Theorem crngringd 19807

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 19806	. 2 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2110  Ringcrg 19794  CRingccrg 19795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-iota 6390  df-fv 6440  df-cring 19797

This theorem is referenced by:  crnggrpd  19808  asclrhm  21105  evlslem1  21303  recvs  24320  frobrhm  31494  znfermltl  31571  idlsrgmulrssin  31667  ply1fermltl  31679  zarclsun  31829  zarmxt1  31839  zarcmplem  31840  pwsgprod  40278  mplascl0  40279  evl0  40280  evlsval3  40281  evlsbagval  40284  evlsexpval  40285  mhphf  40294  mhphf4  40297