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Theorem srgmgp 13151
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 13148 . 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 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   .rcmulr 12537   0gc0g 12705   Mndcmnd 12817  CMndccmn 13088  mulGrpcmgp 13130  SRingcsrg 13146
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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908
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 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-riota 5831  df-ov 5878  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-plusg 12549  df-mulr 12550  df-0g 12707  df-srg 13147
This theorem is referenced by:  srgcl  13153  srgass  13154  srgideu  13155  srgidcl  13159  srgidmlem  13161  srg1zr  13170  srgpcomp  13173  srgpcompp  13174  srgpcomppsc  13175  srg1expzeq1  13178
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