MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmodvsdi Structured version   Unicode version

Theorem lmodvsdi 15973
Description: Distributive law for scalar product. (ax-hvdistr1 22511 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 2436 . . . . . . . . 9  |-  ( +g  `  F )  =  ( +g  `  F )
7 eqid 2436 . . . . . . . . 9  |-  ( .r
`  F )  =  ( .r `  F
)
8 eqid 2436 . . . . . . . . 9  |-  ( 1r
`  F )  =  ( 1r `  F
)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 15955 . . . . . . . 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 446 . . . . . . 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 970 . . . . . 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 1155 . . . . 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 796 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
( Y  e.  V  /\  X  e.  V
)  ->  ( R  .x.  ( X  .+  Y
) )  =  ( ( R  .x.  X
)  .+  ( R  .x.  Y ) ) ) )
1413exp4b 591 . . 3  |-  ( W  e.  LMod  ->  ( R  e.  K  ->  ( Y  e.  V  ->  ( X  e.  V  -> 
( R  .x.  ( X  .+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) ) ) ) ) )
1514com34 79 . 2  |-  ( W  e.  LMod  ->  ( R  e.  K  ->  ( X  e.  V  ->  ( Y  e.  V  -> 
( R  .x.  ( X  .+  Y ) )  =  ( ( R 
.x.  X )  .+  ( R  .x.  Y ) ) ) ) ) )
16153imp2 1168 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533   1rcur 15662   LModclmod 15950
This theorem is referenced by:  lmodcom  15990  lmodsubdi  16001  lmodvsghm  16005  islss3  16035  prdslmodd  16045  lmodvsinv2  16113  lmhmplusg  16120  lsmcl  16155  pj1lmhm  16172  lspfixed  16200  lspsolvlem  16214  mendlmod  27478  lshpkrlem4  29911  baerlem5alem1  32506  baerlem5blem1  32507  hdmap14lem8  32676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-lmod 15952
  Copyright terms: Public domain W3C validator