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Theorem srgmgp 13082
Description: A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
Hypothesis
Ref Expression
srgmgp.g  |-  G  =  (mulGrp `  R )
Assertion
Ref Expression
srgmgp  |-  ( R  e. SRing  ->  G  e.  Mnd )

Proof of Theorem srgmgp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 srgmgp.g . . 3  |-  G  =  (mulGrp `  R )
3 eqid 2177 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2177 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2177 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
61, 2, 3, 4, 5issrg 13079 . 2  |-  ( R  e. SRing 
<->  ( R  e. CMnd  /\  G  e.  Mnd  /\  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 ) ) )  /\  ( ( ( 0g `  R ) ( .r `  R
) x )  =  ( 0g `  R
)  /\  ( x
( .r `  R
) ( 0g `  R ) )  =  ( 0g `  R
) ) ) ) )
76simp2bi 1013 1  |-  ( R  e. SRing  ->  G  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   .rcmulr 12529   0gc0g 12693   Mndcmnd 12749  CMndccmn 13019  mulGrpcmgp 13061  SRingcsrg 13077
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-riota 5828  df-ov 5875  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-mulr 12542  df-0g 12695  df-srg 13078
This theorem is referenced by:  srgcl  13084  srgass  13085  srgideu  13086  srgidcl  13090  srgidmlem  13092  srg1zr  13101  srgpcomp  13104  srgpcompp  13105  srgpcomppsc  13106  srg1expzeq1  13109
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