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Theorem lmodvsass 15938
Description: Associative law for scalar product. (ax-hvmulass 22471 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
lmodvsass.v  |-  V  =  ( Base `  W
)
lmodvsass.f  |-  F  =  (Scalar `  W )
lmodvsass.s  |-  .x.  =  ( .s `  W )
lmodvsass.k  |-  K  =  ( Base `  F
)
lmodvsass.t  |-  .X.  =  ( .r `  F )
Assertion
Ref Expression
lmodvsass  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
)  ->  ( ( Q  .X.  R )  .x.  X )  =  ( Q  .x.  ( R 
.x.  X ) ) )

Proof of Theorem lmodvsass
StepHypRef Expression
1 lmodvsass.v . . . . . . . 8  |-  V  =  ( Base `  W
)
2 eqid 2412 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
3 lmodvsass.s . . . . . . . 8  |-  .x.  =  ( .s `  W )
4 lmodvsass.f . . . . . . . 8  |-  F  =  (Scalar `  W )
5 lmodvsass.k . . . . . . . 8  |-  K  =  ( Base `  F
)
6 eqid 2412 . . . . . . . 8  |-  ( +g  `  F )  =  ( +g  `  F )
7 lmodvsass.t . . . . . . . 8  |-  .X.  =  ( .r `  F )
8 eqid 2412 . . . . . . . 8  |-  ( 1r
`  F )  =  ( 1r `  F
)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 15918 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  (
( ( R  .x.  X )  e.  V  /\  ( R  .x.  ( X ( +g  `  W
) X ) )  =  ( ( R 
.x.  X ) ( +g  `  W ) ( R  .x.  X
) )  /\  (
( Q ( +g  `  F ) R ) 
.x.  X )  =  ( ( Q  .x.  X ) ( +g  `  W ) ( R 
.x.  X ) ) )  /\  ( ( ( Q  .X.  R
)  .x.  X )  =  ( Q  .x.  ( R  .x.  X ) )  /\  ( ( 1r `  F ) 
.x.  X )  =  X ) ) )
109simprd 450 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  (
( ( Q  .X.  R )  .x.  X
)  =  ( Q 
.x.  ( R  .x.  X ) )  /\  ( ( 1r `  F )  .x.  X
)  =  X ) )
1110simpld 446 . . . . 5  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K )  /\  ( X  e.  V  /\  X  e.  V
) )  ->  (
( Q  .X.  R
)  .x.  X )  =  ( Q  .x.  ( R  .x.  X ) ) )
12113expa 1153 . . . 4  |-  ( ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K
) )  /\  ( X  e.  V  /\  X  e.  V )
)  ->  ( ( Q  .X.  R )  .x.  X )  =  ( Q  .x.  ( R 
.x.  X ) ) )
1312anabsan2 796 . . 3  |-  ( ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K
) )  /\  X  e.  V )  ->  (
( Q  .X.  R
)  .x.  X )  =  ( Q  .x.  ( R  .x.  X ) ) )
1413exp42 595 . 2  |-  ( W  e.  LMod  ->  ( Q  e.  K  ->  ( R  e.  K  ->  ( X  e.  V  -> 
( ( Q  .X.  R )  .x.  X
)  =  ( Q 
.x.  ( R  .x.  X ) ) ) ) ) )
15143imp2 1168 1  |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
)  ->  ( ( Q  .X.  R )  .x.  X )  =  ( Q  .x.  ( R 
.x.  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048   Basecbs 13432   +g cplusg 13492   .rcmulr 13493  Scalarcsca 13495   .scvsca 13496   1rcur 15625   LModclmod 15913
This theorem is referenced by:  lmodvs0  15947  lmodvsneg  15951  lmodsubvs  15963  lmodsubdi  15964  lmodsubdir  15965  islss3  15998  lss1d  16002  prdslmodd  16008  lmodvsinv  16075  lmhmvsca  16084  lvecvs0or  16143  lssvs0or  16145  lvecinv  16148  lspsnvs  16149  lspfixed  16163  lspsolvlem  16177  lspsolv  16178  asclrhm  16363  mplmon2mul  16524  clmvsass  19073  frlmup1  27126  mendlmod  27377  lshpkrlem4  29608  lcdvsass  32102  baerlem3lem1  32202  hgmapmul  32393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-nul 4306
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051  df-lmod 15915
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