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Theorem lmodvsdir 17316
Description: Distributive law for scalar product (right-distributivity). (ax-hvdistr1 25598 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 2467 . . . . . . . 8  |-  ( .r
`  F )  =  ( .r `  F
)
8 eqid 2467 . . . . . . . 8  |-  ( 1r
`  F )  =  ( 1r `  F
)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 17297 . . . . . . 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 459 . . . . . 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 1010 . . . . 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 1196 . . . 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 820 . . 3  |-  ( ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K
) )  /\  X  e.  V )  ->  (
( Q  .+^  R ) 
.x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) )
1413exp42 611 . 2  |-  ( W  e.  LMod  ->  ( Q  e.  K  ->  ( R  e.  K  ->  ( X  e.  V  -> 
( ( Q  .+^  R )  .x.  X )  =  ( ( Q 
.x.  X )  .+  ( R  .x.  X ) ) ) ) ) )
15143imp2 1211 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   .rcmulr 14549  Scalarcsca 14551   .scvsca 14552   1rcur 16940   LModclmod 17292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-lmod 17294
This theorem is referenced by:  lmod0vs  17325  lmodvsmmulgdi  17327  lmodvneg1  17333  lmodcom  17336  lmodsubdir  17348  islss3  17385  lss1d  17389  prdslmodd  17395  lspsolvlem  17568  asclghm  17755  frlmup1  18596  scmataddcl  18782  scmatghm  18799  pm2mpghm  19081  clmvsdir  21320  mendlmod  30747  lmodvsmdi  32048  lincsum  32103  ldepsprlem  32146  lshpkrlem4  33910  baerlem3lem1  36504  baerlem5blem1  36506  hgmapadd  36694
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