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Theorem lmodvsdi 17097
Description: Distributive law for scalar product (left-distributivity). (ax-hvdistr1 24582 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 2454 . . . . . . . . 9  |-  ( +g  `  F )  =  ( +g  `  F )
7 eqid 2454 . . . . . . . . 9  |-  ( .r
`  F )  =  ( .r `  F
)
8 eqid 2454 . . . . . . . . 9  |-  ( 1r
`  F )  =  ( 1r `  F
)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 17079 . . . . . . . 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 459 . . . . . . 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 1001 . . . . . 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 1190 . . . . 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 818 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
( Y  e.  V  /\  X  e.  V
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) ) )
1413exp4b 607 . . 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 1203 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   Basecbs 14295   +g cplusg 14360   .rcmulr 14361  Scalarcsca 14363   .scvsca 14364   1rcur 16728   LModclmod 17074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4532
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206  df-lmod 17076
This theorem is referenced by:  lmodcom  17117  lmodsubdi  17128  lmodvsghm  17132  islss3  17166  prdslmodd  17176  lmodvsinv2  17244  lmhmplusg  17251  lsmcl  17290  pj1lmhm  17307  lspfixed  17335  lspsolvlem  17349  mendlmod  29718  lmodvsmdi  30987  lshpkrlem4  33116  baerlem5alem1  35711  baerlem5blem1  35712  hdmap14lem8  35881
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