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Theorem slmdvsdi 26369
Description: Distributive law for scalar product. (ax-hvdistr1 24555 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvsdi.v  |-  V  =  ( Base `  W
)
slmdvsdi.a  |-  .+  =  ( +g  `  W )
slmdvsdi.f  |-  F  =  (Scalar `  W )
slmdvsdi.s  |-  .x.  =  ( .s `  W )
slmdvsdi.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
slmdvsdi  |-  ( ( W  e. SLMod  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) )

Proof of Theorem slmdvsdi
StepHypRef Expression
1 slmdvsdi.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
2 slmdvsdi.a . . . . . . . . 9  |-  .+  =  ( +g  `  W )
3 slmdvsdi.s . . . . . . . . 9  |-  .x.  =  ( .s `  W )
4 eqid 2451 . . . . . . . . 9  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 slmdvsdi.f . . . . . . . . 9  |-  F  =  (Scalar `  W )
6 slmdvsdi.k . . . . . . . . 9  |-  K  =  ( Base `  F
)
7 eqid 2451 . . . . . . . . 9  |-  ( +g  `  F )  =  ( +g  `  F )
8 eqid 2451 . . . . . . . . 9  |-  ( .r
`  F )  =  ( .r `  F
)
9 eqid 2451 . . . . . . . . 9  |-  ( 1r
`  F )  =  ( 1r `  F
)
10 eqid 2451 . . . . . . . . 9  |-  ( 0g
`  F )  =  ( 0g `  F
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 26357 . . . . . . . 8  |-  ( ( W  e. SLMod  /\  ( R  e.  K  /\  R  e.  K )  /\  ( Y  e.  V  /\  X  e.  V
) )  ->  (
( ( R  .x.  X )  e.  V  /\  ( R  .x.  ( X  .+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) )  /\  ( ( R ( +g  `  F
) R )  .x.  X )  =  ( ( R  .x.  X
)  .+  ( 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 ) ) ) )
1211simpld 459 . . . . . . 7  |-  ( ( W  e. SLMod  /\  ( R  e.  K  /\  R  e.  K )  /\  ( Y  e.  V  /\  X  e.  V
) )  ->  (
( R  .x.  X
)  e.  V  /\  ( R  .x.  ( X 
.+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) )  /\  ( ( R ( +g  `  F
) R )  .x.  X )  =  ( ( R  .x.  X
)  .+  ( R  .x.  X ) ) ) )
1312simp2d 1001 . . . . . 6  |-  ( ( W  e. SLMod  /\  ( R  e.  K  /\  R  e.  K )  /\  ( Y  e.  V  /\  X  e.  V
) )  ->  ( R  .x.  ( X  .+  Y ) )  =  ( ( R  .x.  X )  .+  ( R  .x.  Y ) ) )
14133expia 1190 . . . . 5  |-  ( ( W  e. SLMod  /\  ( R  e.  K  /\  R  e.  K )
)  ->  ( ( Y  e.  V  /\  X  e.  V )  ->  ( R  .x.  ( X  .+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) ) ) )
1514anabsan2 818 . . . 4  |-  ( ( W  e. SLMod  /\  R  e.  K )  ->  (
( Y  e.  V  /\  X  e.  V
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) ) )
1615exp4b 607 . . 3  |-  ( W  e. SLMod  ->  ( R  e.  K  ->  ( Y  e.  V  ->  ( X  e.  V  ->  ( R  .x.  ( X  .+  Y ) )  =  ( ( R  .x.  X )  .+  ( R  .x.  Y ) ) ) ) ) )
1716com34 83 . 2  |-  ( W  e. SLMod  ->  ( R  e.  K  ->  ( X  e.  V  ->  ( Y  e.  V  ->  ( R  .x.  ( X  .+  Y ) )  =  ( ( R  .x.  X )  .+  ( R  .x.  Y ) ) ) ) ) )
18173imp2 1203 1  |-  ( ( W  e. SLMod  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   Basecbs 14285   +g cplusg 14349   .rcmulr 14350  Scalarcsca 14352   .scvsca 14353   0gc0g 14489   1rcur 16717  SLModcslmd 26354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196  df-slmd 26355
This theorem is referenced by:  gsumvsca1  26389
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