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Theorem lflset 29871
Description: The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lflset.v  |-  V  =  ( Base `  W
)
lflset.a  |-  .+  =  ( +g  `  W )
lflset.d  |-  D  =  (Scalar `  W )
lflset.s  |-  .x.  =  ( .s `  W )
lflset.k  |-  K  =  ( Base `  D
)
lflset.p  |-  .+^  =  ( +g  `  D )
lflset.t  |-  .X.  =  ( .r `  D )
lflset.f  |-  F  =  (LFnl `  W )
Assertion
Ref Expression
lflset  |-  ( W  e.  X  ->  F  =  { f  e.  ( K  ^m  V )  |  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( f `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  (
f `  x )
)  .+^  ( f `  y ) ) } )
Distinct variable groups:    f, r, K    x, f, y, V   
f, W    x, r,
y, W
Allowed substitution hints:    D( x, y, f, r)    .+ ( x, y, f, r)    .+^ ( x, y, f, r)    .x. ( x, y, f, r)    .X. ( x, y, f, r)    F( x, y, f, r)    K( x, y)    V( r)    X( x, y, f, r)

Proof of Theorem lflset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 lflset.f . . 3  |-  F  =  (LFnl `  W )
3 fveq2 5541 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
4 lflset.d . . . . . . . . 9  |-  D  =  (Scalar `  W )
53, 4syl6eqr 2346 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  D )
65fveq2d 5545 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  D )
)
7 lflset.k . . . . . . 7  |-  K  =  ( Base `  D
)
86, 7syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
9 fveq2 5541 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
10 lflset.v . . . . . . 7  |-  V  =  ( Base `  W
)
119, 10syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
128, 11oveq12d 5892 . . . . 5  |-  ( w  =  W  ->  (
( Base `  (Scalar `  w
) )  ^m  ( Base `  w ) )  =  ( K  ^m  V ) )
13 fveq2 5541 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( +g  `  w )  =  ( +g  `  W
) )
14 lflset.a . . . . . . . . . . . 12  |-  .+  =  ( +g  `  W )
1513, 14syl6eqr 2346 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  w )  = 
.+  )
16 fveq2 5541 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
17 lflset.s . . . . . . . . . . . . 13  |-  .x.  =  ( .s `  W )
1816, 17syl6eqr 2346 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
1918oveqd 5891 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
r ( .s `  w ) x )  =  ( r  .x.  x ) )
20 eqidd 2297 . . . . . . . . . . 11  |-  ( w  =  W  ->  y  =  y )
2115, 19, 20oveq123d 5895 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( r ( .s
`  w ) x ) ( +g  `  w
) y )  =  ( ( r  .x.  x )  .+  y
) )
2221fveq2d 5545 . . . . . . . . 9  |-  ( w  =  W  ->  (
f `  ( (
r ( .s `  w ) x ) ( +g  `  w
) y ) )  =  ( f `  ( ( r  .x.  x )  .+  y
) ) )
235fveq2d 5545 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  ( +g  `  D ) )
24 lflset.p . . . . . . . . . . 11  |-  .+^  =  ( +g  `  D )
2523, 24syl6eqr 2346 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  .+^  )
265fveq2d 5545 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  ( .r `  D ) )
27 lflset.t . . . . . . . . . . . 12  |-  .X.  =  ( .r `  D )
2826, 27syl6eqr 2346 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  .X.  )
2928oveqd 5891 . . . . . . . . . 10  |-  ( w  =  W  ->  (
r ( .r `  (Scalar `  w ) ) ( f `  x
) )  =  ( r  .X.  ( f `  x ) ) )
30 eqidd 2297 . . . . . . . . . 10  |-  ( w  =  W  ->  (
f `  y )  =  ( f `  y ) )
3125, 29, 30oveq123d 5895 . . . . . . . . 9  |-  ( w  =  W  ->  (
( r ( .r
`  (Scalar `  w )
