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Theorem rngabl 38692
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.
StepHypRef Expression
1 eqid 2420 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2420 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
3 eqid 2420 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2420 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isrng 38691 . 2  |-  ( R  e. Rng 
<->  ( R  e.  Abel  /\  (mulGrp `  R )  e. SGrp  /\  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 ) ) ) ) )
65simp1bi 1020 1  |-  ( R  e. Rng  ->  R  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   ` cfv 5592  (class class class)co 6296   Basecbs 15081   +g cplusg 15150   .rcmulr 15151  SGrpcsgrp 16478   Abelcabl 17372  mulGrpcmgp 17664  Rngcrng 38689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-nul 4547
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299  df-rng0 38690
This theorem is referenced by:  isringrng  38696  rnglz  38699  isrnghm  38707  isrnghmd  38717  idrnghm  38723  c0rnghm  38728  zrrnghm  38732
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