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Theorem lmodvsghm 17349
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 2462 . 2  |-  ( +g  `  W )  =  ( +g  `  W )
3 lmodgrp 17297 . . 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 17307 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  x  e.  V )  ->  ( R  .x.  x )  e.  V )
983expa 1191 . . 3  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  x  e.  V
)  ->  ( R  .x.  x )  e.  V
)
10 eqid 2462 . . 3  |-  ( x  e.  V  |->  ( R 
.x.  x ) )  =  ( x  e.  V  |->  ( R  .x.  x ) )
119, 10fmptd 6038 . 2  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
x  e.  V  |->  ( R  .x.  x ) ) : V --> V )
121, 2, 5, 6, 7lmodvsdi 17313 . . . . 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 1209 . . . 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 17304 . . . . . 6  |-  ( ( W  e.  LMod  /\  y  e.  V  /\  z  e.  V )  ->  (
y ( +g  `  W
) z )  e.  V )
16153expb 1192 . . . . 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 6302 . . . . 5  |-  ( R 
.x.  ( y ( +g  `  W ) z ) )  e. 
_V
2018, 10, 19fvmpt 5943 . . . 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 6302 . . . . . 6  |-  ( R 
.x.  y )  e. 
_V
2422, 10, 23fvmpt 5943 . . . . 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 6302 . . . . . 6  |-  ( R 
.x.  z )  e. 
_V
2725, 10, 26fvmpt 5943 . . . . 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 2513 . 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 16066 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 1374    e. wcel 1762    |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546  Scalarcsca 14549   .scvsca 14550   Grpcgrp 15718    GrpHom cghm 16054   LModclmod 17290
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 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-mnd 15723  df-grp 15853  df-ghm 16055  df-lmod 17292
This theorem is referenced by:  gsumvsmul  17352  gsumvsmulOLD  17353  lmhmvsca  17469
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