) ( f `  x ) ) ( +g  `  (Scalar `  w ) ) ( f `  y ) )  =  ( ( r  .X.  ( f `  x ) )  .+^  ( f `  y
) ) )
3222, 31eqeq12d 2310 . . . . . . . 8  |-  ( w  =  W  ->  (
( f `  (
( r ( .s
`  w ) x ) ( +g  `  w
) y ) )  =  ( ( r ( .r `  (Scalar `  w ) ) ( f `  x ) ) ( +g  `  (Scalar `  w ) ) ( f `  y ) )  <->  ( f `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  (
f `  x )
)  .+^  ( f `  y ) ) ) )
3311, 32raleqbidv 2761 . . . . . . 7  |-  ( w  =  W  ->  ( A. y  e.  ( Base `  w ) ( f `  ( ( r ( .s `  w ) x ) ( +g  `  w
) y ) )  =  ( ( r ( .r `  (Scalar `  w ) ) ( f `  x ) ) ( +g  `  (Scalar `  w ) ) ( f `  y ) )  <->  A. y  e.  V  ( f `  (
( r  .x.  x
)  .+  y )
)  =  ( ( r  .X.  ( f `  x ) )  .+^  ( f `  y
) ) ) )
3411, 33raleqbidv 2761 . . . . . 6  |-  ( w  =  W  ->  ( A. x  e.  ( Base `  w ) A. y  e.  ( Base `  w ) ( f `
 ( ( r ( .s `  w
) x ) ( +g  `  w ) y ) )  =  ( ( r ( .r `  (Scalar `  w ) ) ( f `  x ) ) ( +g  `  (Scalar `  w ) ) ( f `  y ) )  <->  A. x  e.  V  A. y  e.  V  ( f `  (
( r  .x.  x
)  .+  y )
)  =  ( ( r  .X.  ( f `  x ) )  .+^  ( f `  y
) ) ) )
358, 34raleqbidv 2761 . . . . 5  |-  ( w  =  W  ->  ( A. r  e.  ( Base `  (Scalar `  w
) ) A. x  e.  ( Base `  w
) A. y  e.  ( Base `  w
) ( f `  ( ( r ( .s `  w ) x ) ( +g  `  w ) y ) )  =  ( ( r ( .r `  (Scalar `  w ) ) ( f `  x
) ) ( +g  `  (Scalar `  w )
) ( f `  y ) )  <->  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( f `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  (
f `  x )
)  .+^  ( f `  y ) ) ) )
3612, 35rabeqbidv 2796 . . . 4  |-  ( w  =  W  ->  { f  e.  ( ( Base `  (Scalar `  w )
)  ^m  ( Base `  w ) )  | 
A. r  e.  (
Base `  (Scalar `  w
) ) A. x  e.  ( Base `  w
) A. y  e.  ( Base `  w
) ( f `  ( ( r ( .s `  w ) x ) ( +g  `  w ) y ) )  =  ( ( r ( .r `  (Scalar `  w ) ) ( f `  x
) ) ( +g  `  (Scalar `  w )
) ( f `  y ) ) }  =  { f  e.  ( K  ^m  V
)  |  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( f `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  (
f `  x )
)  .+^  ( f `  y ) ) } )
37 df-lfl 29870 . . . 4  |- LFnl  =  ( w  e.  _V  |->  { f  e.  ( (
Base `  (Scalar `  w
) )  ^m  ( Base `  w ) )  |  A. r  e.  ( Base `  (Scalar `  w ) ) A. x  e.  ( Base `  w ) A. y  e.  ( Base `  w
) ( f `  ( ( r ( .s `  w ) x ) ( +g  `  w ) y ) )  =  ( ( r ( .r `  (Scalar `  w ) ) ( f `  x
) ) ( +g  `  (Scalar `  w )
) ( f `  y ) ) } )
38 ovex 5899 . . . . 5  |-  ( K  ^m  V )  e. 
_V
3938rabex 4181 . . . 4  |-  { f  e.  ( K  ^m  V )  |  A. r  e.  K  A. x  e.  V  A. y  e.  V  (
f `  ( (
r  .x.  x )  .+  y ) )  =  ( ( r  .X.  ( f `  x
) )  .+^  ( f `
 y ) ) }  e.  _V
4036, 37, 39fvmpt 5618 . . 3  |-  ( W  e.  _V  ->  (LFnl `  W )  =  {
f  e.  ( K  ^m  V )  | 
A. r  e.  K  A. x  e.  V  A. y  e.  V  ( f `  (
( r  .x.  x
)  .+  y )
)  =  ( ( r  .X.  ( f `  x ) )  .+^  ( f `  y
) ) } )
412, 40syl5eq 2340 . 2  |-  ( W  e.  _V  ->  F  =  { f  e.  ( K  ^m  V )  |  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( f `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  (
f `  x )
)  .+^  ( f `  y ) ) } )
421, 41syl 15 1  |-  ( W  e.  X  ->  F  =  { f  e.  ( K  ^m  V )  |  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( f `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  (
f `  x )
)  .+^  ( f `  y ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228  LFnlclfn 29869
This theorem is referenced by:  islfl  29872
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-lfl 29870
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