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Theorem lmodvsghm 17439
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 2441 . 2  |-  ( +g  `  W )  =  ( +g  `  W )
3 lmodgrp 17387 . . 3  |-  ( W  e.  LMod  ->  W  e. 
Grp )
43adantr 465 . 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 17397 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  x  e.  V )  ->  ( R  .x.  x )  e.  V )
983expa 1195 . . 3  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  x  e.  V
)  ->  ( R  .x.  x )  e.  V
)
10 eqid 2441 . . 3  |-  ( x  e.  V  |->  ( R 
.x.  x ) )  =  ( x  e.  V  |->  ( R  .x.  x ) )
119, 10fmptd 6036 . 2  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
x  e.  V  |->  ( R  .x.  x ) ) : V --> V )
121, 2, 5, 6, 7lmodvsdi 17403 . . . . 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 1213 . . . 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 595 . . 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 17394 . . . . . 6  |-  ( ( W  e.  LMod  /\  y  e.  V  /\  z  e.  V )  ->  (
y ( +g  `  W
) z )  e.  V )
16153expb 1196 . . . . 5  |-  ( ( W  e.  LMod  /\  (
y  e.  V  /\  z  e.  V )
)  ->  ( y
( +g  `  W ) z )  e.  V
)
1716adantlr 714 . . . 4  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  ( y  e.  V  /\  z  e.  V ) )  -> 
( y ( +g  `  W ) z )  e.  V )
18 oveq2 6285 . . . . 5  |-  ( x  =  ( y ( +g  `  W ) z )  ->  ( R  .x.  x )  =  ( R  .x.  (
y ( +g  `  W
) z ) ) )
19 ovex 6305 . . . . 5  |-  ( R 
.x.  ( y ( +g  `  W ) z ) )  e. 
_V
2018, 10, 19fvmpt 5937 . . . 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 6285 . . . . . 6  |-  ( x  =  y  ->  ( R  .x.  x )  =  ( R  .x.  y
) )
23 ovex 6305 . . . . . 6  |-  ( R 
.x.  y )  e. 
_V
2422, 10, 23fvmpt 5937 . . . . 5  |-  ( y  e.  V  ->  (
( x  e.  V  |->  ( R  .x.  x
) ) `  y
)  =  ( R 
.x.  y ) )
25 oveq2 6285 . . . . . 6  |-  ( x  =  z  ->  ( R  .x.  x )  =  ( R  .x.  z
) )
26 ovex 6305 . . . . . 6  |-  ( R 
.x.  z )  e. 
_V
2725, 10, 26fvmpt 5937 . . . . 5  |-  ( z  e.  V  ->  (
( x  e.  V  |->  ( R  .x.  x
) ) `  z
)  =  ( R 
.x.  z ) )
2824, 27oveqan12d 6296 . . . 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 466 . . 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 2492 . 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 16145 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 1381    e. wcel 1802    |-> cmpt 4491   ` cfv 5574  (class class class)co 6277   Basecbs 14504   +g cplusg 14569  Scalarcsca 14572   .scvsca 14573   Grpcgrp 15922    GrpHom cghm 16133   LModclmod 17380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-ghm 16134  df-lmod 17382
This theorem is referenced by:  gsumvsmul  17442  gsumvsmulOLD  17443  lmhmvsca  17559
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