MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmodvsghm Unicode version

Theorem lmodvsghm 15702
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 2296 . 2  |-  ( +g  `  W )  =  ( +g  `  W )
3 lmodgrp 15650 . . 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 15660 . . . 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 2296 . . 3  |-  ( x  e.  V  |->  ( R 
.x.  x ) )  =  ( x  e.  V  |->  ( R  .x.  x ) )
119, 10fmptd 5700 . 2  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  (
x  e.  V  |->  ( R  .x.  x ) ) : V --> V )
121, 2, 5, 6, 7lmodvsdi 15666 . . . . 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 15657 . . . . . 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 5882 . . . . 5  |-  ( x  =  ( y ( +g  `  W ) z )  ->  ( R  .x.  x )  =  ( R  .x.  (
y ( +g  `  W
) z ) ) )
19 ovex 5899 . . . . 5  |-  ( R 
.x.  ( y ( +g  `  W ) z ) )  e. 
_V
2018, 10, 19fvmpt 5618 . . . 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 5882 . . . . . 6  |-  ( x  =  y  ->  ( R  .x.  x )  =  ( R  .x.  y
) )
23 ovex 5899 . . . . . 6  |-  ( R 
.x.  y )  e. 
_V
2422, 10, 23fvmpt 5618 . . . . 5  |-  ( y  e.  V  ->  (
( x  e.  V  |->  ( R  .x.  x
) ) `  y
)  =  ( R 
.x.  y ) )
25 oveq2 5882 . . . . . 6  |-  ( x  =  z  ->  ( R  .x.  x )  =  ( R  .x.  z
) )
26 ovex 5899 . . . . . 6  |-  ( R 
.x.  z )  e. 
_V
2725, 10, 26fvmpt 5618 . . . . 5  |-  ( z  e.  V  ->  (
( x  e.  V  |->  ( R  .x.  x
) ) `  z
)  =  ( R 
.x.  z ) )
2824, 27oveqan12d 5893 . . . 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 2338 . 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 14708 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 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   Grpcgrp 14378    GrpHom cghm 14696   LModclmod 15643
This theorem is referenced by:  lmhmvsca  15818  gsumvsmul  26867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-mnd 14383  df-grp 14505  df-ghm 14697  df-lmod 15645
  Copyright terms: Public domain W3C validator