply1evl  32692  ply1fermltlchr  32707  evls1fldgencl  32804  elirng  32810  0ringirng  32813  irngnzply1lem  32814  evls1maprhm  32819  ply1annidl  32823  ply1annnr  32824  algextdeglem4  32836  zarclsun  32919  zarmxt1  32929  zarcmplem  32930  fldhmf1  41041  crng12d  41172  riccrng1  41180  pwsgprod  41196  evl0  41211  evlsval3  41213  evlsvvvallem  41215  evlsvvvallem2  41216  evlsvvval  41217  evlsbagval  41220  evlsexpval  41221  evlsmaprhm  41224  evlsevl  41225  evlvvval  41227  evlvvvallem  41228  selvvvval  41239  evlselv  41241  selvmul  41243  evlsmhpvvval  41249  mhphf  41251  mhphf4  41254