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Theorem lmodvsdir 15894
Description: Distributive law for scalar product. (ax-hvdistr1 22352 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
lmodvsdir.v  |-  V  =  ( Base `  W
)
lmodvsdir.a  |-  .+  =  ( +g  `  W )
lmodvsdir.f  |-  F  =  (Scalar `  W )
lmodvsdir.s  |-  .x.  =  ( .s `  W )
lmodvsdir.k  |-  K  =  ( Base `  F
)
lmodvsdir.p  |-  .+^  =  ( +g  `  F )
Assertion
Ref Expression
lmodvsdir  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
)  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) )

Proof of Theorem lmodvsdir
StepHypRef Expression
1 lmodvsdir.v . . . . . . . 8  |-  V  =  ( Base `  W
)
2 lmodvsdir.a . . . . . . . 8  |-  .+  =  ( +g  `  W )
3 lmodvsdir.s . . . . . . . 8  |-  .x.  =  ( .s `  W )
4 lmodvsdir.f . . . . . . . 8  |-  F  =  (Scalar `  W )
5 lmodvsdir.k . . . . . . . 8  |-  K  =  ( Base `  F
)
6 lmodvsdir.p . . . . . . . 8  |-  .+^  =  ( +g  `  F )
7 eqid 2380 . . . . . . . 8  |-  ( .r
`  F )  =  ( .r `  F
)
8 eqid 2380 . . . . . . . 8  |-  ( 1r
`  F )  =  ( 1r `  F
)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 15875 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( 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 ) ) )
109simpld 446 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( 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 ) ) ) )
1110simp3d 971 . . . . 5  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  (
( Q  .+^  R ) 
.x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) )
12113expa 1153 . . . 4  |-  ( ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K
) )  /\  ( X  e.  V  /\  X  e.  V )
)  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) )
1312anabsan2 796 . . 3  |-  ( ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K
) )  /\  X  e.  V )  ->  (
( Q  .+^  R ) 
.x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) )
1413exp42 595 . 2  |-  ( W  e.  LMod  ->  ( Q  e.  K  ->  ( R  e.  K  ->  ( X  e.  V  -> 
( ( Q  .+^  R )  .x.  X )  =  ( ( Q 
.x.  X )  .+  ( R  .x.  X ) ) ) ) ) )
15143imp2 1168 1  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
)  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X
)  .+  ( R  .x.  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   Basecbs 13389   +g cplusg 13449   .rcmulr 13450  Scalarcsca 13452   .scvsca 13453   1rcur 15582   LModclmod 15870
This theorem is referenced by:  lmod0vs  15903  lmodvneg1  15907  lmodcom  15910  lmodsubdir  15922  islss3  15955  lss1d  15959  prdslmodd  15965  lspsolvlem  16134  asclghm  16317  clmvsdir  18977  frlmup1  26912  mendlmod  27163  lshpkrlem4  29279  baerlem3lem1  31873  baerlem5blem1  31875  hgmapadd  32063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-nul 4272
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-ov 6016  df-lmod 15872
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