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Theorem lmodvsdi 17853
Description: Distributive law for scalar product (left-distributivity). (ax-hvdistr1 26325 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 2402 . . . . . . . . 9  |-  ( +g  `  F )  =  ( +g  `  F )
7 eqid 2402 . . . . . . . . 9  |-  ( .r
`  F )  =  ( .r `  F
)
8 eqid 2402 . . . . . . . . 9  |-  ( 1r
`  F )  =  ( 1r `  F
)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 17835 . . . . . . . 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 457 . . . . . . 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 1010 . . . . . 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 1199 . . . . 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 823 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
( Y  e.  V  /\  X  e.  V
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) ) )
1413exp4b 605 . . 3  |-  ( W  e.  LMod  ->  ( R  e.  K  ->  ( Y  e.  V  ->  ( X  e.  V  -> 
( R  .x.  ( X  .+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) ) ) ) ) )
1514com34 83 . 2  |-  ( W  e.  LMod  ->  ( R  e.  K  ->  ( X  e.  V  ->  ( Y  e.  V  -> 
( R  .x.  ( X  .+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) ) ) ) ) )
16153imp2 1212 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907   .rcmulr 14908  Scalarcsca 14910   .scvsca 14911   1rcur 17471   LModclmod 17830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-lmod 17832
This theorem is referenced by:  lmodcom  17874  lmodsubdi  17885  lmodvsghm  17889  islss3  17923  prdslmodd  17933  lmodvsinv2  18001  lmhmplusg  18008  lsmcl  18047  pj1lmhm  18064  lspfixed  18092  lspsolvlem  18106  lshpkrlem4  32111  baerlem5alem1  34708  baerlem5blem1  34709  hdmap14lem8  34878  mendlmod  35486  lmodvsmdi  38467
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