Theorem crngringd 14252

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

Hypothesis

Ref	Expression
crngringd.1	|- ( ph -> R e. CRing )

Assertion

Ref	Expression
crngringd	|- ( ph -> R e. Ring )

Proof of Theorem crngringd

Step	Hyp	Ref	Expression
1		crngringd.1	. 2 |- ( ph -> R e. CRing )
2		crngring 14251	. 2 |- ( R e. CRing -> R e. Ring )
3	1, 2	syl 14	1 |- ( ph -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2205   Ringcrg 14239   CRingccrg 14240

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-cring 14242

This theorem is referenced by:  crnggrpd  14253  unitmulclb  14359  rdivmuldivd  14389  idomringd  14526  znrrg  14934  lgseisenlem4  16072