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Theorem lmodvsdi 15666
Description: Distributive law for scalar product. (ax-hvdistr1 21604 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
lmodvsdi.v  |-  V  =  ( Base `  W
)
lmodvsdi.a  |-  .+  =  ( +g  `  W )
lmodvsdi.f  |-  F  =  (Scalar `  W )
lmodvsdi.s  |-  .x.  =  ( .s `  W )
lmodvsdi.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
lmodvsdi  |-  ( ( W  e.  LMod  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) )

Proof of Theorem lmodvsdi
StepHypRef Expression
1 lmodvsdi.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
2 lmodvsdi.a . . . . . . . . 9  |-  .+  =  ( +g  `  W )
3 lmodvsdi.s . . . . . . . . 9  |-  .x.  =  ( .s `  W )
4 lmodvsdi.f . . . . . . . . 9  |-  F  =  (Scalar `  W )
5 lmodvsdi.k . . . . . . . . 9  |-  K  =  ( Base `  F
)
6 eqid 2296 . . . . . . . . 9  |-  ( +g  `  F )  =  ( +g  `  F )
7 eqid 2296 . . . . . . . . 9  |-  ( .r
`  F )  =  ( .r `  F
)
8 eqid 2296 . . . . . . . . 9  |-  ( 1r
`  F )  =  ( 1r `  F
)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 15648 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( 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 ) ) )
109simpld 445 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( 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 ) ) ) )
1110simp2d 968 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( R  e.  K  /\  R  e.  K )  /\  ( Y  e.  V  /\  X  e.  V
) )  ->  ( R  .x.  ( X  .+  Y ) )  =  ( ( R  .x.  X )  .+  ( R  .x.  Y ) ) )
12113expia 1153 . . . . 5  |-  ( ( W  e.  LMod  /\  ( R  e.  K  /\  R  e.  K )
)  ->  ( ( Y  e.  V  /\  X  e.  V )  ->  ( R  .x.  ( X  .+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) ) ) )
1312anabsan2 795 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
( Y  e.  V  /\  X  e.  V
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) ) )
1413exp4b 590 . . 3  |-  ( W  e.  LMod  ->  ( R  e.  K  ->  ( Y  e.  V  ->  ( X  e.  V  -> 
( R  .x.  ( X  .+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) ) ) ) ) )
1514com34 77 . 2  |-  ( W  e.  LMod  ->  ( R  e.  K  ->  ( X  e.  V  ->  ( Y  e.  V  -> 
( R  .x.  ( X  .+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) ) ) ) ) )
16153imp2 1166 1  |-  ( ( W  e.  LMod  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   1rcur 15355   LModclmod 15643
This theorem is referenced by:  lmodvsdi1OLD  15667  lmodcom  15687  lmodsubdi  15698  lmodvsghm  15702  islss3  15732  prdslmodd  15742  lmodvsinv2  15810  lmhmplusg  15817  lsmcl  15852  pj1lmhm  15869  lspfixed  15897  lspsolvlem  15911  mendlmod  27604  lshpkrlem4  29925  baerlem5alem1  32520  baerlem5blem1  32521  hdmap14lem8  32690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-lmod 15645
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