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Theorem slmdvscl 27407
Description: Closure of scalar product for a semiring left module. (hvmulcl 25594 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvscl.v  |-  V  =  ( Base `  W
)
slmdvscl.f  |-  F  =  (Scalar `  W )
slmdvscl.s  |-  .x.  =  ( .s `  W )
slmdvscl.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
slmdvscl  |-  ( ( W  e. SLMod  /\  R  e.  K  /\  X  e.  V )  ->  ( R  .x.  X )  e.  V )

Proof of Theorem slmdvscl
StepHypRef Expression
1 biid 236 . 2  |-  ( W  e. SLMod 
<->  W  e. SLMod )
2 pm4.24 643 . 2  |-  ( R  e.  K  <->  ( R  e.  K  /\  R  e.  K ) )
3 pm4.24 643 . 2  |-  ( X  e.  V  <->  ( X  e.  V  /\  X  e.  V ) )
4 slmdvscl.v . . . . 5  |-  V  =  ( Base `  W
)
5 eqid 2462 . . . . 5  |-  ( +g  `  W )  =  ( +g  `  W )
6 slmdvscl.s . . . . 5  |-  .x.  =  ( .s `  W )
7 eqid 2462 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
8 slmdvscl.f . . . . 5  |-  F  =  (Scalar `  W )
9 slmdvscl.k . . . . 5  |-  K  =  ( Base `  F
)
10 eqid 2462 . . . . 5  |-  ( +g  `  F )  =  ( +g  `  F )
11 eqid 2462 . . . . 5  |-  ( .r
`  F )  =  ( .r `  F
)
12 eqid 2462 . . . . 5  |-  ( 1r
`  F )  =  ( 1r `  F
)
13 eqid 2462 . . . . 5  |-  ( 0g
`  F )  =  ( 0g `  F
)
144, 5, 6, 7, 8, 9, 10, 11, 12, 13slmdlema 27396 . . . 4  |-  ( ( W  e. SLMod  /\  ( R  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  (
( ( R  .x.  X )  e.  V  /\  ( R  .x.  ( X ( +g  `  W
) X ) )  =  ( ( R 
.x.  X ) ( +g  `  W ) ( R  .x.  X
) )  /\  (
( R ( +g  `  F ) R ) 
.x.  X )  =  ( ( R  .x.  X ) ( +g  `  W ) ( R 
.x.  X ) ) )  /\  ( ( ( R ( .r
`  F ) R )  .x.  X )  =  ( R  .x.  ( R  .x.  X ) )  /\  ( ( 1r `  F ) 
.x.  X )  =  X  /\  ( ( 0g `  F ) 
.x.  X )  =  ( 0g `  W
) ) ) )
1514simpld 459 . . 3  |-  ( ( W  e. SLMod  /\  ( R  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  (
( R  .x.  X
)  e.  V  /\  ( R  .x.  ( X ( +g  `  W
) X ) )  =  ( ( R 
.x.  X ) ( +g  `  W ) ( R  .x.  X
) )  /\  (
( R ( +g  `  F ) R ) 
.x.  X )  =  ( ( R  .x.  X ) ( +g  `  W ) ( R 
.x.  X ) ) ) )
1615simp1d 1003 . 2  |-  ( ( W  e. SLMod  /\  ( R  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  ( R  .x.  X )  e.  V )
171, 2, 3, 16syl3anb 1266 1  |-  ( ( W  e. SLMod  /\  R  e.  K  /\  X  e.  V )  ->  ( R  .x.  X )  e.  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546   .rcmulr 14547  Scalarcsca 14549   .scvsca 14550   0gc0g 14686   1rcur 16938  SLModcslmd 27393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-nul 4571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-iota 5544  df-fv 5589  df-ov 6280  df-slmd 27394
This theorem is referenced by:  gsumvsca1  27424  gsumvsca2  27425
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