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Theorem slmdvsdir 28224
Description: Distributive law for scalar product. (ax-hvdistr1 26352 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 2404 . . . . . . . 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 2404 . . . . . . . 8  |-  ( .r
`  F )  =  ( .r `  F
)
9 eqid 2404 . . . . . . . 8  |-  ( 1r
`  F )  =  ( 1r `  F
)
10 eqid 2404 . . . . . . . 8  |-  ( 0g
`  F )  =  ( 0g `  F
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 28211 . . . . . . 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 1013 . . . . 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 1199 . . . 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 825 . . 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 1214 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 976    = wceq 1407    e. wcel 1844   ` cfv 5571  (class class class)co 6280   Basecbs 14843   +g cplusg 14911   .rcmulr 14912  Scalarcsca 14914   .scvsca 14915   0gc0g 15056   1rcur 17475  SLModcslmd 28208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-nul 4527
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-iota 5535  df-fv 5579  df-ov 6283  df-slmd 28209
This theorem is referenced by:  gsumvsca2  28239
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