Theorem crngringd 13972

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 13971	. 2 |- ( R e. CRing -> R e. Ring )
3	1, 2	syl 14	1 |- ( ph -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2200   Ringcrg 13959   CRingccrg 13960

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-cring 13962

This theorem is referenced by:  crnggrpd  13973  unitmulclb  14078  rdivmuldivd  14108  idomringd  14243  znrrg  14624  lgseisenlem4  15752