Theorem crngringd 12985

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 12984	. 2 |- ( R e. CRing -> R e. Ring )
3	1, 2	syl 14	1 |- ( ph -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2146   Ringcrg 12972   CRingccrg 12973

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-rab 2462  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-cring 12975

This theorem is referenced by:  crnggrpd  12986