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Theorem rngmgp 32522
Description: A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
Hypothesis
Ref Expression
rngmgp.g  |-  G  =  (mulGrp `  R )
Assertion
Ref Expression
rngmgp  |-  ( R  e. Rng  ->  G  e. SGrp )

Proof of Theorem rngmgp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 rngmgp.g . . 3  |-  G  =  (mulGrp `  R )
3 eqid 2443 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2443 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isrng 32520 . 2  |-  ( R  e. Rng 
<->  ( R  e.  Abel  /\  G  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 ) ) ) ) )
65simp2bi 1013 1  |-  ( R  e. Rng  ->  G  e. SGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   ` cfv 5578  (class class class)co 6281   Basecbs 14614   +g cplusg 14679   .rcmulr 14680  SGrpcsgrp 15889   Abelcabl 16778  mulGrpcmgp 17120  Rngcrng 32518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-rng0 32519
This theorem is referenced by:  isringrng  32525  isrnghmmul  32534  idrnghm  32549
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