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Theorem srgmgp 16735
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 2454 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 srgmgp.g . . 3  |-  G  =  (mulGrp `  R )
3 eqid 2454 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2454 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2454 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
61, 2, 3, 4, 5issrg 16732 . 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 1004 1  |-  ( R  e. SRing  ->  G  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   .rcmulr 14359   0gc0g 14498   Mndcmnd 15529  CMndccmn 16399  mulGrpcmgp 16714  SRingcsrg 16730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-iota 5490  df-fv 5535  df-ov 6204  df-srg 16731
This theorem is referenced by:  srgcl  16737  srgass  16738  srgideu  16739  srgidcl  16742  srgidmlem  16744  srgpcomp  16754  srgpcompp  16755  srgpcomppsc  16756  srg1expzeq1  16761  srgbinomlem1  16762  srgbinomlem4  16765  srgbinomlem  16766
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