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Theorem slmdvsdi 27992
Description: Distributive law for scalar product. (ax-hvdistr1 26123 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 2454 . . . . . . . . 9  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 slmdvsdi.f . . . . . . . . 9  |-  F  =  (Scalar `  W )
6 slmdvsdi.k . . . . . . . . 9  |-  K  =  ( Base `  F
)
7 eqid 2454 . . . . . . . . 9  |-  ( +g  `  F )  =  ( +g  `  F )
8 eqid 2454 . . . . . . . . 9  |-  ( .r
`  F )  =  ( .r `  F
)
9 eqid 2454 . . . . . . . . 9  |-  ( 1r
`  F )  =  ( 1r `  F
)
10 eqid 2454 . . . . . . . . 9  |-  ( 0g
`  F )  =  ( 0g `  F
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 27980 . . . . . . . 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 457 . . . . . . 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 1007 . . . . . 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 1196 . . . . 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 820 . . . 4  |-  ( ( W  e. SLMod  /\  R  e.  K )  ->  (
( Y  e.  V  /\  X  e.  V
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) ) )
1615exp4b 605 . . 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 1209 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   .rcmulr 14785  Scalarcsca 14787   .scvsca 14788   0gc0g 14929   1rcur 17348  SLModcslmd 27977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-slmd 27978
This theorem is referenced by:  gsumvsca1  28008
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