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Theorem srgcmn 16947
Description: A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
Assertion
Ref Expression
srgcmn  |-  ( R  e. SRing  ->  R  e. CMnd )

Proof of Theorem srgcmn
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2467 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
3 eqid 2467 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2467 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2467 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
61, 2, 3, 4, 5issrg 16946 . 2  |-  ( R  e. SRing 
<->  ( R  e. CMnd  /\  (mulGrp `  R )  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
) ) ) ) )
76simp1bi 1011 1  |-  ( R  e. SRing  ->  R  e. CMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   .rcmulr 14549   0gc0g 14688   Mndcmnd 15719  CMndccmn 16591  mulGrpcmgp 16928  SRingcsrg 16944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-srg 16945
This theorem is referenced by:  srgmnd  16948  srgcom  16963  srgsummulcr  16973  sgsummulcl  16974  srgbinomlem3  16978  srgbinomlem4  16979  srgbinomlem  16980  gsumvsca2  27434
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