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Theorem lmodvsghm 15686
Description: Scalar multiplication of the vector space by a fixed scalar is an automorphism of the addiive 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 2283 . 2  |-  ( +g  `  W )  =  ( +g  `  W )
3 lmodgrp 15634 . . 3  |-  ( W  e.  LMod  ->  W  e. 
Grp )
43adantr 451 . 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 15644 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  x  e.  V )  ->  ( R  .x.  x )  e.  V )
983expa 1151 . . 3  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  x  e.  V
)  ->  ( R  .x.  x )  e.  V
)
10 eqid 2283 . . 3  |-  ( x  e.  V  |->  ( R 
.x.  x ) )  =  ( x  e.  V  |->  ( R  .x.  x ) )
119, 10fmptd 5684 . 2  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
x  e.  V  |->  ( R  .x.  x ) ) : V --> V )
121, 2, 5, 6, 7lmodvsdi 15650 . . . . 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 1169 . . . 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 578 . . 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 15641 . . . . . 6  |-  ( ( W  e.  LMod  /\  y  e.  V  /\  z  e.  V )  ->  (
y ( +g  `  W
) z )  e.  V )
16153expb 1152 . . . . 5  |-  ( ( W  e.  LMod  /\  (
y  e.  V  /\  z  e.  V )
)  ->  ( y
( +g  `  W ) z )  e.  V
)
1716adantlr 695 . . . 4  |-  ( ( ( W  e.  LMod  /\  R  e.  K )  /\  ( y  e.  V  /\  z  e.  V ) )  -> 
( y ( +g  `  W ) z )  e.  V )
18 oveq2 5866 . . . . 5  |-  ( x  =  ( y ( +g  `  W ) z )  ->  ( R  .x.  x )  =  ( R  .x.  (
y ( +g  `  W
) z ) ) )
19 ovex 5883 . . . . 5  |-  ( R 
.x.  ( y ( +g  `  W ) z ) )  e. 
_V
2018, 10, 19fvmpt 5602 . . . 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 15 . . 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 5866 . . . . . 6  |-  ( x  =  y  ->  ( R  .x.  x )  =  ( R  .x.  y
) )
23 ovex 5883 . . . . . 6  |-  ( R 
.x.  y )  e. 
_V
2422, 10, 23fvmpt 5602 . . . . 5  |-  ( y  e.  V  ->  (
( x  e.  V  |->  ( R  .x.  x
) ) `  y
)  =  ( R 
.x.  y ) )
25 oveq2 5866 . . . . . 6  |-  ( x  =  z  ->  ( R  .x.  x )  =  ( R  .x.  z
) )
26 ovex 5883 . . . . . 6  |-  ( R 
.x.  z )  e. 
_V
2725, 10, 26fvmpt 5602 . . . . 5  |-  ( z  e.  V  ->  (
( x  e.  V  |->  ( R  .x.  x
) ) `  z
)  =  ( R 
.x.  z ) )
2824, 27oveqan12d 5877 . . . 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 452 . . 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 2325 . 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 14692 1  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
x  e.  V  |->  ( R  .x.  x ) )  e.  ( W 
GrpHom  W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   Grpcgrp 14362    GrpHom cghm 14680   LModclmod 15627
This theorem is referenced by:  lmhmvsca  15802  gsumvsmul  26764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mnd 14367  df-grp 14489  df-ghm 14681  df-lmod 15629
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