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Theorem lmodvsdi 15650
Description: Distributive law for scalar product. (ax-hvdistr1 21588 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 2283 . . . . . . . . 9  |-  ( +g  `  F )  =  ( +g  `  F )
7 eqid 2283 . . . . . . . . 9  |-  ( .r
`  F )  =  ( .r `  F
)
8 eqid 2283 . . . . . . . . 9  |-  ( 1r
`  F )  =  ( 1r `  F
)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 15632 . . . . . . . 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   1rcur 15339   LModclmod 15627
This theorem is referenced by:  lmodvsdi1OLD  15651  lmodcom  15671  lmodsubdi  15682  lmodvsghm  15686  islss3  15716  prdslmodd  15726  lmodvsinv2  15794  lmhmplusg  15801  lsmcl  15836  pj1lmhm  15853  lspfixed  15881  lspsolvlem  15895  mendlmod  27501  lshpkrlem4  29303  baerlem5alem1  31898  baerlem5blem1  31899  hdmap14lem8  32068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-lmod 15629
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