MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  srgcmn Structured version   Unicode version

Theorem srgcmn 17480
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 2402 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2402 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
3 eqid 2402 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2402 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2402 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
61, 2, 3, 4, 5issrg 17479 . 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 1012 1  |-  ( R  e. SRing  ->  R  e. CMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   .rcmulr 14910   0gc0g 15054   Mndcmnd 16243  CMndccmn 17122  mulGrpcmgp 17461  SRingcsrg 17477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-ov 6281  df-srg 17478
This theorem is referenced by:  srgmnd  17481  srgcom  17496  srgsummulcr  17508  sgsummulcl  17509  srgbinomlem3  17513  srgbinomlem4  17514  srgbinomlem  17515  gsumvsca2  28226
  Copyright terms: Public domain W3C validator