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Theorem slmdvscl 27630
Description: Closure of scalar product for a semiring left module. (hvmulcl 25802 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 2443 . . . . 5  |-  ( +g  `  W )  =  ( +g  `  W )
6 slmdvscl.s . . . . 5  |-  .x.  =  ( .s `  W )
7 eqid 2443 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
8 slmdvscl.f . . . . 5  |-  F  =  (Scalar `  W )
9 slmdvscl.k . . . . 5  |-  K  =  ( Base `  F
)
10 eqid 2443 . . . . 5  |-  ( +g  `  F )  =  ( +g  `  F )
11 eqid 2443 . . . . 5  |-  ( .r
`  F )  =  ( .r `  F
)
12 eqid 2443 . . . . 5  |-  ( 1r
`  F )  =  ( 1r `  F
)
13 eqid 2443 . . . . 5  |-  ( 0g
`  F )  =  ( 0g `  F
)
144, 5, 6, 7, 8, 9, 10, 11, 12, 13slmdlema 27619 . . . 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 1009 . 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 1272 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 974    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574   .rcmulr 14575  Scalarcsca 14577   .scvsca 14578   0gc0g 14714   1rcur 17027  SLModcslmd 27616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-slmd 27617
This theorem is referenced by:  gsumvsca1  27646  gsumvsca2  27647
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