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Theorem slmdsrg 28361
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 2429 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2429 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
3 eqid 2429 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
4 eqid 2429 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 slmdsrg.1 . . 3  |-  F  =  (Scalar `  W )
6 eqid 2429 . . 3  |-  ( Base `  F )  =  (
Base `  F )
7 eqid 2429 . . 3  |-  ( +g  `  F )  =  ( +g  `  F )
8 eqid 2429 . . 3  |-  ( .r
`  F )  =  ( .r `  F
)
9 eqid 2429 . . 3  |-  ( 1r
`  F )  =  ( 1r `  F
)
10 eqid 2429 . . 3  |-  ( 0g
`  F )  =  ( 0g `  F
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 28356 . 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 1021 1  |-  ( W  e. SLMod  ->  F  e. SRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   .rcmulr 15153  Scalarcsca 15155   .scvsca 15156   0gc0g 15297  CMndccmn 17365   1rcur 17670  SRingcsrg 17674  SLModcslmd 28354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-nul 4556
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609  df-ov 6308  df-slmd 28355
This theorem is referenced by:  slmdacl  28363  slmdmcl  28364  slmdsn0  28365  slmd0cl  28372  slmd1cl  28373  slmdvs0  28379  gsumvsca2  28385
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