Theorem rngabl 14096

Description: A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)

Assertion

Ref	Expression
rngabl	|- ( R e. Rng -> R e. Abel )

Proof of Theorem rngabl

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. . 3 |- ( Base ` R ) = ( Base ` R )
2		eqid 2234	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
3		eqid 2234	. . 3 |- ( +g ` R ) = ( +g ` R )
4		eqid 2234	. . 3 |- ( .r ` R ) = ( .r ` R )
5	1, 2, 3, 4	isrng 14095	. 2 |- ( R e. Rng <-> ( R e. Abel /\ (mulGrp ` R ) e. Smgrp /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) )
6	5	simp1bi 1039	1 |- ( R e. Rng -> R e. Abel )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   .rcmulr 13308  Smgrpcsgrp 13631   Abelcabl 14019  mulGrpcmgp 14081  Rngcrng 14093

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-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-ov 6055  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-plusg 13320  df-mulr 13321  df-rng 14094

This theorem is referenced by:  rnggrp  14099  rnglz  14106  rngansg  14111  rngressid  14115  imasrng  14117  opprrng  14238  subrngringnsg  14367  issubrng2  14372  rnglidlrng  14663  2idlcpblrng  14688  qus2idrng  14690