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Theorem slmdsrg 27440
Description: The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypothesis
Ref Expression
slmdsrg.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
slmdsrg  |-  ( W  e. SLMod  ->  F  e. SRing )

Proof of Theorem slmdsrg
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2467 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
3 eqid 2467 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
4 eqid 2467 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 slmdsrg.1 . . 3  |-  F  =  (Scalar `  W )
6 eqid 2467 . . 3  |-  ( Base `  F )  =  (
Base `  F )
7 eqid 2467 . . 3  |-  ( +g  `  F )  =  ( +g  `  F )
8 eqid 2467 . . 3  |-  ( .r
`  F )  =  ( .r `  F
)
9 eqid 2467 . . 3  |-  ( 1r
`  F )  =  ( 1r `  F
)
10 eqid 2467 . . 3  |-  ( 0g
`  F )  =  ( 0g `  F
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 27435 . 2  |-  ( W  e. SLMod 
<->  ( W  e. CMnd  /\  F  e. SRing  /\  A. w  e.  ( Base `  F
) A. z  e.  ( Base `  F
) A. x  e.  ( Base `  W
) A. y  e.  ( Base `  W
) ( ( ( z ( .s `  W ) y )  e.  ( Base `  W
)  /\  ( z
( .s `  W
) ( y ( +g  `  W ) x ) )  =  ( ( z ( .s `  W ) y ) ( +g  `  W ) ( z ( .s `  W
) x ) )  /\  ( ( w ( +g  `  F
) z ) ( .s `  W ) y )  =  ( ( w ( .s
`  W ) y ) ( +g  `  W
) ( z ( .s `  W ) y ) ) )  /\  ( ( ( w ( .r `  F ) z ) ( .s `  W
) y )  =  ( w ( .s
`  W ) ( z ( .s `  W ) y ) )  /\  ( ( 1r `  F ) ( .s `  W
) y )  =  y  /\  ( ( 0g `  F ) ( .s `  W
) y )  =  ( 0g `  W
) ) ) ) )
1211simp2bi 1012 1  |-  ( W  e. SLMod  ->  F  e. SRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555   .rcmulr 14556  Scalarcsca 14558   .scvsca 14559   0gc0g 14695  CMndccmn 16604   1rcur 16955  SRingcsrg 16959  SLModcslmd 27433
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 5551  df-fv 5596  df-ov 6287  df-slmd 27434
This theorem is referenced by:  slmdacl  27442  slmdmcl  27443  slmdsn0  27444  slmd0cl  27451  slmd1cl  27452  slmdvs0  27458  gsumvsca2  27465
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