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Theorem srgmgp 17357
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 17354 . 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 1010 1  |-  ( R  e. SRing  ->  G  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   .rcmulr 14785   0gc0g 14929   Mndcmnd 16118  CMndccmn 16997  mulGrpcmgp 17336  SRingcsrg 17352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-srg 17353
This theorem is referenced by:  srgcl  17359  srgass  17360  srgideu  17361  srgidcl  17364  srgidmlem  17366  srg1zr  17375  srgpcomp  17378  srgpcompp  17379  srgpcomppsc  17380  srg1expzeq1  17385  srgbinomlem1  17386  srgbinomlem4  17389  srgbinomlem  17390
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