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Theorem lmodvsghm 16932
Description: Scalar multiplication of the vector space by a fixed scalar is an automorphism of the additive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)
Hypotheses
Ref Expression
lmodvsghm.v  |-  V  =  ( Base `  W
)
lmodvsghm.f  |-  F  =  (Scalar `  W )
lmodvsghm.s  |-  .x.  =  ( .s `  W )
lmodvsghm.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
lmodvsghm  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
x  e.  V  |->  ( R  .x.  x ) )  e.  ( W 
GrpHom  W ) )
Distinct variable groups:    x, K    x, R    x,  .x.    x, V   
x, W
Allowed substitution hint:    F( x)

Proof of Theorem lmodvsghm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmodvsghm.v . 2  |-  V  =  ( Base `  W
)
2 eqid 2435 . 2  |-  ( +g  `  W )  =  ( +g  `  W )
3 lmodgrp 16881 . . 3  |-  ( W  e.  LMod  ->  W  e. 
Grp )
43adantr 462 . 2  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  W  e.  Grp )
5 lmodvsghm.f . . . . 5  |-  F  =  (Scalar `  W )
6 lmodvsghm.s . . . . 5  |-  .x.  =  ( .s `  W )
7 lmodvsghm.k . . . . 5  |-  K  =  ( Base `  F
)
81, 5, 6, 7lmodvscl 16891 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  x  e.  V )  ->  ( R  .x.  x )  e.  V )
983expa 1182 . . 3  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  x  e.  V
)  ->  ( R  .x.  x )  e.  V
)
10 eqid 2435 . . 3  |-  ( x  e.  V  |->  ( R 
.x.  x ) )  =  ( x  e.  V  |->  ( R  .x.  x ) )
119, 10fmptd 5857 . 2  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
x  e.  V  |->  ( R  .x.  x ) ) : V --> V )
121, 2, 5, 6, 7lmodvsdi 16897 . . . . 5  |-  ( ( W  e.  LMod  /\  ( R  e.  K  /\  y  e.  V  /\  z  e.  V )
)  ->  ( R  .x.  ( y ( +g  `  W ) z ) )  =  ( ( R  .x.  y ) ( +g  `  W
) ( R  .x.  z ) ) )
13123exp2 1200 . . . 4  |-  ( W  e.  LMod  ->  ( R  e.  K  ->  (
y  e.  V  -> 
( z  e.  V  ->  ( R  .x.  (
y ( +g  `  W
) z ) )  =  ( ( R 
.x.  y ) ( +g  `  W ) ( R  .x.  z
) ) ) ) ) )
1413imp43 592 . . 3  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  ( y  e.  V  /\  z  e.  V ) )  -> 
( R  .x.  (
y ( +g  `  W
) z ) )  =  ( ( R 
.x.  y ) ( +g  `  W ) ( R  .x.  z
) ) )
151, 2lmodvacl 16888 . . . . . 6  |-  ( ( W  e.  LMod  /\  y  e.  V  /\  z  e.  V )  ->  (
y ( +g  `  W
) z )  e.  V )
16153expb 1183 . . . . 5  |-  ( ( W  e.  LMod  /\  (
y  e.  V  /\  z  e.  V )
)  ->  ( y
( +g  `  W ) z )  e.  V
)
1716adantlr 709 . . . 4  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  ( y  e.  V  /\  z  e.  V ) )  -> 
( y ( +g  `  W ) z )  e.  V )
18 oveq2 6090 . . . . 5  |-  ( x  =  ( y ( +g  `  W ) z )  ->  ( R  .x.  x )  =  ( R  .x.  (
y ( +g  `  W
) z ) ) )
19 ovex 6107 . . . . 5  |-  ( R 
.x.  ( y ( +g  `  W ) z ) )  e. 
_V
2018, 10, 19fvmpt 5764 . . . 4  |-  ( ( y ( +g  `  W
) z )  e.  V  ->  ( (
x  e.  V  |->  ( R  .x.  x ) ) `  ( y ( +g  `  W
) z ) )  =  ( R  .x.  ( y ( +g  `  W ) z ) ) )
2117, 20syl 16 . . 3  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  ( y  e.  V  /\  z  e.  V ) )  -> 
( ( x  e.  V  |->  ( R  .x.  x ) ) `  ( y ( +g  `  W ) z ) )  =  ( R 
.x.  ( y ( +g  `  W ) z ) ) )
22 oveq2 6090 . . . . . 6  |-  ( x  =  y  ->  ( R  .x.  x )  =  ( R  .x.  y
) )
23 ovex 6107 . . . . . 6  |-  ( R 
.x.  y )  e. 
_V
2422, 10, 23fvmpt 5764 . . . . 5  |-  ( y  e.  V  ->  (
( x  e.  V  |->  ( R  .x.  x
) ) `  y
)  =  ( R 
.x.  y ) )
25 oveq2 6090 . . . . . 6  |-  ( x  =  z  ->  ( R  .x.  x )  =  ( R  .x.  z
) )
26 ovex 6107 . . . . . 6  |-  ( R 
.x.  z )  e. 
_V
2725, 10, 26fvmpt 5764 . . . . 5  |-  ( z  e.  V  ->  (
( x  e.  V  |->  ( R  .x.  x
) ) `  z
)  =  ( R 
.x.  z ) )
2824, 27oveqan12d 6101 . . . 4  |-  ( ( y  e.  V  /\  z  e.  V )  ->  ( ( ( x  e.  V  |->  ( R 
.x.  x ) ) `
 y ) ( +g  `  W ) ( ( x  e.  V  |->  ( R  .x.  x ) ) `  z ) )  =  ( ( R  .x.  y ) ( +g  `  W ) ( R 
.x.  z ) ) )
2928adantl 463 . . 3  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  ( y  e.  V  /\  z  e.  V ) )  -> 
( ( ( x  e.  V  |->  ( R 
.x.  x ) ) `
 y ) ( +g  `  W ) ( ( x  e.  V  |->  ( R  .x.  x ) ) `  z ) )  =  ( ( R  .x.  y ) ( +g  `  W ) ( R 
.x.  z ) ) )
3014, 21, 293eqtr4d 2477 . 2  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  ( y  e.  V  /\  z  e.  V ) )  -> 
( ( x  e.  V  |->  ( R  .x.  x ) ) `  ( y ( +g  `  W ) z ) )  =  ( ( ( x  e.  V  |->  ( R  .x.  x
) ) `  y
) ( +g  `  W
) ( ( x  e.  V  |->  ( R 
.x.  x ) ) `
 z ) ) )
311, 1, 2, 2, 4, 4, 11, 30isghmd 15738 1  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
x  e.  V  |->  ( R  .x.  x ) )  e.  ( W 
GrpHom  W ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1757    e. cmpt 4340   ` cfv 5408  (class class class)co 6082   Basecbs 14159   +g cplusg 14223  Scalarcsca 14226   .scvsca 14227   Grpcgrp 15395    GrpHom cghm 15726   LModclmod 16874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-id 4625  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-mnd 15400  df-grp 15527  df-ghm 15727  df-lmod 16876
This theorem is referenced by:  gsumvsmul  16935  gsumvsmulOLD  16936  lmhmvsca  17050
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