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Theorem slmdvsdir 27407
Description: Distributive law for scalar product. (ax-hvdistr1 25587 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvsdir.v  |-  V  =  ( Base `  W
)
slmdvsdir.a  |-  .+  =  ( +g  `  W )
slmdvsdir.f  |-  F  =  (Scalar `  W )
slmdvsdir.s  |-  .x.  =  ( .s `  W )
slmdvsdir.k  |-  K  =  ( Base `  F
)
slmdvsdir.p  |-  .+^  =  ( +g  `  F )
Assertion
Ref Expression
slmdvsdir  |-  ( ( W  e. SLMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
)  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) )

Proof of Theorem slmdvsdir
StepHypRef Expression
1 slmdvsdir.v . . . . . . . 8  |-  V  =  ( Base `  W
)
2 slmdvsdir.a . . . . . . . 8  |-  .+  =  ( +g  `  W )
3 slmdvsdir.s . . . . . . . 8  |-  .x.  =  ( .s `  W )
4 eqid 2460 . . . . . . . 8  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 slmdvsdir.f . . . . . . . 8  |-  F  =  (Scalar `  W )
6 slmdvsdir.k . . . . . . . 8  |-  K  =  ( Base `  F
)
7 slmdvsdir.p . . . . . . . 8  |-  .+^  =  ( +g  `  F )
8 eqid 2460 . . . . . . . 8  |-  ( .r
`  F )  =  ( .r `  F
)
9 eqid 2460 . . . . . . . 8  |-  ( 1r
`  F )  =  ( 1r `  F
)
10 eqid 2460 . . . . . . . 8  |-  ( 0g
`  F )  =  ( 0g `  F
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 27394 . . . . . . 7  |-  ( ( W  e. SLMod  /\  ( Q  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  (
( ( R  .x.  X )  e.  V  /\  ( R  .x.  ( X  .+  X ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  X ) )  /\  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) )  /\  ( ( ( Q ( .r `  F ) R ) 
.x.  X )  =  ( Q  .x.  ( R  .x.  X ) )  /\  ( ( 1r
`  F )  .x.  X )  =  X  /\  ( ( 0g
`  F )  .x.  X )  =  ( 0g `  W ) ) ) )
1211simpld 459 . . . . . 6  |-  ( ( W  e. SLMod  /\  ( Q  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  (
( R  .x.  X
)  e.  V  /\  ( R  .x.  ( X 
.+  X ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  X ) )  /\  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) ) )
1312simp3d 1005 . . . . 5  |-  ( ( W  e. SLMod  /\  ( Q  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  (
( Q  .+^  R ) 
.x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) )
14133expa 1191 . . . 4  |-  ( ( ( W  e. SLMod  /\  ( Q  e.  K  /\  R  e.  K )
)  /\  ( X  e.  V  /\  X  e.  V ) )  -> 
( ( Q  .+^  R )  .x.  X )  =  ( ( Q 
.x.  X )  .+  ( R  .x.  X ) ) )
1514anabsan2 819 . . 3  |-  ( ( ( W  e. SLMod  /\  ( Q  e.  K  /\  R  e.  K )
)  /\  X  e.  V )  ->  (
( Q  .+^  R ) 
.x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) )
1615exp42 611 . 2  |-  ( W  e. SLMod  ->  ( Q  e.  K  ->  ( R  e.  K  ->  ( X  e.  V  ->  (
( Q  .+^  R ) 
.x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) ) ) ) )
17163imp2 1206 1  |-  ( ( W  e. SLMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
)  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   .rcmulr 14545  Scalarcsca 14547   .scvsca 14548   0gc0g 14684   1rcur 16936  SLModcslmd 27391
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 1961  ax-ext 2438  ax-nul 4569
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 2272  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-ov 6278  df-slmd 27392
This theorem is referenced by:  gsumvsca2  27423
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