Theorem crngringd 13256

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 13255	. 2 |- ( R e. CRing -> R e. Ring )
3	1, 2	syl 14	1 |- ( ph -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2158   Ringcrg 13243   CRingccrg 13244

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-rab 2474  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-cring 13246

This theorem is referenced by:  crnggrpd  13257  unitmulclb  13357  rdivmuldivd  13387