Theorem crngringd 19796

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 19795	. 2 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2106  Ringcrg 19783  CRingccrg 19784

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-cring 19786

This theorem is referenced by:  crnggrpd  19797  asclrhm  21094  evlslem1  21292  recvs  24309  frobrhm  31485  znfermltl  31562  idlsrgmulrssin  31658  ply1fermltl  31670  zarclsun  31820  zarmxt1  31830  zarcmplem  31831  pwsgprod  40269  mplascl0  40270  evl0  40271  evlsval3  40272  evlsbagval  40275  evlsexpval  40276  mhphf  40285  mhphf4  40288